init: magnetohydrodynamics with physicsnemo

.gitignore

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+checkpoints/
+logs/
+outputs/
+launch.log
+
+# Byte-compiled / optimized / DLL files
+**__pycache__/** 
+*.py[codz]
+*$py.class
+
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+share/python-wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+MANIFEST
+
+# PyInstaller
+#   Usually these files are written by a python script from a template
+#   before PyInstaller builds the exe, so as to inject date/other infos into it.
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.nox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*.cover
+*.py.cover
+.hypothesis/
+.pytest_cache/
+cover/
+
+# Translations
+*.mo
+*.pot
+
+# Django stuff:
+*.log
+local_settings.py
+db.sqlite3
+db.sqlite3-journal
+
+# Flask stuff:
+instance/
+.webassets-cache
+
+# Scrapy stuff:
+.scrapy
+
+# Sphinx documentation
+docs/_build/
+
+# PyBuilder
+.pybuilder/
+target/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# IPython
+profile_default/
+ipython_config.py
+
+# pyenv
+#   For a library or package, you might want to ignore these files since the code is
+#   intended to run in multiple environments; otherwise, check them in:
+# .python-version
+
+# pipenv
+#   According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
+#   However, in case of collaboration, if having platform-specific dependencies or dependencies
+#   having no cross-platform support, pipenv may install dependencies that don't work, or not
+#   install all needed dependencies.
+# Pipfile.lock
+
+# UV
+#   Similar to Pipfile.lock, it is generally recommended to include uv.lock in version control.
+#   This is especially recommended for binary packages to ensure reproducibility, and is more
+#   commonly ignored for libraries.
+# uv.lock
+
+# poetry
+#   Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
+#   This is especially recommended for binary packages to ensure reproducibility, and is more
+#   commonly ignored for libraries.
+#   https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
+# poetry.lock
+# poetry.toml
+
+# pdm
+#   Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
+#   pdm recommends including project-wide configuration in pdm.toml, but excluding .pdm-python.
+#   https://pdm-project.org/en/latest/usage/project/#working-with-version-control
+# pdm.lock
+# pdm.toml
+.pdm-python
+.pdm-build/
+
+# pixi
+#   Similar to Pipfile.lock, it is generally recommended to include pixi.lock in version control.
+# pixi.lock
+#   Pixi creates a virtual environment in the .pixi directory, just like venv module creates one
+#   in the .venv directory. It is recommended not to include this directory in version control.
+.pixi
+
+# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
+__pypackages__/
+
+# Celery stuff
+celerybeat-schedule
+celerybeat.pid
+
+# Redis
+*.rdb
+*.aof
+*.pid
+
+# RabbitMQ
+mnesia/
+rabbitmq/
+rabbitmq-data/
+
+# ActiveMQ
+activemq-data/
+
+# SageMath parsed files
+*.sage.py
+
+# Environments
+.env
+.envrc
+.venv
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# Spyder project settings
+.spyderproject
+.spyproject
+
+# Rope project settings
+.ropeproject
+
+# mkdocs documentation
+/site
+
+# mypy
+.mypy_cache/
+.dmypy.json
+dmypy.json
+
+# Pyre type checker
+.pyre/
+
+# pytype static type analyzer
+.pytype/
+
+# Cython debug symbols
+cython_debug/
+
+# PyCharm
+#   JetBrains specific template is maintained in a separate JetBrains.gitignore that can
+#   be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
+#   and can be added to the global gitignore or merged into this file.  For a more nuclear
+#   option (not recommended) you can uncomment the following to ignore the entire idea folder.
+# .idea/
+
+# Abstra
+#   Abstra is an AI-powered process automation framework.
+#   Ignore directories containing user credentials, local state, and settings.
+#   Learn more at https://abstra.io/docs
+.abstra/
+
+# Visual Studio Code
+#   Visual Studio Code specific template is maintained in a separate VisualStudioCode.gitignore 
+#   that can be found at https://github.com/github/gitignore/blob/main/Global/VisualStudioCode.gitignore
+#   and can be added to the global gitignore or merged into this file. However, if you prefer, 
+#   you could uncomment the following to ignore the entire vscode folder
+# .vscode/
+
+# Ruff stuff:
+.ruff_cache/
+
+# PyPI configuration file
+.pypirc
+
+# Marimo
+marimo/_static/
+marimo/_lsp/
+__marimo__/
+
+# Streamlit
+.streamlit/secrets.toml

.pre-commit-config.yaml

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+repos:
+- repo: https://github.com/pre-commit/pre-commit-hooks
+  rev: v4.4.0
+  hooks:
+  -   id: trailing-whitespace
+  -   id: end-of-file-fixer
+  -   id: check-yaml
+  -   id: debug-statements
+- repo: https://github.com/astral-sh/ruff-pre-commit
+  rev: v0.4.0
+  hooks:
+    - id: ruff
+      args: [ --fix ]
+      types_or: [ python, pyi, jupyter ]
+    - id: ruff-format
+      types_or: [ python, pyi, jupyter ]

Dockerfile

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+FROM nvcr.io/nvidia/physicsnemo/physicsnemo:25.08
+
+ENV DEBIAN_FRONTEND=noninteractive
+
+USER root
+# Create non-root user and set up directories
+RUN useradd -m -u 1001 user && \
+    mkdir -p /home/user/.cache /home/user/.config /home/user/.local /home/user/.local/share/jupyter && \
+    chmod -R 777 /home/user && \
+    mkdir /mhd-demo && chown user:user /mhd-demo && chmod 777 /mhd-demo
+
+USER user
+ENV HOME=/home/user
+ENV PATH=/home/user/.local/bin:$PATH
+WORKDIR $HOME/app
+
+
+# Upgrade pip
+RUN python -m pip install --upgrade pip
+
+# # Copy all files at once
+COPY --chown=user on_startup.sh README.md start_server.sh requirements.txt ./
+COPY --chown=user login.html /usr/local/lib/python3.12/dist-packages/jupyter_server/templates/login.html
+COPY --chown=user magnetohydrodynamics.ipynb /mhd-demo/
+COPY --chown=user mhd /mhd-demo/mhd/
+
+
+RUN chmod +x start_server.sh && \
+    chmod -R 777 /mhd-demo/ && \
+    pip install -r requirements.txt
+
+EXPOSE 7860
+CMD ["./start_server.sh"]

Dockerfile.dedalus

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+FROM nvidia/cuda:12.8.1-cudnn-runtime-ubuntu24.04 
+
+ENV DEBIAN_FRONTEND=noninteractive
+
+RUN apt-get update -qq && \
+    apt-get autoremove -y -qq && \
+    apt-get install -y -qq apt-file \
+                        vim \
+                        wget \
+                        git \
+                        software-properties-common \
+                        make \
+                        g++ \
+                        gcc \
+                        gpg-agent && \
+    apt-get clean && rm -rf /var/cache/apt/archives /var/lib/apt/lists/*
+
+RUN useradd -m -u 1001 user && \
+    mkdir -p /home/user/.cache /home/user/.config /home/user/.local && \
+    chmod -R 777 /home/user && \
+    mkdir /mhd-demo && chown user:user /mhd-demo && chmod 777 /mhd-demo
+
+USER user
+ENV HOME=/home/user
+ENV PATH=/home/user/.local/bin:$PATH
+WORKDIR $HOME/app
+
+ENV CONDA_DIR=$HOME/conda
+ENV PATH=$CONDA_DIR/bin:$PATH
+RUN wget --quiet https://repo.anaconda.com/miniconda/Miniconda3-latest-Linux-x86_64.sh -O ~/miniconda.sh && \
+    /bin/bash ~/miniconda.sh -b -p $HOME/conda && \
+    rm ~/miniconda.sh && \
+    conda config --add channels conda-forge && \
+    conda config --set channel_priority strict && \
+    conda tos accept --override-channels --channel https://repo.anaconda.com/pkgs/main && \
+    conda tos accept --override-channels --channel https://repo.anaconda.com/pkgs/r && \
+    conda create -n env python=3.12 -y --quiet && \
+    conda run -n env conda env config vars set OMP_NUM_THREADS=1 && \
+    conda run -n env conda env config vars set NUMEXPR_MAX_THREADS=1 && \
+    conda run -n env conda install -c conda-forge dedalus jupyter jupyterlab torch hydra-core imageio -y --quiet
+
+ENV PATH=$HOME/conda/envs/env/bin:$PATH
+
+# # Copy all files at once
+COPY --chown=user on_startup.sh README.md start_server.sh requirements.txt ./
+COPY --chown=user magnetohydrodynamics.ipynb mhd /mhd-demo/
+
+RUN chmod +x start_server.sh && \
+    chmod -R 777 /mhd-demo/
+
+EXPOSE 7860
+CMD ["./start_server.sh"]

README.md

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+---
+title: Modeling Magnetohydrodynamics with PhysicsNeMo
+emoji: 🟢
+colorFrom: green
+colorTo: gray
+sdk: docker
+app_port: 7860
+pinned: false
+tags:
+  - physics
+  - cfd
+  - machine-learning
+  - neural-operators
+  - magnetohydrodynamics
+  - scientific-computing
+---
+

login.html

@@ -0,0 +1,68 @@
+{% extends "page.html" %}
+
+
+{% block stylesheet %}
+{% endblock %}
+
+{% block site %}
+
+<div id="jupyter-main-app" class="container">
+    
+    <img src="https://huggingface.co/front/assets/huggingface_logo-noborder.svg" alt="Hugging Face Logo">
+    <h4>Welcome to JupyterLab</h4>
+    
+    <h5>The default token is <span style="color:orange;">huggingface</span></h5>
+  
+    {% if login_available %}
+    {# login_available means password-login is allowed. Show the form. #}
+    <div class="row">
+        <div class="navbar col-sm-8">
+            <div class="navbar-inner">
+                <div class="container">
+                    <div class="center-nav">
+                        <form action="{{base_url}}login?next={{next}}" method="post" class="navbar-form pull-left">
+                            {{ xsrf_form_html() | safe }}
+                            {% if token_available %}
+                            <label for="password_input"><strong>{% trans %}Jupyter token <span title="This is the secret you set up when deploying your JupyterLab space">ⓘ</span> {% endtrans
+                                    %}</strong></label>
+                            {% else %}
+                            <label for="password_input"><strong>{% trans %}Jupyter password:{% endtrans %}</strong></label>
+                            {% endif %}
+                            <input type="password" name="password" id="password_input" class="form-control">
+                            <button type="submit" class="btn btn-default" id="login_submit">{% trans %}Log in{% endtrans
+                                %}</button>
+                        </form>
+                    </div>
+                </div>
+            </div>
+        </div>
+    </div>
+    {% else %}
+    <p>{% trans %}No login available, you shouldn't be seeing this page.{% endtrans %}</p>
+    {% endif %}
+
+      <h5>If you don't have the credentials for this Jupyter space, <a target="_blank" href="https://huggingface.co/spaces/SpacesExamples/jupyterlab?duplicate=true">create your own.</a></h5>
+    <br>
+  
+    <p>This template was created by <a href="https://twitter.com/camenduru" target="_blank" >camenduru</a> and <a href="https://huggingface.co/nateraw" target="_blank" >nateraw</a>, with contributions of <a href="https://huggingface.co/osanseviero" target="_blank" >osanseviero</a> and <a href="https://huggingface.co/azzr" target="_blank" >azzr</a> </p>
+    {% if message %}
+    <div class="row">
+        {% for key in message %}
+        <div class="message {{key}}">
+            {{message[key]}}
+        </div>
+        {% endfor %}
+    </div>
+    {% endif %}
+    {% if token_available %}
+    {% block token_message %}
+    
+    {% endblock token_message %}
+    {% endif %}
+</div>
+
+{% endblock %}
+
+
+{% block script %}
+{% endblock %}

magnetohydrodynamics.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "3902fdc0",
+   "metadata": {},
+   "source": [
+    "# Modeling Magnetohydrodynamics with Physics Informed Neural Operators"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "982a76df",
+   "metadata": {},
+   "source": [
+    "In this notebook, we will study the application of physics informed data-driven modeling to the incompressible magnetohydrodynamics (MHD) equations representing an incompressible fluid in the presence of a magnetic field $\\mathbf{B}$. Our model will be built using a Tensor Factorized Fourier Neural Operator (tFNO), and trained in conjunction with the PDEs representing our system. The model is physics-informed during training by encoding known information about the physical system into the loss functions,  enabling generalization of the resulting model to a variety of settings in the solution space. Specifically, the PDEs and initial conditions are used as soft constraints learned by the neural network as its trains. Models covering different data regimes governed by the Reynolds number are trained using transfer learning to showcase how our model may be applied to both laminar and turbulent flows. The AI-accelerated surrogate model is compared to classical simulations to compare its throughput and accuracy.\n",
+    "\n",
+    "Note that while the majority of the code needed to run this example is provided in the notebook, the lower barrier to entry for training and evaluating models will be to run the scripts in the source directory, and the material referenced here should be used as a base for learning the underlying components leading to model training and evaluation. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "19f35947",
+   "metadata": {},
+   "source": [
+    "#### Learning Outcomes\n",
+    "* How to apply physics constraints to neural networks\n",
+    "* Learn how the Tensor Factorized Fourier Neural Operator can be applied to physics based problems\n",
+    "* Learn how to define PDEs with PhysicsNeMo\n",
+    "* Train PINOs with PhysicsNeMo Core\n",
+    "* Learn how data driven modeling can help build computationally efficient surrogates for physics problems"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "03be824a",
+   "metadata": {},
+   "source": [
+    "## Pre-Requisites\n",
+    "This workshop is derived primarily from the informative paper [Magnetohydrodynamics with physics informed neural operators\n",
+    "](https://iopscience.iop.org/article/10.1088/2632-2153/ace30a)[1]. Reading the paper will provide both context and an overview of what will be presented in this workshop. Additionally, the paper serves as a great reference if more details are needed on any specific section. It is encouraged to read through the paper before continuing.\n",
+    "\n",
+    "[1] Rosofsky, S. G., & Huerta, E. A. (2023). Magnetohydrodynamics with physics informed neural operators. Machine Learning: Science and Technology, 4(3), 035002."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e5a666a2",
+   "metadata": {},
+   "source": [
+    "## Problem Overview\n",
+    "\n",
+    "To examine the properties of PINOs with multiple complex equations, we examined the ability of the networks to reproduce the incompressible magnetohydrodynamics (MHD) equations representing an incompressible fluid in the presence of a magnetic field $\\mathbf{B}$. These equations are present in several astrophysical phenomena, including black hole accretion and binary neutron star mergers. Additionally, MDH has applications to nuclear power engineering, and plasma modeling. \n",
+    "\n",
+    "These equations for incompressible MHD are given by:\n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "\\partial_t \\mathbf{u}+\\mathbf{u} \\cdot \\nabla \\mathbf{u} &=\n",
+    "-\\nabla \\left( p+\\frac{B^2}{2} \\right)/\\rho_0 +\\mathbf{B}\n",
+    "\\cdot \\nabla \\mathbf{B}+\\nu \\nabla^2 \\mathbf{u}, \\\\\n",
+    "\\partial_t \\mathbf{B}+\\mathbf{u} \\cdot \\nabla \\mathbf{B} &=\n",
+    "\\mathbf{B} \\cdot \\nabla \\mathbf{u}+\\eta \\nabla^2 \\mathbf{B}, \\\\\n",
+    "\\nabla \\cdot \\mathbf{u} &= 0, \\\\\n",
+    "\\nabla \\cdot \\mathbf{B} &= 0,\n",
+    "\\end{align*}$$\n",
+    " \n",
+    "where $\\mathbf{u}$ is the velocity field, $p$ is  the pressure, $B^2$ is the magnitude of the magnetic field, $\\rho_0=1$  is the density of the fluid, $\\nu$ is the kinetic viscosity,  and $\\eta$ is the magnetic resistivity.  We have two equations for evolution and two constraint equations.\n",
+    "\n",
+    "\n",
+    "For the magnetic field divergence equation, we can either include it in the loss function or instead evolve the magnetic vector potential $\\mathbf{A}$. This quantity is defined such that\n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "\\mathbf{B} = \\nabla \\times \\mathbf{A},\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "which ensures that the divergence of $\\mathbf{B}$ is zero. By evolving magnetic vector potential $\\mathbf{A}$ instead of the magnetic field $\\mathbf{B}$, we have a new evolution equation for the vector potential $\\mathbf{A}$. This equation is given by \n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "\\partial_t \\mathbf{A} + \\mathbf{u} \\cdot \\nabla \\mathbf{A}=\\eta \\nabla^2 \\mathbf{A}.\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "In practice, using the magnetic vector potential representation leads to better model performance. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5331b9b9",
+   "metadata": {},
+   "source": [
+    "## Data Creation\n",
+    "Note that in this HuggingFace Space, the data are available at `/data/mhd_data`. There are:\n",
+    "1000 samples for Re=100, 100 samples for Re=250 and 100 samples for Re=1,000.\n",
+    "\n",
+    "To train our model, a representative dataset is first created that gives enough coverage of the solution space to train a surrogate model to make predictions on new data points. To obtain interesting results without additional computational difficulty, we will solve the equations in 2D with periodic boundary conditions. This results in solving a total of 3 evolution PDEs at each time step. Two for the velocity evolution, and one for the magnetic vector potential. \n",
+    "\n",
+    "The solution space to this problem can be obtained numerically by solving the PDEs from above with a numerical solver such as `dedalus`. To generate this data, `dedalus` is used to simulate a 2D periodic incompressible MHD flow with a passive tracer field for visualization. The initial flow is in the $x$-direction and depends only on $z$. The problem is non-dimensionalized using the shear-layer spacing and velocity jump, so the resulting viscosity and tracer diffusivity are related to the Reynolds and\n",
+    "Schmidt numbers as:\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "\\nu &= \\frac{1}{\\text{Re}} \\\\\n",
+    "\\eta &= \\frac{1}{\\text{Re}_M} \\\\\n",
+    "D &= \\frac{\\nu}{\\text{Sc}}\n",
+    "\\end{align}$$\n",
+    "\n",
+    "The initial data field for running the simulation is produced using the Gaussian Random Field method in which the radial basis function kernel (RBF) is transformed into Fourier space to obey the desired periodic boundary conditions. Finally, two initial data fields the vorticity potential and magnetic potential are used to guarantee initial velocity and magnetic fields are divergence free. \n",
+    "\n",
+    "The dataset is produced by running 1,000 simulations with different initial conditions, and evolving the system for 1,000 time steps. The time step used is $\\Delta t=0.001s$, however output data is saved at an interval of $t=0.01$ for a total time of $1$ second, resulting in 101 samples per simulation. \n",
+    "\n",
+    "Scripts to generate the dataset are in the `generate_mhd_data` folder. Make sure to source this environment with `source activate env` to make use of the environment.   To generate the dataset, run the command: \n",
+    "```bash\n",
+    "python dedalus_mhd_parallel.py\n",
+    "```\n",
+    "Note that depending on system resources, this process may take up to a few hours to complete. Once data generation is finished, we can exit the env with `conda deactivate`. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69a099f7",
+   "metadata": {},
+   "source": [
+    "## Defining our Constraints - Setting up the PDE\n",
+    "\n",
+    "Constraints are used to define the objectives for training our model. They house a set of nodes from which a computational graph is build for execution as well as the loss function. [PhysicsNeMo Sim](https://docs.nvidia.com/deeplearning/physicsnemo/physicsnemo-sym/index.html) provides utilities tailored for physics-informed machine learning, and uses abstracted APIs that allow users to think and model the problem from the lens of equations, constraints, etc. In this example, we will only leverage the physics-informed utilities to see how we can add physics to an existing data-driven model with ease while still maintaining the flexibility to define our own training loop and other details. The types of constraints used will be problem dependent. For this example, we can define the following constraints: \n",
+    "\n",
+    "**Data Loss**: Obtain simulation data and compare it to the PINO output.\n",
+    "\n",
+    "**PDE Loss**: Use the known PDEs of the system to describe violations of the time evolution of our system\n",
+    "\n",
+    "**Constraint Loss**: This loss describes constraints from the PDE. Specifically, the velocity divergence free condition and magnetic divergence free condition.\n",
+    "\n",
+    "**Initial Condition Loss**: Input field compared to the output at $t=0$\n",
+    "\n",
+    "**Boundary Condition Loss**: Difference in boundary terms. In our case, we have a periodic boundary constraint.\n",
+    "\n",
+    "\n",
+    "\n",
+    "To begin setting up our constraints, we can start by defining the MHD equations using the `PDE` class from `physicsnemo.sym.eq.pde`. The process of converting our PDEs into a form that is compatible with `PhysicsNeMo` involves defining a class to hold our equations, called `MHD_PDE`, and including each term of the equations. Each variable of the equations is set up as a `Sympy` `Function`, which is then used to create an attribute of our `MHD_PDE` class that holds the final `equations`.\n",
+    "\n",
+    "Because we have elected to solve the equations in two dimensions, we only have the input variables $x$, $y$, $t$ and and the Laplacian operator. \n",
+    "\n",
+    "In PhysicsNeMo, it is preferable to represent our equations by isolating our target terms on the left, and moving the rest of the equation to the right-hand-side. To do this, various components of each equation are compartmentalized, and the final set of equations is composed from these parts.\n",
+    "\n",
+    "```python\n",
+    "from physicsnemo.sym.eq.pde import PDE\n",
+    "from sympy import Symbol, Function, Number\n",
+    "\n",
+    "\n",
+    "class MHD_PDE(PDE):\n",
+    "    \"\"\"MHD PDEs using PhysicsNeMo Sym\"\"\"\n",
+    "\n",
+    "    name = \"MHD_PDE\"\n",
+    "\n",
+    "    def __init__(self, nu=1e-4, eta=1e-4, rho0=1.0):\n",
+    "\n",
+    "        # x, y, time\n",
+    "        x, y, t, lap = Symbol(\"x\"), Symbol(\"y\"), Symbol(\"t\"), Symbol(\"lap\")\n",
+    "\n",
+    "        # make input variables\n",
+    "        input_variables = {\"x\": x, \"y\": y, \"t\": t, \"lap\": lap}\n",
+    "\n",
+    "        # make functions\n",
+    "        u = Function(\"u\")(*input_variables)\n",
+    "        v = Function(\"v\")(*input_variables)\n",
+    "        Bx = Function(\"Bx\")(*input_variables)\n",
+    "        By = Function(\"By\")(*input_variables)\n",
+    "        A = Function(\"A\")(*input_variables)\n",
+    "        # pressure\n",
+    "        ptot = Function(\"ptot\")(*input_variables)\n",
+    "\n",
+    "        u_rhs = Function(\"u_rhs\")(*input_variables)\n",
+    "        v_rhs = Function(\"v_rhs\")(*input_variables)\n",
+    "        Bx_rhs = Function(\"Bx_rhs\")(*input_variables)\n",
+    "        By_rhs = Function(\"By_rhs\")(*input_variables)\n",
+    "        A_rhs = Function(\"A_rhs\")(*input_variables)\n",
+    "\n",
+    "        # initialize constants\n",
+    "        nu = Number(nu)\n",
+    "        eta = Number(eta)\n",
+    "        rho0 = Number(rho0)\n",
+    "\n",
+    "        # set equations\n",
+    "        self.equations = {}\n",
+    "\n",
+    "        # u · ∇u\n",
+    "        self.equations[\"vel_grad_u\"] = u * u.diff(x) + v * u.diff(y)\n",
+    "        self.equations[\"vel_grad_v\"] = u * v.diff(x) + v * v.diff(y)\n",
+    "        # B · ∇u\n",
+    "        self.equations[\"B_grad_u\"] = Bx * u.diff(x) + v * Bx.diff(y)\n",
+    "        self.equations[\"B_grad_v\"] = Bx * v.diff(x) + By * v.diff(y)\n",
+    "        # u · ∇B\n",
+    "        self.equations[\"vel_grad_Bx\"] = u * Bx.diff(x) + v * Bx.diff(y)\n",
+    "        self.equations[\"vel_grad_By\"] = u * By.diff(x) + v * By.diff(y)\n",
+    "        # B · ∇B\n",
+    "        self.equations[\"B_grad_Bx\"] = Bx * Bx.diff(x) + By * Bx.diff(y)\n",
+    "        self.equations[\"B_grad_By\"] = Bx * By.diff(x) + By * By.diff(y)\n",
+    "        # ∇ × (u × B) = u(∇ · B) - B(∇ · u) + B · ∇u − u · ∇B\n",
+    "        self.equations[\"uBy_x\"] = u * By.diff(x) + By * u.diff(x)\n",
+    "        self.equations[\"uBy_y\"] = u * By.diff(y) + By * u.diff(y)\n",
+    "        self.equations[\"vBx_x\"] = v * Bx.diff(x) + Bx * v.diff(x)\n",
+    "        self.equations[\"vBx_y\"] = v * Bx.diff(y) + Bx * v.diff(y)\n",
+    "        # ∇ · B \n",
+    "        self.equations[\"div_B\"] = Bx.diff(x) + By.diff(y)\n",
+    "        # ∇ · u \n",
+    "        self.equations[\"div_vel\"] = u.diff(x) + v.diff(y)\n",
+    "\n",
+    "        # RHS of MHD equations\n",
+    "        # = u · ∇u - p/rho + B · ∇B + ν * ∇^2(u)\n",
+    "        self.equations[\"u_rhs\"] = (\n",
+    "            -self.equations[\"vel_grad_u\"]\n",
+    "            - ptot.diff(x) / rho0\n",
+    "            + self.equations[\"B_grad_Bx\"] / rho0\n",
+    "            + nu * u.diff(lap)\n",
+    "        )\n",
+    "        self.equations[\"v_rhs\"] = (\n",
+    "            -self.equations[\"vel_grad_v\"]\n",
+    "            - ptot.diff(y) / rho0\n",
+    "            + self.equations[\"B_grad_By\"] / rho0\n",
+    "            + nu * v.diff(lap)\n",
+    "        )\n",
+    "        # Uses identity above\n",
+    "        # = ∇ × (u × B) + η * ∇^2(B)\n",
+    "        self.equations[\"Bx_rhs\"] = (\n",
+    "            self.equations[\"uBy_y\"] - self.equations[\"vBx_y\"] + eta * Bx.diff(lap)\n",
+    "        )\n",
+    "        self.equations[\"By_rhs\"] = -(\n",
+    "            self.equations[\"uBy_x\"] - self.equations[\"vBx_x\"]\n",
+    "        ) + eta * By.diff(lap)\n",
+    "\n",
+    "        # Final equations move all terms to RHS\n",
+    "        # Node 18, 19, 20, 21\n",
+    "        self.equations[\"Du\"] = u.diff(t) - u_rhs\n",
+    "        self.equations[\"Dv\"] = v.diff(t) - v_rhs\n",
+    "        self.equations[\"DBx\"] = Bx.diff(t) - Bx_rhs\n",
+    "        self.equations[\"DBy\"] = By.diff(t) - By_rhs\n",
+    "\n",
+    "        # Vec potential equations\n",
+    "        # Node 22, 23, 24\n",
+    "        self.equations[\"vel_grad_A\"] = u * A.diff(x) + v * A.diff(y)\n",
+    "        self.equations[\"A_rhs\"] = -self.equations[\"vel_grad_A\"] + eta * A.diff(lap)\n",
+    "        self.equations[\"DA\"] = A.diff(t) - A_rhs\n",
+    "```\n",
+    "\n",
+    "Our model's output can then be used to compute the loss between prediction and true values, and for computing loss based on initial conditions, PDEs, and simulation data.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "677d49b4",
+   "metadata": {
+    "vscode": {
+     "languageId": "plaintext"
+    }
+   },
+   "source": [
+    "## Defining our Constraints - Loss Functions \n",
+    "\n",
+    "Now that we have defined our PDE, we can define all of the constraints that make up the loss function for our problem. The loss functions are defined inside of a class called `LossMHD_PhysicsNeMo`, which can use a weighted sum of individual losses for training. Additionally, all of the fixed and constant parameters needed are added to the class definition.\n",
+    "\n",
+    "```python\n",
+    "import torch\n",
+    "import torch.nn.functional as F\n",
+    "from physicsnemo.models.layers.spectral_layers import fourier_derivatives\n",
+    "\n",
+    "from .losses import (LpLoss, fourier_derivatives_lap, fourier_derivatives_ptot,\n",
+    "                     fourier_derivatives_vec_pot)\n",
+    "from .mhd_pde import MHD_PDE\n",
+    "\n",
+    "\n",
+    "class LossMHDVecPot_PhysicsNeMo(object):\n",
+    "    \"Calculate loss for MHD equations with vector potential, using physicsnemo derivatives\"\n",
+    "\n",
+    "    def __init__(\n",
+    "        self,\n",
+    "        nu=1e-4,\n",
+    "        eta=1e-4,\n",
+    "        rho0=1.0,\n",
+    "        data_weight=1.0,\n",
+    "        ic_weight=1.0,\n",
+    "        pde_weight=1.0,\n",
+    "        constraint_weight=1.0,\n",
+    "        use_data_loss=True,\n",
+    "        use_ic_loss=True,\n",
+    "        use_pde_loss=True,\n",
+    "        use_constraint_loss=True,\n",
+    "        u_weight=1.0,\n",
+    "        v_weight=1.0,\n",
+    "        A_weight=1.0,\n",
+    "        Du_weight=1.0,\n",
+    "        Dv_weight=1.0,\n",
+    "        DA_weight=1.0,\n",
+    "        div_B_weight=1.0,\n",
+    "        div_vel_weight=1.0,\n",
+    "        Lx=1.0,\n",
+    "        Ly=1.0,\n",
+    "        tend=1.0,\n",
+    "        use_weighted_mean=False,\n",
+    "        **kwargs,\n",
+    "    ):  # add **kwargs so that we ignore unexpected kwargs when passing a config dict):\n",
+    "\n",
+    "        self.nu = nu\n",
+    "        self.eta = eta\n",
+    "        self.rho0 = rho0\n",
+    "        self.data_weight = data_weight\n",
+    "        self.ic_weight = ic_weight\n",
+    "        self.pde_weight = pde_weight\n",
+    "        self.constraint_weight = constraint_weight\n",
+    "        self.use_data_loss = use_data_loss\n",
+    "        self.use_ic_loss = use_ic_loss\n",
+    "        self.use_pde_loss = use_pde_loss\n",
+    "        self.use_constraint_loss = use_constraint_loss\n",
+    "        self.u_weight = u_weight\n",
+    "        self.v_weight = v_weight\n",
+    "        self.Du_weight = Du_weight\n",
+    "        self.Dv_weight = Dv_weight\n",
+    "        self.div_B_weight = div_B_weight\n",
+    "        self.div_vel_weight = div_vel_weight\n",
+    "        self.Lx = Lx\n",
+    "        self.Ly = Ly\n",
+    "        self.tend = tend\n",
+    "        self.use_weighted_mean = use_weighted_mean\n",
+    "        self.A_weight = A_weight\n",
+    "        self.DA_weight = DA_weight\n",
+    "        # Define 2D MHD PDEs\n",
+    "        self.mhd_pde_eq = MHD_PDE(self.nu, self.eta, self.rho0)\n",
+    "        self.mhd_pde_node = self.mhd_pde_eq.make_nodes()\n",
+    "\n",
+    "        if not self.use_data_loss:\n",
+    "            self.data_weight = 0\n",
+    "        if not self.use_ic_loss:\n",
+    "            self.ic_weight = 0\n",
+    "        if not self.use_pde_loss:\n",
+    "            self.pde_weight = 0\n",
+    "        if not self.use_constraint_loss:\n",
+    "            self.constraint_weight = 0\n",
+    "\n",
+    "    def __call__(self, pred, true, inputs, return_loss_dict=False):\n",
+    "        loss, loss_dict = self.compute_losses(pred, true, inputs)\n",
+    "        return loss, loss_dict\n",
+    "\n",
+    "    def compute_losses(self, pred, true, inputs):\n",
+    "        \"Compute weighted loss and dictionary\"\n",
+    "        pred = pred.reshape(true.shape)\n",
+    "        u = pred[..., 0]\n",
+    "        v = pred[..., 1]\n",
+    "        A = pred[..., 2]\n",
+    "\n",
+    "        loss_dict = {}\n",
+    "\n",
+    "        # Data\n",
+    "        if self.use_data_loss:\n",
+    "            loss_data, loss_u, loss_v, loss_A = self.data_loss(\n",
+    "                pred, true, return_all_losses=True\n",
+    "            )\n",
+    "            loss_dict[\"loss_data\"] = loss_data\n",
+    "            loss_dict[\"loss_u\"] = loss_u\n",
+    "            loss_dict[\"loss_v\"] = loss_v\n",
+    "            loss_dict[\"loss_A\"] = loss_A\n",
+    "        else:\n",
+    "            loss_data = 0\n",
+    "        # IC\n",
+    "        if self.use_ic_loss:\n",
+    "            loss_ic, loss_u_ic, loss_v_ic, loss_A_ic = self.ic_loss(\n",
+    "                pred, inputs, return_all_losses=True\n",
+    "            )\n",
+    "            loss_dict[\"loss_ic\"] = loss_ic\n",
+    "            loss_dict[\"loss_u_ic\"] = loss_u_ic\n",
+    "            loss_dict[\"loss_v_ic\"] = loss_v_ic\n",
+    "            loss_dict[\"loss_A_ic\"] = loss_A_ic\n",
+    "        else:\n",
+    "            loss_ic = 0\n",
+    "\n",
+    "        # PDE\n",
+    "        if self.use_pde_loss:\n",
+    "            Du, Dv, DA = self.mhd_pde(u, v, A)\n",
+    "            loss_pde, loss_Du, loss_Dv, loss_DA = self.mhd_pde_loss(\n",
+    "                Du, Dv, DA, return_all_losses=True\n",
+    "            )\n",
+    "            loss_dict[\"loss_pde\"] = loss_pde\n",
+    "            loss_dict[\"loss_Du\"] = loss_Du\n",
+    "            loss_dict[\"loss_Dv\"] = loss_Dv\n",
+    "            loss_dict[\"loss_DA\"] = loss_DA\n",
+    "        else:\n",
+    "            loss_pde = 0\n",
+    "\n",
+    "        # Constraints\n",
+    "        if self.use_constraint_loss:\n",
+    "            div_vel, div_B = self.mhd_constraint(u, v, A)\n",
+    "            loss_constraint, loss_div_vel, loss_div_B = self.mhd_constraint_loss(\n",
+    "                div_vel, div_B, return_all_losses=True\n",
+    "            )\n",
+    "            loss_dict[\"loss_constraint\"] = loss_constraint\n",
+    "            loss_dict[\"loss_div_vel\"] = loss_div_vel\n",
+    "            loss_dict[\"loss_div_B\"] = loss_div_B\n",
+    "        else:\n",
+    "            loss_constraint = 0\n",
+    "\n",
+    "        if self.use_weighted_mean:\n",
+    "            weight_sum = (\n",
+    "                self.data_weight\n",
+    "                + self.ic_weight\n",
+    "                + self.pde_weight\n",
+    "                + self.constraint_weight\n",
+    "            )\n",
+    "        else:\n",
+    "            weight_sum = 1.0\n",
+    "\n",
+    "        loss = (\n",
+    "            self.data_weight * loss_data\n",
+    "            + self.ic_weight * loss_ic\n",
+    "            + self.pde_weight * loss_pde\n",
+    "            + self.constraint_weight * loss_constraint\n",
+    "        ) / weight_sum\n",
+    "        loss_dict[\"loss\"] = loss\n",
+    "        return loss, loss_dict\n",
+    "\n",
+    "```\n",
+    "\n",
+    "The MDH equations that we defined before are initialized for use within the  following loss functions. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c6c1a66a",
+   "metadata": {},
+   "source": [
+    "### Data Loss\n",
+    "The data loss is used to compare simulation data to the output of our model. The velocity in $x$ and $y$, as well as magnetic vector potential $\\mathbf{A}$ is directly compared to the ground truth data through the `Lp-Loss`, and the relative mean squared error is returned. \n",
+    "\n",
+    "\n",
+    "```python\n",
+    "def data_loss(self, pred, true, return_all_losses=False):\n",
+    "    \"Compute data loss\"\n",
+    "    lploss = LpLoss(size_average=True)\n",
+    "    u_pred = pred[..., 0]\n",
+    "    v_pred = pred[..., 1]\n",
+    "    A_pred = pred[..., 2]\n",
+    "\n",
+    "    u_true = true[..., 0]\n",
+    "    v_true = true[..., 1]\n",
+    "    A_true = true[..., 2]\n",
+    "\n",
+    "    loss_u = lploss(u_pred, u_true)\n",
+    "    loss_v = lploss(v_pred, v_true)\n",
+    "    loss_A = lploss(A_pred, A_true)\n",
+    "\n",
+    "    if self.use_weighted_mean:\n",
+    "        weight_sum = self.u_weight + self.v_weight + self.A_weight\n",
+    "    else:\n",
+    "        weight_sum = 1.0\n",
+    "\n",
+    "    loss_data = (\n",
+    "        self.u_weight * loss_u + self.v_weight * loss_v + self.A_weight * loss_A\n",
+    "    ) / weight_sum\n",
+    "\n",
+    "    if return_all_losses:\n",
+    "        return loss_data, loss_u, loss_v, loss_A\n",
+    "    else:\n",
+    "        return loss_data\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe2f1f44",
+   "metadata": {},
+   "source": [
+    "## PDE Loss\n",
+    "The PDE loss describes violations of the time evolution of the PDEs and the PINO outputs. In order to make this comparison, the spatial and temporal derivatives of the output fields need to be computed. To do so, Fourier differentiation is used to calculate the spacial derivatives, and second order finite differencing is used for temporal derivatives. The output fields are the velocity in the $x$ direction ($u$), the velocity in the $y$ direction ($v$), and the magnetic vector potential ($\\mathbf{A}$). The PDE loss is then defined as the MSE loss between zero and the PDE, after putting all the terms on the same side of the equation.\n",
+    "\n",
+    "Specifically, this loss covers the following equations: \n",
+    "$$\\begin{align*}\n",
+    "\\partial_t \\mathbf{u}+\\mathbf{u} \\cdot \\nabla \\mathbf{u} &=\n",
+    "-\\nabla \\left( p+\\frac{B^2}{2} \\right)/\\rho_0 +\\mathbf{B}\n",
+    "\\cdot \\nabla \\mathbf{B}+\\nu \\nabla^2 \\mathbf{u}, \\\\\n",
+    "\n",
+    "\\partial_t \\mathbf{A} + \\mathbf{u} \\cdot \\nabla \\mathbf{A} &=\\eta \\nabla^2 \\mathbf{A}\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "```python\n",
+    "def mhd_pde(self, u, v, A, p=None):\n",
+    "    \"Compute PDEs for MHD using vector potential\"\n",
+    "    nt = u.size(1)\n",
+    "    nx = u.size(2)\n",
+    "    ny = u.size(3)\n",
+    "    dt = self.tend / (nt - 1)\n",
+    "\n",
+    "    # compute fourier derivatives\n",
+    "    f_du, _ = fourier_derivatives(u, [self.Lx, self.Ly])\n",
+    "    f_dv, _ = fourier_derivatives(v, [self.Lx, self.Ly])\n",
+    "    f_dBx, f_dBy, f_dA, f_dB, B2_h = fourier_derivatives_vec_pot(\n",
+    "        A, [self.Lx, self.Ly]\n",
+    "    )\n",
+    "\n",
+    "    u_x = f_du[:, 0:nt, :nx, :ny]\n",
+    "    u_y = f_du[:, nt : 2 * nt, :nx, :ny]\n",
+    "    v_x = f_dv[:, 0:nt, :nx, :ny]\n",
+    "    v_y = f_dv[:, nt : 2 * nt, :nx, :ny]\n",
+    "    A_x = f_dA[:, 0:nt, :nx, :ny]\n",
+    "    A_y = f_dA[:, nt : 2 * nt, :nx, :ny]\n",
+    "\n",
+    "    Bx = f_dB[:, 0:nt, :nx, :ny]\n",
+    "    By = f_dB[:, nt : 2 * nt, :nx, :ny]\n",
+    "    Bx_x = f_dBx[:, 0:nt, :nx, :ny]\n",
+    "    Bx_y = f_dBx[:, nt : 2 * nt, :nx, :ny]\n",
+    "    By_x = f_dBy[:, 0:nt, :nx, :ny]\n",
+    "    By_y = f_dBy[:, nt : 2 * nt, :nx, :ny]\n",
+    "\n",
+    "    u_lap = fourier_derivatives_lap(u, [self.Lx, self.Ly])\n",
+    "    v_lap = fourier_derivatives_lap(v, [self.Lx, self.Ly])\n",
+    "    A_lap = fourier_derivatives_lap(A, [self.Lx, self.Ly])\n",
+    "\n",
+    "    # note that for pressure, the zero mode (the mean) cannot be zero for invertability so it is set to 1\n",
+    "    div_vel_grad_vel = u_x**2 + 2 * u_y * v_x + v_y**2\n",
+    "    div_B_grad_B = Bx_x**2 + 2 * Bx_y * By_x + By_y**2\n",
+    "    f_dptot = fourier_derivatives_ptot(\n",
+    "        p, div_vel_grad_vel, div_B_grad_B, B2_h, self.rho0, [self.Lx, self.Ly]\n",
+    "    )\n",
+    "    ptot_x = f_dptot[:, 0:nt, :nx, :ny]\n",
+    "    ptot_y = f_dptot[:, nt : 2 * nt, :nx, :ny]\n",
+    "\n",
+    "    # Plug inputs into dictionary\n",
+    "    all_inputs = {\n",
+    "        \"u\": u,\n",
+    "        \"u__x\": u_x,\n",
+    "        \"u__y\": u_y,\n",
+    "        \"v\": v,\n",
+    "        \"v__x\": v_x,\n",
+    "        \"v__y\": v_y,\n",
+    "        \"Bx\": Bx,\n",
+    "        \"Bx__x\": Bx_x,\n",
+    "        \"Bx__y\": Bx_y,\n",
+    "        \"By\": By,\n",
+    "        \"By__x\": By_x,\n",
+    "        \"By__y\": By_y,\n",
+    "        \"A__x\": A_x,\n",
+    "        \"A__y\": A_y,\n",
+    "        \"ptot__x\": ptot_x,\n",
+    "        \"ptot__y\": ptot_y,\n",
+    "        \"u__lap\": u_lap,\n",
+    "        \"v__lap\": v_lap,\n",
+    "        \"A__lap\": A_lap,\n",
+    "    }\n",
+    "\n",
+    "    # Substitute values into PDE equations\n",
+    "    u_rhs = self.mhd_pde_node[14].evaluate(all_inputs)[\"u_rhs\"]\n",
+    "    v_rhs = self.mhd_pde_node[15].evaluate(all_inputs)[\"v_rhs\"]\n",
+    "    A_rhs = self.mhd_pde_node[23].evaluate(all_inputs)[\"A_rhs\"]\n",
+    "\n",
+    "    u_t = self.Du_t(u, dt)\n",
+    "    v_t = self.Du_t(v, dt)\n",
+    "    A_t = self.Du_t(A, dt)\n",
+    "\n",
+    "    # Find difference\n",
+    "    Du = self.mhd_pde_node[18].evaluate({\"u__t\": u_t, \"u_rhs\": u_rhs[:, 1:-1]})[\n",
+    "        \"Du\"\n",
+    "    ]\n",
+    "    Dv = self.mhd_pde_node[19].evaluate({\"v__t\": v_t, \"v_rhs\": v_rhs[:, 1:-1]})[\n",
+    "        \"Dv\"\n",
+    "    ]\n",
+    "    DA = self.mhd_pde_node[24].evaluate({\"A__t\": A_t, \"A_rhs\": A_rhs[:, 1:-1]})[\n",
+    "        \"DA\"\n",
+    "    ]\n",
+    "\n",
+    "    return Du, Dv, DA\n",
+    "\n",
+    "\n",
+    "def mhd_pde_loss(self, Du, Dv, DA, return_all_losses=None):\n",
+    "    \"Compute PDE loss\"\n",
+    "    Du_val = torch.zeros_like(Du)\n",
+    "    Dv_val = torch.zeros_like(Dv)\n",
+    "    DA_val = torch.zeros_like(DA)\n",
+    "\n",
+    "    loss_Du = F.mse_loss(Du, Du_val)\n",
+    "    loss_Dv = F.mse_loss(Dv, Dv_val)\n",
+    "    loss_DA = F.mse_loss(DA, DA_val)\n",
+    "\n",
+    "    if self.use_weighted_mean:\n",
+    "        weight_sum = self.Du_weight + self.Dv_weight + self.DA_weight\n",
+    "    else:\n",
+    "        weight_sum = 1.0\n",
+    "\n",
+    "    loss_pde = (\n",
+    "        self.Du_weight * loss_Du\n",
+    "        + self.Dv_weight * loss_Dv\n",
+    "        + self.DA_weight * loss_DA\n",
+    "    ) / weight_sum\n",
+    "\n",
+    "    if return_all_losses:\n",
+    "        return loss_pde, loss_Du, loss_Dv, loss_DA\n",
+    "    else:\n",
+    "        return loss_pde\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cd40c1ed",
+   "metadata": {},
+   "source": [
+    "## Constraint Loss\n",
+    "The constraint illustrates the deviations of the velocity divergence free condition and the magnetic divergence free condition. These conditions are implemented similarly to the PDE loss, but without time derivative terms. The constraint loss is then the MSE between each of the constraint equations and zero.  \n",
+    "\n",
+    "Specifically, the equations used for constraint loss are:\n",
+    "$$\\begin{align*}\n",
+    "\\nabla \\cdot \\mathbf{u} &= 0, \\\\\n",
+    "\\nabla \\cdot \\mathbf{B} &= 0\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "\n",
+    "```python\n",
+    "def mhd_constraint(self, u, v, A):\n",
+    "    \"Compute constraints\"\n",
+    "    nt = u.size(1)\n",
+    "    nx = u.size(2)\n",
+    "    ny = u.size(3)\n",
+    "\n",
+    "    f_du, _ = fourier_derivatives(u, [self.Lx, self.Ly])\n",
+    "    f_dv, _ = fourier_derivatives(v, [self.Lx, self.Ly])\n",
+    "    f_dBx, f_dBy, _, _, _ = fourier_derivatives_vec_pot(A, [self.Lx, self.Ly])\n",
+    "\n",
+    "    u_x = f_du[:, 0:nt, :nx, :ny]\n",
+    "    v_y = f_dv[:, nt : 2 * nt, :nx, :ny]\n",
+    "    Bx_x = f_dBx[:, 0:nt, :nx, :ny]\n",
+    "    By_y = f_dBy[:, nt : 2 * nt, :nx, :ny]\n",
+    "\n",
+    "    div_B = self.mhd_pde_node[12].evaluate({\"Bx__x\": Bx_x, \"By__y\": By_y})[\"div_B\"]\n",
+    "    div_vel = self.mhd_pde_node[13].evaluate({\"u__x\": u_x, \"v__y\": v_y})[\"div_vel\"]\n",
+    "\n",
+    "    return div_vel, div_B\n",
+    "\n",
+    "def mhd_constraint_loss(self, div_vel, div_B, return_all_losses=False):\n",
+    "        \"Compute constraint loss\"\n",
+    "        div_vel_val = torch.zeros_like(div_vel)\n",
+    "        div_B_val = torch.zeros_like(div_B)\n",
+    "\n",
+    "        loss_div_vel = F.mse_loss(div_vel, div_vel_val)\n",
+    "        loss_div_B = F.mse_loss(div_B, div_B_val)\n",
+    "\n",
+    "        if self.use_weighted_mean:\n",
+    "            weight_sum = self.div_vel_weight + self.div_B_weight\n",
+    "        else:\n",
+    "            weight_sum = 1.0\n",
+    "\n",
+    "        loss_constraint = (\n",
+    "            self.div_vel_weight * loss_div_vel + self.div_B_weight * loss_div_B\n",
+    "        ) / weight_sum\n",
+    "\n",
+    "        if return_all_losses:\n",
+    "            return loss_constraint, loss_div_vel, loss_div_B\n",
+    "        else:\n",
+    "            return loss_constraint\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "627ae018",
+   "metadata": {},
+   "source": [
+    "## Initial Condition Loss\n",
+    "The initial condition loss encourages the model to associate the input field with the output field specifically at $t=0$. This constraint can usually be achieved with data loss, however this approach emphasized the importance of correct initial condition prediction, and enables training in the absence of data. Training without data and the significance of the initial condition term stem from the PDE loss term. \n",
+    "\n",
+    "```python\n",
+    "def ic_loss(self, pred, input, return_all_losses=False):\n",
+    "    \"Compute initial condition loss\"\n",
+    "    lploss = LpLoss(size_average=True)\n",
+    "    ic_pred = pred[:, 0]\n",
+    "    ic_true = input[:, 0, ..., 3:]\n",
+    "    u_ic_pred = ic_pred[..., 0]\n",
+    "    v_ic_pred = ic_pred[..., 1]\n",
+    "    A_ic_pred = ic_pred[..., 2]\n",
+    "\n",
+    "    u_ic_true = ic_true[..., 0]\n",
+    "    v_ic_true = ic_true[..., 1]\n",
+    "    A_ic_true = ic_true[..., 2]\n",
+    "\n",
+    "    loss_u_ic = lploss(u_ic_pred, u_ic_true)\n",
+    "    loss_v_ic = lploss(v_ic_pred, v_ic_true)\n",
+    "    loss_A_ic = lploss(A_ic_pred, A_ic_true)\n",
+    "\n",
+    "    if self.use_weighted_mean:\n",
+    "        weight_sum = self.u_weight + self.v_weight + self.A_weight\n",
+    "    else:\n",
+    "        weight_sum = 1.0\n",
+    "\n",
+    "    loss_ic = (\n",
+    "        self.u_weight * loss_u_ic\n",
+    "        + self.v_weight * loss_v_ic\n",
+    "        + self.A_weight * loss_A_ic\n",
+    "    ) / weight_sum\n",
+    "\n",
+    "    if return_all_losses:\n",
+    "        return loss_ic, loss_u_ic, loss_v_ic, loss_A_ic\n",
+    "    else:\n",
+    "        return loss_ic\n",
+    "```\n",
+    "\n",
+    "Similar to the initial condition loss, boundary condition loss can be used to describe violations of the boundary terms. In this specific case, the tFNO architecture ensures that the periodic boundary conditions are satisfied, thus the term is not used in this example. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "58809a2f",
+   "metadata": {},
+   "source": [
+    "In theory, training can be done by correctly predicting the initial conditions, boundary conditions and correctly evolving the PDE forward in time. In practice, having data helps the model converge more quickly. However, an incorrect initial condition results in the PDE evolving the wrong state forward in time, which is why it is emphasized as its own term. The initial condition loss is calculated by taking the input fields and computing the relative MSE with output fields at $t=0$. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f14c9c48",
+   "metadata": {},
+   "source": [
+    "## Dataset and Dataloaders\n",
+    "To use the data we have generated, we need to define a dataset and a dataloader that can ingest the files and parse them based on the relevant content. \n",
+    "\n",
+    "```python\n",
+    "import glob\n",
+    "import os\n",
+    "\n",
+    "import h5py\n",
+    "from torch.utils import data\n",
+    "\n",
+    "\n",
+    "class Dedalus2DDataset(data.Dataset):\n",
+    "    \"Dataset for MHD 2D Dataset\"\n",
+    "\n",
+    "    def __init__(\n",
+    "        self,\n",
+    "        data_path,\n",
+    "        output_names=\"output-\",\n",
+    "        field_names=[\"magnetic field\", \"velocity\"],\n",
+    "        num_train=None,\n",
+    "        num_test=None,\n",
+    "        num=None,\n",
+    "        use_train=True,\n",
+    "    ):\n",
+    "        self.data_path = data_path\n",
+    "        output_names = \"output-\" + \"?\"*len(str(len(os.listdir(data_path))))\n",
+    "        self.output_names = output_names\n",
+    "        raw_path = os.path.join(data_path, output_names, \"*.h5\")\n",
+    "        files_raw = sorted(glob.glob(raw_path))\n",
+    "        self.files_raw = files_raw\n",
+    "        self.num_files_raw = num_files_raw = len(files_raw)\n",
+    "        self.field_names = field_names\n",
+    "        self.use_train = use_train\n",
+    "\n",
+    "        # Handle num parameter: -1 means use full dataset, otherwise limit to specified number\n",
+    "        if num is not None and num > 0:\n",
+    "            num_files_raw = min(num, num_files_raw)\n",
+    "            files_raw = files_raw[:num_files_raw]\n",
+    "            self.files_raw = files_raw\n",
+    "            self.num_files_raw = num_files_raw\n",
+    "\n",
+    "        # Handle percentage-based splits\n",
+    "        if num_train is not None and num_train <= 1.0:\n",
+    "            # num_train is a percentage\n",
+    "            num_train = int(num_train * num_files_raw)\n",
+    "        elif num_train is None or num_train > num_files_raw:\n",
+    "            num_train = num_files_raw\n",
+    "\n",
+    "        if num_test is not None and num_test <= 1.0:\n",
+    "            # num_test is a percentage\n",
+    "            num_test = int(num_test * num_files_raw)\n",
+    "        elif num_test is None or num_test > (num_files_raw - num_train):\n",
+    "            num_test = num_files_raw - num_train\n",
+    "\n",
+    "        self.num_train = num_train\n",
+    "        self.train_files = self.files_raw[:num_train]\n",
+    "        self.num_test = num_test\n",
+    "        self.test_end = test_end = num_train + num_test\n",
+    "        self.test_files = self.files_raw[num_train:test_end]\n",
+    "        \n",
+    "        if (self.use_train) or (self.test_files is None):\n",
+    "            files = self.train_files\n",
+    "        else:\n",
+    "            files = self.test_files\n",
+    "        self.files = files\n",
+    "        self.num_files = num_files = len(files)\n",
+    "\n",
+    "    def __len__(self):\n",
+    "        length = len(self.files)\n",
+    "        return length\n",
+    "\n",
+    "    def __getitem__(self, index):\n",
+    "        \"Gets item for dataloader\"\n",
+    "        file = self.files[index]\n",
+    "\n",
+    "        field_names = self.field_names\n",
+    "        fields = {}\n",
+    "        coords = []\n",
+    "        with h5py.File(file, mode=\"r\") as h5file:\n",
+    "            data_file = h5file[\"tasks\"]\n",
+    "            keys = list(data_file.keys())\n",
+    "            if field_names is None:\n",
+    "                field_names = keys\n",
+    "            for field_name in field_names:\n",
+    "                if field_name in data_file:\n",
+    "                    field = data_file[field_name][:]\n",
+    "                    fields[field_name] = field\n",
+    "                else:\n",
+    "                    print(f\"field name {field_name} not found\")\n",
+    "        dataset = fields\n",
+    "        return dataset\n",
+    "\n",
+    "    def get_coords(self, index):\n",
+    "        \"Gets coordinates of t, x, y for dataloader\"\n",
+    "        file = self.files[index]\n",
+    "        with h5py.File(file, mode=\"r\") as h5file:\n",
+    "            data_file = h5file[\"tasks\"]\n",
+    "            keys = list(data_file.keys())\n",
+    "            dims = data_file[keys[0]].dims\n",
+    "\n",
+    "            ndims = len(dims)\n",
+    "            t = dims[0][\"sim_time\"][:]\n",
+    "            x = dims[ndims - 2][0][:]\n",
+    "            y = dims[ndims - 1][0][:]\n",
+    "        return t, x, y\n",
+    "```\n",
+    "\n",
+    "And the dataloader which is sampled from during training.\n",
+    "\n",
+    "```python\n",
+    "class MHDDataloaderVecPot(Dataset):\n",
+    "    \"Dataloader for MHD Dataset with vector potential\"\n",
+    "\n",
+    "    def __init__(\n",
+    "        self, dataset: Dedalus2DDataset, sub_x=1, sub_t=1, ind_x=None, ind_t=None\n",
+    "    ):\n",
+    "        self.dataset = dataset\n",
+    "        self.sub_x = sub_x\n",
+    "        self.sub_t = sub_t\n",
+    "        self.ind_x = ind_x\n",
+    "        self.ind_t = ind_t\n",
+    "        t, x, y = dataset.get_coords(0)\n",
+    "        self.x = x[:ind_x:sub_x]\n",
+    "        self.y = y[:ind_x:sub_x]\n",
+    "        self.t = t[:ind_t:sub_t]\n",
+    "        self.nx = len(self.x)\n",
+    "        self.ny = len(self.y)\n",
+    "        self.nt = len(self.t)\n",
+    "        self.num = num = len(self.dataset)\n",
+    "        self.x_slice = slice(0, self.ind_x, self.sub_x)\n",
+    "        self.t_slice = slice(0, self.ind_t, self.sub_t)\n",
+    "\n",
+    "    def __len__(self):\n",
+    "        length = len(self.dataset)\n",
+    "        return length\n",
+    "\n",
+    "    def __getitem__(self, index):\n",
+    "        \"Gets input of dataloader, including data, t, x, and y\"\n",
+    "        fields = self.dataset[index]\n",
+    "\n",
+    "        # Data includes velocity and vector potential\n",
+    "        velocity = fields[\"velocity\"]\n",
+    "        vector_potential = fields[\"vector potential\"]\n",
+    "\n",
+    "        u = torch.from_numpy(\n",
+    "            velocity[\n",
+    "                : self.ind_t : self.sub_t,\n",
+    "                0,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "            ]\n",
+    "        )\n",
+    "        v = torch.from_numpy(\n",
+    "            velocity[\n",
+    "                : self.ind_t : self.sub_t,\n",
+    "                1,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "            ]\n",
+    "        )\n",
+    "        A = torch.from_numpy(\n",
+    "            vector_potential[\n",
+    "                : self.ind_t : self.sub_t,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "                : self.ind_x : self.sub_x,\n",
+    "            ]\n",
+    "        )\n",
+    "\n",
+    "        # shape is now (self.nt, self.nx, self.ny, nfields)\n",
+    "        data = torch.stack([u, v, A], dim=-1)\n",
+    "        data0 = data[0].reshape(1, self.nx, self.ny, -1).repeat(self.nt, 1, 1, 1)\n",
+    "\n",
+    "        grid_t = (\n",
+    "            torch.from_numpy(self.t)\n",
+    "            .reshape(self.nt, 1, 1, 1)\n",
+    "            .repeat(1, self.nx, self.ny, 1)\n",
+    "        )\n",
+    "        grid_x = (\n",
+    "            torch.from_numpy(self.x)\n",
+    "            .reshape(1, self.nx, 1, 1)\n",
+    "            .repeat(self.nt, 1, self.ny, 1)\n",
+    "        )\n",
+    "        grid_y = (\n",
+    "            torch.from_numpy(self.y)\n",
+    "            .reshape(1, 1, self.ny, 1)\n",
+    "            .repeat(self.nt, self.nx, 1, 1)\n",
+    "        )\n",
+    "\n",
+    "        inputs = torch.cat([grid_t, grid_x, grid_y, data0], dim=-1)\n",
+    "        outputs = data\n",
+    "\n",
+    "        return inputs, outputs\n",
+    "    \n",
+    "    def create_dataloader(\n",
+    "        self,\n",
+    "        batch_size=1,\n",
+    "        shuffle=False,\n",
+    "        num_workers=0,\n",
+    "        pin_memory=False,\n",
+    "        distributed=False,\n",
+    "    ):\n",
+    "        \"Creates dataloader and sampler based on whether distributed training is on\"\n",
+    "        if distributed:\n",
+    "            sampler = torch.utils.data.DistributedSampler(self)\n",
+    "            dataloader = DataLoader(\n",
+    "                self,\n",
+    "                batch_size=batch_size,\n",
+    "                shuffle=False,\n",
+    "                sampler=sampler,\n",
+    "                num_workers=num_workers,\n",
+    "                pin_memory=pin_memory,\n",
+    "            )\n",
+    "        else:\n",
+    "            sampler = None\n",
+    "            dataloader = DataLoader(\n",
+    "                self,\n",
+    "                batch_size=batch_size,\n",
+    "                shuffle=shuffle,\n",
+    "                num_workers=num_workers,\n",
+    "                pin_memory=pin_memory,\n",
+    "            )\n",
+    "\n",
+    "        return dataloader, sampler\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf33d763",
+   "metadata": {},
+   "source": [
+    "## Model Architecture\n",
+    "<div style=\"display: flex; justify-content: center; gap: 10px;\">\n",
+    "  <figure style=\"text-align: center;\">\n",
+    "    <img src=\"https://raw.githubusercontent.com/openhackathons-org/End-to-End-AI-for-Science/main/workspace/python/jupyter_notebook/MagnetoHydrodynamics/images/model_arch.png\" style=\"width: 100%; height: auto;\">\n",
+    "    <figcaption>Model architecture overview.</figcaption>\n",
+    "  </figure>\n",
+    "</div>\n",
+    "\n",
+    "<!-- ![model_arch](images/model_arch.png) -->\n",
+    "\n",
+    "Our PINO model is composed of Tensor Factorized Neural Operators as the core component. Input fields are fed in as the input, which are composed of $u$, $v$, and $A$ initial conditions. The data is first lifted into a higher dimension representation by the neural network, P1. The data then enters the Fourier layers ($F_1$,...,$F_n$). Each Fourier layer consists of a sequence of non-logical integral operators, and nonlinear activation functions. $T_1$ represents a linear transform that employs CP decomposed tensors as weights, and $T_2$ represents a local linear transform. $\\sigma$ is the activation function, and $\\mathcal{F}$, $\\mathcal{F}^{-1}$ represent the Fourier transfrom and inverse Fourier transform respectively. At the end, $P_2$ projects back down into the input space, producing the output shown on the right which describe the\n",
+    "time evolution of the system. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5dacfbfb",
+   "metadata": {},
+   "source": [
+    "## Training our Model\n",
+    "\n",
+    "PhysicsNeMo has two distinct styles, namely Core and Sym. PhysicsNeMo Sym is a framework providing pythonic APIs, algorithms and utilities to be used with PhysicsNeMo Core, while PhysicsNeMo Core interoperates with PyTorch directly. Working with PhysicsNeMo Core looks and feels more like a PyTorch workflow with some key utils like models, utils, and datapipes imported directly from `physicsnemo` itself. While some components of this workflow so far have borrowed from PhysicsNeMo Sym (`MHD_PDE`), the training workflow for this problem will be build primarily using the Core style. This will provide more flexibility over our training loop, and allow for further customizations to our workflow. The training script follows the standard flow of training models using pytorch. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3339921a",
+   "metadata": {},
+   "source": [
+    "## Hydra Config\n",
+    "\n",
+    "Training in PhysicsNeMo is facilitated by Hydra configs, which allow us to set and manager parameters from a single file, updating parameters for components such as our model, datasets, optimizer, logger, loss function, and dataloaders. The first step in getting set up for training is defining this yaml file and loading the config.\n",
+    "\n",
+    "\n",
+    "```yaml \n",
+    "## Training options\n",
+    "# Reynolds number parameter\n",
+    "reynolds_number: 100\n",
+    "\n",
+    "load_ckpt: False\n",
+    "output_dir: './checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/figures/'\n",
+    "\n",
+    "###################\n",
+    "## Model options\n",
+    "model_params:\n",
+    "  layers: 8\n",
+    "  modes: 8\n",
+    "  num_fno_layers: 4\n",
+    "  fc_dim: 128\n",
+    "  decoder_layers: 1\n",
+    "  in_dim: 6 # 3 + in_fields\n",
+    "  out_dim: 3\n",
+    "  dimension: 3\n",
+    "  activation: 'gelu'\n",
+    "  pad_x: 5\n",
+    "  pad_y: 0\n",
+    "  pad_z: 0\n",
+    "  input_norm: [1.0, 1.0, 1.0, 1.0, 1.0, 0.00025]\n",
+    "  output_norm: [1.0, 1.0, 0.00025]\n",
+    "\n",
+    "  #TensorLy arguments\n",
+    "  rank: 0.5\n",
+    "  factorization: 'cp'\n",
+    "  fixed_rank_modes: null\n",
+    "  decomposition_kwargs: {}\n",
+    "\n",
+    "###################\n",
+    "## Dataset options\n",
+    "dataset_params:\n",
+    "  data_dir: '/Datasets/mhd_data/simulation_outputs_Re${reynolds_number}'\n",
+    "  field_names: ['velocity', 'vector potential']\n",
+    "  output_names: 'output-????'\n",
+    "  dataset_type: 'mhd'\n",
+    "  name: 'MHDVecPot_TFNO_Re${reynolds_number}'\n",
+    "  num: -1  # -1 means use full dataset, otherwise specify total number\n",
+    "  num_train: 0.8  # percentage of dataset for training\n",
+    "  num_test: 0.2   # percentage of dataset for testing\n",
+    "  sub_x: 1\n",
+    "  sub_t: 1\n",
+    "  ind_x: null\n",
+    "  ind_t: null\n",
+    "  nin: 3\n",
+    "  nout: 3\n",
+    "  fields: ['u', 'v', 'A']\n",
+    "\n",
+    "###################\n",
+    "## Dataloader options\n",
+    "train_loader_params:\n",
+    "  batch_size: 1\n",
+    "  shuffle: True\n",
+    "  num_workers: 4\n",
+    "  pin_memory: True\n",
+    "\n",
+    "val_loader_params:\n",
+    "  batch_size: 1\n",
+    "  shuffle: False\n",
+    "  num_workers: 4\n",
+    "  pin_memory: True\n",
+    "\n",
+    "test_loader_params:\n",
+    "  batch_size: 1\n",
+    "  shuffle: False\n",
+    "  num_workers: 4\n",
+    "  pin_memory: True\n",
+    "\n",
+    "###################\n",
+    "## Loss options\n",
+    "loss_params:\n",
+    "  nu: 0.004\n",
+    "  eta: 0.004\n",
+    "  rho0: 1.0\n",
+    "\n",
+    "  data_weight: 5.0\n",
+    "  ic_weight: 1.0\n",
+    "  pde_weight: 1.0\n",
+    "  constraint_weight: 10.0\n",
+    "\n",
+    "  use_data_loss: True\n",
+    "  use_ic_loss: True\n",
+    "  use_pde_loss: True\n",
+    "  use_constraint_loss: True\n",
+    "\n",
+    "  u_weight: 1.0\n",
+    "  v_weight: 1.0\n",
+    "  A_weight: 1.0\n",
+    "\n",
+    "  Du_weight: 1.0\n",
+    "  Dv_weight: 1.0\n",
+    "  DA_weight: 1_000_000\n",
+    "\n",
+    "  div_B_weight: 1.0\n",
+    "  div_vel_weight: 1.0\n",
+    "\n",
+    "  Lx: 1.0\n",
+    "  Ly: 1.0\n",
+    "  tend: 1.0\n",
+    "\n",
+    "  use_weighted_mean: False\n",
+    "\n",
+    "###################\n",
+    "## Optimizer options\n",
+    "optimizer_params:\n",
+    "  betas: [0.9, 0.999]\n",
+    "  lr: 5.0e-4\n",
+    "  milestones: [20, 40, 60, 80, 100]\n",
+    "  gamma: 0.5\n",
+    "\n",
+    "\n",
+    "###################\n",
+    "## Train params\n",
+    "train_params:\n",
+    "  epochs: 100\n",
+    "  ckpt_freq: 10\n",
+    "  ckpt_path: 'checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/'\n",
+    "\n",
+    "###################\n",
+    "## log params\n",
+    "log_params:\n",
+    "  log_dir: 'logs'\n",
+    "  log_project: 'MHD_PINO'\n",
+    "  log_group: 'MHDVecPot_TFNO_Re${reynolds_number}'\n",
+    "  log_num_plots: 1\n",
+    "  log_plot_freq: 5\n",
+    "  log_plot_types: ['ic', 'pred', 'true', 'error']\n",
+    "\n",
+    "test:\n",
+    "  batchsize: 1\n",
+    "  ckpt_path: 'checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/'\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c9c370fa",
+   "metadata": {},
+   "source": [
+    "## Training Setup\n",
+    "\n",
+    "We begin with importing the required modules, capturing our hydra config, and initializing some utilities to facilitate the model training. Most of this initial setup is \n",
+    "\n",
+    "```python\n",
+    "import os\n",
+    "\n",
+    "import hydra\n",
+    "from omegaconf import ListConfig, OmegaConf\n",
+    "import torch\n",
+    "from omegaconf import DictConfig\n",
+    "from physicsnemo.distributed import DistributedManager\n",
+    "from physicsnemo.launch.logging import LaunchLogger, PythonLogger\n",
+    "from physicsnemo.launch.utils import load_checkpoint, save_checkpoint\n",
+    "from physicsnemo.sym.hydra import to_absolute_path\n",
+    "from torch.nn.parallel import DistributedDataParallel\n",
+    "from torch.optim import AdamW\n",
+    "\n",
+    "from dataloaders import Dedalus2DDataset, MHDDataloaderVecPot\n",
+    "from losses import LossMHDVecPot_PhysicsNeMo\n",
+    "from tfno import TFNO\n",
+    "from utils.plot_utils import plot_predictions_mhd, plot_predictions_mhd_plotly\n",
+    "\n",
+    "dtype = torch.float\n",
+    "torch.set_default_dtype(dtype)\n",
+    "\n",
+    "\n",
+    "@hydra.main(\n",
+    "    version_base=\"1.3\", config_path=\"config\", config_name=\"train_mhd_vec_pot_tfno.yaml\"\n",
+    ")\n",
+    "def main(cfg: DictConfig) -> None:\n",
+    "    DistributedManager.initialize()  # Only call this once in the entire script!\n",
+    "    dist = DistributedManager()  # call if required elsewhere\n",
+    "    cfg = OmegaConf.to_container(cfg, resolve=True)\n",
+    "\n",
+    "    # initialize monitoring\n",
+    "    log = PythonLogger(name=\"mhd_pino\")\n",
+    "    log.file_logging()\n",
+    "\n",
+    "    log_params = cfg[\"log_params\"]\n",
+    "\n",
+    "    # Load config file parameters\n",
+    "    model_params = cfg[\"model_params\"]\n",
+    "    dataset_params = cfg[\"dataset_params\"]\n",
+    "    train_loader_params = cfg[\"train_loader_params\"]\n",
+    "    val_loader_params = cfg[\"val_loader_params\"]\n",
+    "    loss_params = cfg[\"loss_params\"]\n",
+    "    optimizer_params = cfg[\"optimizer_params\"]\n",
+    "    train_params = cfg[\"train_params\"]\n",
+    "\n",
+    "    load_ckpt = cfg[\"load_ckpt\"]\n",
+    "    output_dir = cfg[\"output_dir\"]\n",
+    "\n",
+    "    output_dir = to_absolute_path(output_dir)\n",
+    "    os.makedirs(output_dir, exist_ok=True)\n",
+    "\n",
+    "    data_dir = dataset_params[\"data_dir\"]\n",
+    "    ckpt_path = train_params[\"ckpt_path\"]\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bacc38c7",
+   "metadata": {},
+   "source": [
+    "## Datasets and Dataloaders\n",
+    "\n",
+    "Datasets and dataloaders are initialized using parameters from the hydra config.\n",
+    "\n",
+    "```python\n",
+    "# Construct dataloaders\n",
+    "dataset_train = Dedalus2DDataset(\n",
+    "    dataset_params[\"data_dir\"],\n",
+    "    output_names=dataset_params[\"output_names\"],\n",
+    "    field_names=dataset_params[\"field_names\"],\n",
+    "    num_train=dataset_params[\"num_train\"],\n",
+    "    num_test=dataset_params[\"num_test\"],\n",
+    "    num=dataset_params[\"num\"],\n",
+    "    use_train=True,\n",
+    ")\n",
+    "dataset_val = Dedalus2DDataset(\n",
+    "    data_dir,\n",
+    "    output_names=dataset_params[\"output_names\"],\n",
+    "    field_names=dataset_params[\"field_names\"],\n",
+    "    num_train=dataset_params[\"num_train\"],\n",
+    "    num_test=dataset_params[\"num_test\"],\n",
+    "    num=dataset_params[\"num\"],\n",
+    "    use_train=False,\n",
+    ")\n",
+    "\n",
+    "mhd_dataloader_train = MHDDataloaderVecPot(\n",
+    "    dataset_train,\n",
+    "    sub_x=dataset_params[\"sub_x\"],\n",
+    "    sub_t=dataset_params[\"sub_t\"],\n",
+    "    ind_x=dataset_params[\"ind_x\"],\n",
+    "    ind_t=dataset_params[\"ind_t\"],\n",
+    ")\n",
+    "mhd_dataloader_val = MHDDataloaderVecPot(\n",
+    "    dataset_val,\n",
+    "    sub_x=dataset_params[\"sub_x\"],\n",
+    "    sub_t=dataset_params[\"sub_t\"],\n",
+    "    ind_x=dataset_params[\"ind_x\"],\n",
+    "    ind_t=dataset_params[\"ind_t\"],\n",
+    ")\n",
+    "\n",
+    "dataloader_train, sampler_train = mhd_dataloader_train.create_dataloader(\n",
+    "    batch_size=train_loader_params[\"batch_size\"],\n",
+    "    shuffle=train_loader_params[\"shuffle\"],\n",
+    "    num_workers=train_loader_params[\"num_workers\"],\n",
+    "    pin_memory=train_loader_params[\"pin_memory\"],\n",
+    "    distributed=dist.distributed,\n",
+    ")\n",
+    "dataloader_val, sampler_val = mhd_dataloader_val.create_dataloader(\n",
+    "    batch_size=val_loader_params[\"batch_size\"],\n",
+    "    shuffle=val_loader_params[\"shuffle\"],\n",
+    "    num_workers=val_loader_params[\"num_workers\"],\n",
+    "    pin_memory=val_loader_params[\"pin_memory\"],\n",
+    "    distributed=dist.distributed,\n",
+    ")\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4826da07",
+   "metadata": {},
+   "source": [
+    "## Model Construction\n",
+    "For a relatively simple model such as `FNO`, we can directly use an architecture pre-defined by PhysicsNeMo. Hyper-parameters are set directly from the hydra config, and makes it straight forward to configure hyper parameter optimization if necessary. For a more complex model such as `tFNO`, we can leverage a combination of PhysicsNeMo primitives and third party packages to build a model in pytorch. \n",
+    "\n",
+    "```python\n",
+    "# Define the model\n",
+    "model = TFNO(\n",
+    "    in_channels=model_params[\"in_dim\"],\n",
+    "    out_channels=model_params[\"out_dim\"],\n",
+    "    decoder_layers=model_params[\"decoder_layers\"],\n",
+    "    decoder_layer_size=model_params[\"fc_dim\"],\n",
+    "    dimension=model_params[\"dimension\"],\n",
+    "    latent_channels=model_params[\"layers\"],\n",
+    "    num_fno_layers=model_params[\"num_fno_layers\"],\n",
+    "    num_fno_modes=model_params[\"modes\"],\n",
+    "    padding=[model_params[\"pad_z\"], model_params[\"pad_y\"], model_params[\"pad_x\"]],\n",
+    "    rank=model_params[\"rank\"],\n",
+    "    factorization=model_params[\"factorization\"],\n",
+    "    fixed_rank_modes=model_params[\"fixed_rank_modes\"],\n",
+    "    decomposition_kwargs=model_params[\"decomposition_kwargs\"],\n",
+    ").to(dist.device)\n",
+    "# Set up DistributedDataParallel if using more than a single process.\n",
+    "# The `distributed` property of DistributedManager can be used to\n",
+    "# check this.\n",
+    "if dist.distributed:\n",
+    "    ddps = torch.cuda.Stream()\n",
+    "    with torch.cuda.stream(ddps):\n",
+    "        model = DistributedDataParallel(\n",
+    "            model,\n",
+    "            device_ids=[dist.local_rank],  # Set the device_id to be\n",
+    "            # the local rank of this process on\n",
+    "            # this node\n",
+    "            output_device=dist.device,\n",
+    "            broadcast_buffers=dist.broadcast_buffers,\n",
+    "            find_unused_parameters=dist.find_unused_parameters,\n",
+    "        )\n",
+    "    torch.cuda.current_stream().wait_stream(ddps)\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "99604fbc",
+   "metadata": {},
+   "source": [
+    "## Optimizer, Scheduler, Loss Functions and Check-pointing\n",
+    "\n",
+    "\n",
+    "```python\n",
+    "# Construct optimizer and scheduler\n",
+    "optimizer = AdamW(\n",
+    "    model.parameters(),\n",
+    "    betas=optimizer_params[\"betas\"],\n",
+    "    lr=optimizer_params[\"lr\"],\n",
+    "    weight_decay=0.1,\n",
+    ")\n",
+    "\n",
+    "scheduler = torch.optim.lr_scheduler.MultiStepLR(\n",
+    "    optimizer,\n",
+    "    milestones=optimizer_params[\"milestones\"],\n",
+    "    gamma=optimizer_params[\"gamma\"],\n",
+    ")\n",
+    "\n",
+    "# Construct Loss class\n",
+    "mhd_loss = LossMHDVecPot_PhysicsNeMo(**loss_params)\n",
+    "\n",
+    "# Load model from checkpoint (if exists)\n",
+    "loaded_epoch = 0\n",
+    "if load_ckpt:\n",
+    "    loaded_epoch = load_checkpoint(\n",
+    "        ckpt_path, model, optimizer, scheduler, device=dist.device\n",
+    "    )\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a775b128",
+   "metadata": {},
+   "source": [
+    "## Training Loop\n",
+    "Finally, the main training loop iterates through the dataset for our defined number of epochs, saving checkpoints and visualizations of our training along the way.\n",
+    "\n",
+    "```python\n",
+    "# Training Loop\n",
+    "epochs = train_params[\"epochs\"]\n",
+    "ckpt_freq = train_params[\"ckpt_freq\"]\n",
+    "names = dataset_params[\"fields\"]\n",
+    "input_norm = torch.tensor(model_params[\"input_norm\"]).to(dist.device)\n",
+    "output_norm = torch.tensor(model_params[\"output_norm\"]).to(dist.device)\n",
+    "for epoch in range(max(1, loaded_epoch + 1), epochs + 1):\n",
+    "    with LaunchLogger(\n",
+    "        \"train\",\n",
+    "        epoch=epoch,\n",
+    "        num_mini_batch=len(dataloader_train),\n",
+    "        epoch_alert_freq=1,\n",
+    "    ) as log:\n",
+    "        if dist.distributed:\n",
+    "            sampler_train.set_epoch(epoch)\n",
+    "\n",
+    "        # Train Loop\n",
+    "        model.train()\n",
+    "\n",
+    "        for i, (inputs, outputs) in enumerate(dataloader_train):\n",
+    "            inputs = inputs.type(torch.FloatTensor).to(dist.device)\n",
+    "            outputs = outputs.type(torch.FloatTensor).to(dist.device)\n",
+    "            # Zero Gradients\n",
+    "            optimizer.zero_grad()\n",
+    "            # Compute Predictions\n",
+    "            pred = (\n",
+    "                model((inputs / input_norm).permute(0, 4, 1, 2, 3)).permute(\n",
+    "                    0, 2, 3, 4, 1\n",
+    "                )\n",
+    "                * output_norm\n",
+    "            )\n",
+    "            # Compute Loss\n",
+    "            loss, loss_dict = mhd_loss(pred, outputs, inputs, return_loss_dict=True)\n",
+    "            # Compute Gradients for Back Propagation\n",
+    "            loss.backward()\n",
+    "            # Update Weights\n",
+    "            optimizer.step()\n",
+    "\n",
+    "            log.log_minibatch(loss_dict)\n",
+    "\n",
+    "        log.log_epoch({\"Learning Rate\": optimizer.param_groups[0][\"lr\"]})\n",
+    "        scheduler.step()\n",
+    "\n",
+    "    with LaunchLogger(\"valid\", epoch=epoch) as log:\n",
+    "        # Val loop\n",
+    "        model.eval()\n",
+    "        plot_count = 0\n",
+    "        with torch.no_grad():\n",
+    "            for i, (inputs, outputs) in enumerate(dataloader_val):\n",
+    "                inputs = inputs.type(dtype).to(dist.device)\n",
+    "                outputs = outputs.type(dtype).to(dist.device)\n",
+    "\n",
+    "                # Compute Predictions\n",
+    "                pred = (\n",
+    "                    model((inputs / input_norm).permute(0, 4, 1, 2, 3)).permute(\n",
+    "                        0, 2, 3, 4, 1\n",
+    "                    )\n",
+    "                    * output_norm\n",
+    "                )\n",
+    "                # Compute Loss\n",
+    "                loss, loss_dict = mhd_loss(\n",
+    "                    pred, outputs, inputs, return_loss_dict=True\n",
+    "                )\n",
+    "\n",
+    "                log.log_minibatch(loss_dict)\n",
+    "\n",
+    "                # Get prediction plots to log\n",
+    "                # Do for number of batches specified in the config file\n",
+    "                if (i < log_params[\"log_num_plots\"]) and (\n",
+    "                    epoch % log_params[\"log_plot_freq\"] == 0\n",
+    "                ):\n",
+    "                    # Add all predictions in batch\n",
+    "                    for j, _ in enumerate(pred):\n",
+    "                        # Make plots for each field\n",
+    "                        for index, name in enumerate(names):\n",
+    "                            # Generate figure\n",
+    "                            _ = plot_predictions_mhd_plotly(\n",
+    "                                pred[j].cpu(),\n",
+    "                                outputs[j].cpu(),\n",
+    "                                inputs[j].cpu(),\n",
+    "                                index=index,\n",
+    "                                name=name,\n",
+    "                            )\n",
+    "                        plot_count += 1\n",
+    "\n",
+    "                # Get prediction plots and save images locally\n",
+    "                if (i < 2) and (epoch % log_params[\"log_plot_freq\"] == 0):\n",
+    "                    # Add all predictions in batch\n",
+    "                    for j, _ in enumerate(pred):\n",
+    "                        # Generate figure\n",
+    "                        plot_predictions_mhd(\n",
+    "                            pred[j].cpu(),\n",
+    "                            outputs[j].cpu(),\n",
+    "                            inputs[j].cpu(),\n",
+    "                            names=names,\n",
+    "                            save_path=os.path.join(\n",
+    "                                output_dir,\n",
+    "                                \"MHD_physicsnemo\" + \"_\" + str(dist.rank),\n",
+    "                            ),\n",
+    "                            save_suffix=i,\n",
+    "                        )\n",
+    "\n",
+    "        if epoch % ckpt_freq == 0 and dist.rank == 0:\n",
+    "            save_checkpoint(ckpt_path, model, optimizer, scheduler, epoch=epoch)\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ca49a425",
+   "metadata": {},
+   "source": [
+    "## Running the Training Script\n",
+    "\n",
+    "The full set of python code to start training is available in the folder `./mhd`. Configs, data generation, dataloaders, loss functions, model architectures, and training scripts are all available here. If utilizing the scripts outside of this HuggingFace Space, you can launch training with:\n",
+    "\n",
+    "```bash\n",
+    "torchrun --standalone --nnodes=1 --nproc_per_node=1 train_mhd_vec_pot_tfno.py\n",
+    "```\n",
+    "\n",
+    "With the default set of parameters, the model will take up around 5.2GB of GPU memory, and a full training run up to 100 epochs will take around 1.5 hours."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "39c969ce",
+   "metadata": {},
+   "source": [
+    "## End-to-End Training\n",
+    "\n",
+    "All of the code that was detailed above is available to explore in the \"./mhd\" folder. There are also two scripts that execute the end-to-end workflow for training and evaluation. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6013cf5d-8232-45bf-8e79-1757bb29d3fe",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "!python mhd/train_mhd_vec_pot_tfno.py"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "daf003ec",
+   "metadata": {},
+   "source": [
+    "## Transfer Learning to New Reynolds Number\n",
+    "In practice, our system may not follow smooth, laminar flows described with low Reynolds numbers. In MHD systems, much of the magnetic field energy is stored at high wave numbers, which occur at smaller scales. Models must then be able to characterize high frequency features in order to successfully reproduce the trajectories of the system. These turbulent flows at higher Reynolds number are simulated, which will in turn produce higher frequency features that a model trained on smooth flows may not be able to resolve with good accuracy. To this end, transfer learning can be used to take a base model and adapt it to the new data domain by using a pre-trained checkpoint as the starting point of a new iteration of model training. \n",
+    "\n",
+    "To run transfer learning, we need a dataset of points from our new domain. For example, our default model is trained on data using $Re=100$, so we can use the model checkpoint from this domain to start off transferlerning to a new dataset with $Re=250$. In the Hydra config, we can update the following parameters:\n",
+    "\n",
+    "```yaml\n",
+    "load_ckpt: True\n",
+    "output_dir: \"/path/to/new/output_dir\"\n",
+    "\n",
+    "dataset_params:\n",
+    "  data_dir: \"/path/to/new/dataset\"\n",
+    "  name: 'Dataset Name'\n",
+    "\n",
+    "train_params:\n",
+    "    ckpt_path: \"/path/to/starting_checkpoint\"\n",
+    "```\n",
+    "\n",
+    "<div style=\"display: flex; justify-content: center; gap: 10px;\">\n",
+    "  <figure style=\"text-align: center;\">\n",
+    "    <img src=\"https://raw.githubusercontent.com/openhackathons-org/End-to-End-AI-for-Science/main/workspace/python/jupyter_notebook/MagnetoHydrodynamics/images/high_frequency.png\" style=\"width: 100%; height: auto;\">\n",
+    "    <figcaption>Predictions with a large Reynolds Number.</figcaption>\n",
+    "  </figure>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "93c7d926",
+   "metadata": {},
+   "source": [
+    "## Evaluation\n",
+    "When solving the MHD equations with `dedalus`, the average time per simulation is about 37 seconds. On the other hand, our physics informed model has an average inference time of 0.15 seconds, a 246x speedup. This comes at the cost of decreased accuracy in our solution, as it is an approximation to the system equations. Furthermore, our models performance will vary, depending on the Reynolds number. \n",
+    "\n",
+    "Evaluation can be run a few different ways. If there are many systems to evaluate, we can load them into a dataloader and do batch processing. In this example, we will use a standalone script, which is a stripped down version of the training script that will run our model with a single sample. \n",
+    "\n",
+    "To run evaluation we can use the following command, which pulls in a config that points to a specific pre-trained checkpoint and dataset. The  config is found in `eval_mhd_vec_pot_tfno.yaml`\n",
+    "\n",
+    "\n",
+    "```bash\n",
+    "torchrun --standalone --nnodes=1 --nproc_per_node=1 evaluate_mhd_vec_pot_tfno.py\n",
+    "```\n",
+    "\n",
+    "In evaluations, our model is able to accurately simulate flows at $Re<250$. Specifically, for $Re=100$, our surrogate model has less than 4% error at $t=1$ for all fields. At $Re=250$, the velocity field and vector potential potential are accurately described, with MSEs <7% and <10%, respectively. At higher Reynolds numbers, our model starts to break down. An example for $Re=100$ is shown below, as well as some plots showing $MSE$ vs $Re$.\n",
+    "\n",
+    "<div style=\"display: flex; justify-content: center; gap: 10px;\">\n",
+    "  <figure style=\"text-align: center;\">\n",
+    "    <img src=\"https://raw.githubusercontent.com/openhackathons-org/End-to-End-AI-for-Science/main/workspace/python/jupyter_notebook/MagnetoHydrodynamics/images/re100.png\" style=\"width: 100%; height: auto;\">\n",
+    "    <figcaption>Predictions with a low Reynolds Number.</figcaption>\n",
+    "  </figure>\n",
+    "</div>\n",
+    "<div style=\"display: flex; justify-content: center; gap: 10px;\">\n",
+    "  <figure style=\"text-align: center;\">\n",
+    "    <img src=\"https://raw.githubusercontent.com/openhackathons-org/End-to-End-AI-for-Science/main/workspace/python/jupyter_notebook/MagnetoHydrodynamics/images/mse_vs_re.png\" style=\"width: 100%; height: auto;\">\n",
+    "    <figcaption>Error vs. Reynolds Number.</figcaption>\n",
+    "  </figure>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9fec5e93",
+   "metadata": {},
+   "source": [
+    "## End-to-End Evaluation\n",
+    "To run evaluation, use the following script:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b44a1bfd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "!python mhd/evaluate_mhd_vec_pot_tfno.py"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26acf9de",
+   "metadata": {},
+   "source": [
+    "## Shortcomings and areas for improvement\n",
+    "\n",
+    "Physics informed machine learning shows promising results when applied to certain regions of parameter space as governed by the Reynolds number. While models such as tFNOs are able to accurately capture and simulate systems, they do not always perform well when the underlying physics begin to shift into regions of high frequency features. A tradeoff is present in accuracy and throughput, where these AI surrogate models accelerate simulations over 200x, however they remain accuracy for only the low Reynolds number parameter space. To this end, applying physics informed ML to the MHD equations shows both promise and room for improvement. For example, increased model size, additional physical loss functions from energy spectra, and higher resolution datasets may be a few areas in which the development and application of these models may be improved. In conclusion, the efficacy of physics informed machine learning has been shown to the modeling of magnetohydrodynamics, and researchers, scientists, and engineers are encouraged to build on this foundation to enhance these techniques further. "
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

mhd/.gitignore

@@ -0,0 +1,8 @@
+*_results
+outputs
+logs
+mhd_data
+checkpoints 
+README.md
+launch.log
+requirements.txt

mhd/config/eval_mhd_vec_pot_tfno.yaml

@@ -0,0 +1,139 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+
+## Training options
+# Reynolds number parameter
+reynolds_number: 100
+
+load_ckpt: False
+use_log: True
+output_dir: './checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/figures/'
+derivative: 'physicsnemo'
+
+###################
+## Model options
+model_params:
+  layers: 8
+  modes: 8
+  num_fno_layers: 4
+  fc_dim: 128
+  decoder_layers: 1
+  in_dim: 6 # 3 + in_fields
+  out_dim: 3
+  dimension: 3
+  activation: 'gelu'
+  pad_x: 5
+  pad_y: 0
+  pad_z: 0
+  input_norm: [1.0, 1.0, 1.0, 1.0, 1.0, 0.00025]
+  output_norm: [1.0, 1.0, 0.00025]
+
+  #TensorLy arguments
+  rank: 0.5
+  factorization: 'cp'
+  fixed_rank_modes: null
+
+###################
+## Dataset options
+dataset_params:
+  data_dir: '/data/mhd_data/simulation_outputs_Re${reynolds_number}'
+  field_names: ['velocity', 'vector potential']
+  output_names: 'output-????'
+  dataset_type: 'mhd'
+  name: 'MHDVecPot_TFNO_Re${reynolds_number}'
+  num: -1  # -1 means use full dataset for evaluation
+  num_train: 0.8  # percentage of dataset for training (not used in eval)
+  num_test: 0.2   # percentage of dataset for testing (not used in eval)
+  sub_x: 1
+  sub_t: 1
+  ind_x: null
+  ind_t: null
+  nin: 3
+  nout: 3
+  fields: ['u', 'v', 'A']
+
+###################
+## Dataloader options
+test_loader_params:
+  batch_size: 1
+  shuffle: False
+  num_workers: 4
+  pin_memory: True
+
+###################
+## Loss options
+loss_params:
+  nu: 0.004
+  eta: 0.004
+  rho0: 1.0
+
+  data_weight: 5.0
+  ic_weight: 1.0
+  pde_weight: 1.0
+  constraint_weight: 10.0
+
+  use_data_loss: True
+  use_ic_loss: True
+  use_pde_loss: True
+  use_constraint_loss: True
+
+  u_weight: 1.0
+  v_weight: 1.0
+  A_weight: 1.0
+
+  Du_weight: 1.0
+  Dv_weight: 1.0
+  DA_weight: 1_000_000
+
+  div_B_weight: 1.0
+  div_vel_weight: 1.0
+
+  Lx: 1.0
+  Ly: 1.0
+  tend: 1.0
+
+  use_weighted_mean: False
+
+###################
+## Optimizer options
+optimizer_params:
+  betas: [0.9, 0.999]
+  lr: 5.0e-4
+  milestones: [20, 40, 60, 80, 100]
+  gamma: 0.5
+
+
+###################
+## Train params
+train_params:
+  epochs: 100
+  ckpt_freq: 50
+  ckpt_path: 'checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/'
+
+###################
+## log params
+log_params:
+  log_dir: 'logs'
+  log_project: 'MHD_PINO'
+  log_group: 'MHDVecPot_TFNO_Re${reynolds_number}'
+  log_num_plots: 1
+  log_plot_freq: 5
+  log_plot_types: ['ic', 'pred', 'true', 'error']
+
+test:
+  batchsize: 1
+  ckpt_path: 'checkpoints/MHDVecPot_TFNO/MHDVecPot_TFNO_PINO_Re${reynolds_number}/'

mhd/config/train_mhd_vec_pot_tfno.yaml

@@ -0,0 +1,150 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+
+## Training options
+# Reynolds number parameter
+reynolds_number: 100
+
+load_ckpt: False
+output_dir: './outputs/Re${reynolds_number}/figures/'
+
+###################
+## Model options
+model_params:
+  layers: 8
+  modes: 8
+  num_fno_layers: 4
+  fc_dim: 128
+  decoder_layers: 1
+  in_dim: 6 # 3 + in_fields
+  out_dim: 3
+  dimension: 3
+  activation: 'gelu'
+  pad_x: 5
+  pad_y: 0
+  pad_z: 0
+  input_norm: [1.0, 1.0, 1.0, 1.0, 1.0, 0.00025]
+  output_norm: [1.0, 1.0, 0.00025]
+
+  #TensorLy arguments
+  rank: 0.5
+  factorization: 'cp'
+  fixed_rank_modes: null
+  decomposition_kwargs: {}
+
+###################
+## Dataset options
+dataset_params:
+  data_dir: '/data/mhd_data/simulation_outputs_Re${reynolds_number}'
+  field_names: ['velocity', 'vector potential']
+  output_names: 'output-????'
+  dataset_type: 'mhd'
+  name: 'MHDVecPot_TFNO_Re${reynolds_number}'
+  num: -1  # -1 means use full dataset, otherwise specify total number
+  num_train: 0.8  # percentage of dataset for training
+  num_test: 0.2   # percentage of dataset for testing
+  sub_x: 1
+  sub_t: 1
+  ind_x: null
+  ind_t: null
+  nin: 3
+  nout: 3
+  fields: ['u', 'v', 'A']
+
+###################
+## Dataloader options
+train_loader_params:
+  batch_size: 4
+  shuffle: True
+  num_workers: 8
+  pin_memory: True
+
+val_loader_params:
+  batch_size: 4
+  shuffle: False
+  num_workers: 8
+  pin_memory: True
+
+test_loader_params:
+  batch_size: 4
+  shuffle: False
+  num_workers: 8
+  pin_memory: True
+
+###################
+## Loss options
+loss_params:
+  nu: 0.004
+  eta: 0.004
+  rho0: 1.0
+
+  data_weight: 5.0
+  ic_weight: 1.0
+  pde_weight: 1.0
+  constraint_weight: 10.0
+
+  use_data_loss: True
+  use_ic_loss: True
+  use_pde_loss: True
+  use_constraint_loss: True
+
+  u_weight: 1.0
+  v_weight: 1.0
+  A_weight: 1.0
+
+  Du_weight: 1.0
+  Dv_weight: 1.0
+  DA_weight: 1_000_000
+
+  div_B_weight: 1.0
+  div_vel_weight: 1.0
+
+  Lx: 1.0
+  Ly: 1.0
+  tend: 1.0
+
+  use_weighted_mean: False
+
+###################
+## Optimizer options
+optimizer_params:
+  betas: [0.9, 0.999]
+  lr: 5.0e-4
+  milestones: [20, 40, 60, 80, 100]
+  gamma: 0.5
+
+
+###################
+## Train params
+train_params:
+  epochs: 100
+  ckpt_freq: 10
+  ckpt_path: './outputs/checkpoints/Re${reynolds_number}/'
+
+###################
+## log params
+log_params:
+  log_dir: 'logs'
+  log_project: 'MHD_PINO'
+  log_group: 'MHDVecPot_TFNO_Re${reynolds_number}'
+  log_num_plots: 1
+  log_plot_freq: 5
+  log_plot_types: ['ic', 'pred', 'true', 'error']
+
+test:
+  batchsize: 1
+  ckpt_path: './outputs/checkpoints/Re${reynolds_number}/'

mhd/dataloaders/__init__.py

@@ -0,0 +1,18 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from .datasets import Dedalus2DDataset
+from .dataloaders import MHDDataloader, MHDDataloaderVecPot

mhd/dataloaders/dataloaders.py

@@ -0,0 +1,277 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import torch
+from IPython.display import display
+from torch.utils.data import DataLoader, Dataset
+
+try:
+    from .datasets import Dedalus2DDataset
+except:
+    from datasets import Dedalus2DDataset
+
+
+class MHDDataloader(Dataset):
+    "Dataloader for MHD Dataset with magnetic field"
+
+    def __init__(
+        self, dataset: Dedalus2DDataset, sub_x=1, sub_t=1, ind_x=None, ind_t=None
+    ):
+        self.dataset = dataset
+        self.sub_x = sub_x
+        self.sub_t = sub_t
+        self.ind_x = ind_x
+        self.ind_t = ind_t
+        t, x, y = dataset.get_coords(0)
+        self.x = x[:ind_x:sub_x]
+        self.y = y[:ind_x:sub_x]
+        self.t = t[:ind_t:sub_t]
+        self.nx = len(self.x)
+        self.ny = len(self.y)
+        self.nt = len(self.t)
+        self.num = num = len(self.dataset)
+        self.x_slice = slice(0, self.ind_x, self.sub_x)
+        self.t_slice = slice(0, self.ind_t, self.sub_t)
+
+    def __len__(self):
+        length = len(self.dataset)
+        return length
+
+    def __getitem__(self, index):
+        "Gets input of dataloader, including data, t, x, and y"
+        fields = self.dataset[index]
+
+        # Data includes velocity and magnetic field
+        velocity = fields["velocity"]
+        magnetic_field = fields["magnetic field"]
+
+        u = torch.from_numpy(
+            velocity[
+                : self.ind_t : self.sub_t,
+                0,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+        v = torch.from_numpy(
+            velocity[
+                : self.ind_t : self.sub_t,
+                1,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+        Bx = torch.from_numpy(
+            magnetic_field[
+                : self.ind_t : self.sub_t,
+                0,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+        By = torch.from_numpy(
+            magnetic_field[
+                : self.ind_t : self.sub_t,
+                1,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+
+        # shape is now (nt, nx, ny, nfields)
+        data = torch.stack([u, v, Bx, By], dim=-1)
+        data0 = data[0].reshape(1, self.nx, self.ny, -1).repeat(self.nt, 1, 1, 1)
+
+        grid_t = (
+            torch.from_numpy(self.t)
+            .reshape(self.nt, 1, 1, 1)
+            .repeat(1, self.nx, self.ny, 1)
+        )
+        grid_x = (
+            torch.from_numpy(self.x)
+            .reshape(1, self.nx, 1, 1)
+            .repeat(self.nt, 1, self.ny, 1)
+        )
+        grid_y = (
+            torch.from_numpy(self.y)
+            .reshape(1, 1, self.ny, 1)
+            .repeat(self.nt, self.nx, 1, 1)
+        )
+
+        inputs = torch.cat([grid_t, grid_x, grid_y, data0], dim=-1)
+        outputs = data
+
+        return inputs, outputs
+
+    def create_dataloader(
+        self,
+        batch_size=1,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=False,
+        distributed=False,
+    ):
+        "Creates dataloader and sampler based on whether distributed training is on"
+        if distributed:
+            sampler = torch.utils.data.DistributedSampler(self)
+            dataloader = DataLoader(
+                self,
+                batch_size=batch_size,
+                shuffle=False,
+                sampler=sampler,
+                num_workers=num_workers,
+                pin_memory=pin_memory,
+            )
+        else:
+            sampler = None
+            dataloader = DataLoader(
+                self,
+                batch_size=batch_size,
+                shuffle=shuffle,
+                num_workers=num_workers,
+                pin_memory=pin_memory,
+            )
+
+        return dataloader, sampler
+
+
+class MHDDataloaderVecPot(MHDDataloader):
+    "Dataloader for MHD Dataset with vector potential"
+
+    def __init__(
+        self, dataset: Dedalus2DDataset, sub_x=1, sub_t=1, ind_x=None, ind_t=None
+    ):
+        self.dataset = dataset
+        self.sub_x = sub_x
+        self.sub_t = sub_t
+        self.ind_x = ind_x
+        self.ind_t = ind_t
+        t, x, y = dataset.get_coords(0)
+        self.x = x[:ind_x:sub_x]
+        self.y = y[:ind_x:sub_x]
+        self.t = t[:ind_t:sub_t]
+        self.nx = len(self.x)
+        self.ny = len(self.y)
+        self.nt = len(self.t)
+        self.num = num = len(self.dataset)
+        self.x_slice = slice(0, self.ind_x, self.sub_x)
+        self.t_slice = slice(0, self.ind_t, self.sub_t)
+
+    def __len__(self):
+        length = len(self.dataset)
+        return length
+
+    def __getitem__(self, index):
+        "Gets input of dataloader, including data, t, x, and y"
+        fields = self.dataset[index]
+
+        # Data includes velocity and vector potential
+        velocity = fields["velocity"]
+        vector_potential = fields["vector potential"]
+
+        u = torch.from_numpy(
+            velocity[
+                : self.ind_t : self.sub_t,
+                0,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+        v = torch.from_numpy(
+            velocity[
+                : self.ind_t : self.sub_t,
+                1,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+        A = torch.from_numpy(
+            vector_potential[
+                : self.ind_t : self.sub_t,
+                : self.ind_x : self.sub_x,
+                : self.ind_x : self.sub_x,
+            ]
+        )
+
+        # shape is now (self.nt, self.nx, self.ny, nfields)
+        data = torch.stack([u, v, A], dim=-1)
+        data0 = data[0].reshape(1, self.nx, self.ny, -1).repeat(self.nt, 1, 1, 1)
+
+        grid_t = (
+            torch.from_numpy(self.t)
+            .reshape(self.nt, 1, 1, 1)
+            .repeat(1, self.nx, self.ny, 1)
+        )
+        grid_x = (
+            torch.from_numpy(self.x)
+            .reshape(1, self.nx, 1, 1)
+            .repeat(self.nt, 1, self.ny, 1)
+        )
+        grid_y = (
+            torch.from_numpy(self.y)
+            .reshape(1, 1, self.ny, 1)
+            .repeat(self.nt, self.nx, 1, 1)
+        )
+
+        inputs = torch.cat([grid_t, grid_x, grid_y, data0], dim=-1)
+        outputs = data
+
+        return inputs, outputs
+    
+    def create_dataloader(
+        self,
+        batch_size=1,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=False,
+        distributed=False,
+    ):
+        "Creates dataloader and sampler based on whether distributed training is on"
+        if distributed:
+            sampler = torch.utils.data.DistributedSampler(self)
+            dataloader = DataLoader(
+                self,
+                batch_size=batch_size,
+                shuffle=False,
+                sampler=sampler,
+                num_workers=num_workers,
+                pin_memory=pin_memory,
+            )
+        else:
+            sampler = None
+            dataloader = DataLoader(
+                self,
+                batch_size=batch_size,
+                shuffle=shuffle,
+                num_workers=num_workers,
+                pin_memory=pin_memory,
+            )
+
+        return dataloader, sampler
+
+
+
+if __name__ == "__main__":
+    dataset = Dedalus2DDataset(
+        data_path="../mhd_data/simulation_outputs_Re250",
+        output_names="output-????",
+        field_names=["magnetic field", "velocity", "vector potential"],
+    )
+    mhd_dataloader = MHDDataloader(dataset)
+    mhd_vec_pot_dataloader = MHDDataloaderVecPot(dataset)
+
+    data = mhd_dataloader[0]
+    display(data)

mhd/dataloaders/datasets.py

@@ -0,0 +1,117 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import glob
+import os
+
+import h5py
+from torch.utils import data
+
+
+class Dedalus2DDataset(data.Dataset):
+    "Dataset for MHD 2D Dataset"
+
+    def __init__(
+        self,
+        data_path,
+        output_names="output-",
+        field_names=["magnetic field", "velocity"],
+        num_train=None,
+        num_test=None,
+        num=None,
+        use_train=True,
+    ):
+        self.data_path = data_path
+        output_names = "output-" + "?"*len(str(len(os.listdir(data_path))))
+        self.output_names = output_names
+        raw_path = os.path.join(data_path, output_names, "*.h5")
+        files_raw = sorted(glob.glob(raw_path))
+        self.files_raw = files_raw
+        self.num_files_raw = num_files_raw = len(files_raw)
+        self.field_names = field_names
+        self.use_train = use_train
+
+        # Handle num parameter: -1 means use full dataset, otherwise limit to specified number
+        if num is not None and num > 0:
+            num_files_raw = min(num, num_files_raw)
+            files_raw = files_raw[:num_files_raw]
+            self.files_raw = files_raw
+            self.num_files_raw = num_files_raw
+
+        # Handle percentage-based splits
+        if num_train is not None and num_train <= 1.0:
+            # num_train is a percentage
+            num_train = int(num_train * num_files_raw)
+        elif num_train is None or num_train > num_files_raw:
+            num_train = num_files_raw
+
+        if num_test is not None and num_test <= 1.0:
+            # num_test is a percentage
+            num_test = int(num_test * num_files_raw)
+        elif num_test is None or num_test > (num_files_raw - num_train):
+            num_test = num_files_raw - num_train
+
+        self.num_train = num_train
+        self.train_files = self.files_raw[:num_train]
+        self.num_test = num_test
+        self.test_end = test_end = num_train + num_test
+        self.test_files = self.files_raw[num_train:test_end]
+        
+        if (self.use_train) or (self.test_files is None):
+            files = self.train_files
+        else:
+            files = self.test_files
+        self.files = files
+        self.num_files = num_files = len(files)
+
+    def __len__(self):
+        length = len(self.files)
+        return length
+
+    def __getitem__(self, index):
+        "Gets item for dataloader"
+        file = self.files[index]
+
+        field_names = self.field_names
+        fields = {}
+        coords = []
+        with h5py.File(file, mode="r") as h5file:
+            data_file = h5file["tasks"]
+            keys = list(data_file.keys())
+            if field_names is None:
+                field_names = keys
+            for field_name in field_names:
+                if field_name in data_file:
+                    field = data_file[field_name][:]
+                    fields[field_name] = field
+                else:
+                    print(f"field name {field_name} not found")
+        dataset = fields
+        return dataset
+
+    def get_coords(self, index):
+        "Gets coordinates of t, x, y for dataloader"
+        file = self.files[index]
+        with h5py.File(file, mode="r") as h5file:
+            data_file = h5file["tasks"]
+            keys = list(data_file.keys())
+            dims = data_file[keys[0]].dims
+
+            ndims = len(dims)
+            t = dims[0]["sim_time"][:]
+            x = dims[ndims - 2][0][:]
+            y = dims[ndims - 1][0][:]
+        return t, x, y

mhd/evaluate_mhd_vec_pot_tfno.py

@@ -0,0 +1,205 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import os
+
+import hydra
+import torch
+from omegaconf import DictConfig, OmegaConf
+from physicsnemo.distributed import DistributedManager
+from physicsnemo.launch.logging import LaunchLogger, PythonLogger
+from physicsnemo.sym.hydra import to_absolute_path
+from torch.nn.parallel import DistributedDataParallel
+from torch.optim import AdamW
+import time
+
+from dataloaders import Dedalus2DDataset, MHDDataloaderVecPot
+from losses import LossMHDVecPot_PhysicsNeMo
+from tfno import TFNO
+from utils.plot_utils import plot_predictions_mhd, plot_predictions_mhd_plotly
+
+dtype = torch.float
+torch.set_default_dtype(dtype)
+
+
+@hydra.main(
+    version_base="1.3", config_path="config", config_name="eval_mhd_vec_pot_tfno.yaml"
+)
+def main(cfg: DictConfig) -> None:
+    DistributedManager.initialize()  # Only call this once in the entire script!
+    dist = DistributedManager()  # call if required elsewhere
+    cfg = OmegaConf.to_container(cfg, resolve=True)
+    # initialize monitoring
+    log = PythonLogger(name="mhd_pino")
+    log.file_logging()
+    # Load config file parameters
+    model_params = cfg["model_params"]
+    dataset_params = cfg["dataset_params"]
+    test_loader_params = cfg["test_loader_params"]
+    loss_params = cfg["loss_params"]
+    optimizer_params = cfg["optimizer_params"]
+
+    output_dir = cfg["output_dir"]
+    test_params = cfg["test"]
+    load_checkpoint = cfg.get("load_ckpt", False)
+
+    output_dir = to_absolute_path(output_dir)
+    os.makedirs(output_dir, exist_ok=True)
+
+    data_dir = dataset_params["data_dir"]
+
+    # Construct dataloaders
+    dataset_test = Dedalus2DDataset(
+        data_dir,
+        output_names=dataset_params["output_names"],
+        field_names=dataset_params["field_names"],
+        num_train=dataset_params["num_train"],
+        num_test=dataset_params["num_test"],
+        num=dataset_params["num"],
+        use_train=False,
+    )
+    mhd_dataloader_test = MHDDataloaderVecPot(
+        dataset_test,
+        sub_x=dataset_params["sub_x"],
+        sub_t=dataset_params["sub_t"],
+        ind_x=dataset_params["ind_x"],
+        ind_t=dataset_params["ind_t"],
+    )
+    dataloader_test, sampler_test = mhd_dataloader_test.create_dataloader(
+        batch_size=test_loader_params["batch_size"],
+        shuffle=test_loader_params["shuffle"],
+        num_workers=test_loader_params["num_workers"],
+        pin_memory=test_loader_params["pin_memory"],
+        distributed=dist.distributed,
+    )
+
+    # define FNO model
+    model = TFNO(
+        in_channels=model_params["in_dim"],
+        out_channels=model_params["out_dim"],
+        decoder_layers=model_params["decoder_layers"],
+        decoder_layer_size=model_params["fc_dim"],
+        dimension=model_params["dimension"],
+        latent_channels=model_params["layers"],
+        num_fno_layers=model_params["num_fno_layers"],
+        num_fno_modes=model_params["modes"],
+        padding=[model_params["pad_z"], model_params["pad_y"], model_params["pad_x"]],
+        rank=model_params["rank"],
+        factorization=model_params["factorization"],
+        fixed_rank_modes=model_params["fixed_rank_modes"],
+    ).to(dist.device)
+
+    # Set up DistributedDataParallel if using more than a single process.
+    # The `distributed` property of DistributedManager can be used to
+    # check this.
+    if dist.distributed:
+        ddps = torch.cuda.Stream()
+        with torch.cuda.stream(ddps):
+            model = DistributedDataParallel(
+                model,
+                device_ids=[dist.local_rank],  # Set the device_id to be
+                # the local rank of this process on
+                # this node
+                output_device=dist.device,
+                broadcast_buffers=dist.broadcast_buffers,
+                find_unused_parameters=dist.find_unused_parameters,
+            )
+        torch.cuda.current_stream().wait_stream(ddps)
+
+    # Construct optimizer and scheduler
+    optimizer = AdamW(
+        model.parameters(),
+        betas=optimizer_params["betas"],
+        lr=optimizer_params["lr"],
+        weight_decay=0.1,
+    )
+
+    scheduler = torch.optim.lr_scheduler.MultiStepLR(
+        optimizer,
+        milestones=optimizer_params["milestones"],
+        gamma=optimizer_params["gamma"],
+    )
+
+    # Construct Loss class
+    mhd_loss = LossMHDVecPot_PhysicsNeMo(**loss_params)
+
+    # Load model from checkpoint (if exists)
+    if load_checkpoint:
+        _ = load_checkpoint(
+            test_params["ckpt_path"], model, optimizer, scheduler, device=dist.device
+        )
+
+    # Eval Loop
+    names = dataset_params["fields"]
+    input_norm = torch.tensor(model_params["input_norm"]).to(dist.device)
+    output_norm = torch.tensor(model_params["output_norm"]).to(dist.device)
+
+    with LaunchLogger("test") as log:
+        # Val loop
+        model.eval()
+        plot_count = 0
+        with torch.no_grad():
+            for i, (inputs, outputs) in enumerate(dataloader_test):
+                inputs = inputs.type(dtype).to(dist.device)
+                outputs = outputs.type(dtype).to(dist.device)
+                start_time = time.time()
+                # Compute Predictions
+                pred = (
+                    model((inputs / input_norm).permute(0, 4, 1, 2, 3)).permute(
+                        0, 2, 3, 4, 1
+                    )
+                    * output_norm
+                )
+                end_time = time.time()
+                print(f"Inference Time: {end_time-start_time}")
+                # Compute Loss
+                loss, loss_dict = mhd_loss(pred, outputs, inputs, return_loss_dict=True)
+
+                log.log_minibatch(loss_dict)
+
+                # Get prediction plots
+                for j, _ in enumerate(pred):
+                    # Make plots for each field
+                    for index, name in enumerate(names):
+                        # Generate figure
+                        _ = plot_predictions_mhd_plotly(
+                            pred[j].cpu(),
+                            outputs[j].cpu(),
+                            inputs[j].cpu(),
+                            index=index,
+                            name=name,
+                        )
+
+                    plot_count += 1
+
+                # Get prediction plots and save images locally
+                for j, _ in enumerate(pred):
+                    # Generate figure
+                    plot_predictions_mhd(
+                        pred[j].cpu(),
+                        outputs[j].cpu(),
+                        inputs[j].cpu(),
+                        names=names,
+                        save_path=os.path.join(
+                            output_dir,
+                            "MHD_eval_" + str(dist.rank),
+                        ),
+                        save_suffix=i,
+                    )
+
+
+if __name__ == "__main__":
+    main()

mhd/generate_mhd_data/dedalus_mhd_parallel.py

@@ -0,0 +1,352 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+"""
+Dedalus script simulating a 2D periodic incompressible MHD flow with a passive
+tracer field for visualization. This script demonstrates solving a 2D periodic
+initial value problem. This script is meant to be ran in parallel, and uses the
+built-in analysis framework to save data snapshots to HDF5 files. 
+The simulation should take at least 100 gpu-minutes to run. 
+
+The initial flow is in the x-direction and depends only on z. The problem is
+non-dimensionalized usign the shear-layer spacing and velocity jump, so the
+resulting viscosity and tracer diffusivity are related to the Reynolds and
+Schmidt numbers as:
+
+    nu = 1 / Re
+    eta = 1 / ReM
+    D = nu / Schmidt
+
+To run this script:
+    $ python dedalus_mhd_parallel.py
+"""
+
+
+import os
+import glob
+import h5py
+import numpy as np
+import functools
+from functools import partial
+import matplotlib
+import matplotlib.pyplot as plt
+import argparse
+import multiprocessing as mp
+import dedalus
+import dedalus.public as d3
+from dedalus.extras import plot_tools
+import pathlib
+from docopt import docopt
+from dedalus.tools import logging
+from dedalus.tools import post
+from dedalus.tools.parallel import Sync
+import logging
+import math
+from IPython.display import display
+import imageio
+from importlib import reload
+from my_random_fields import GRF_Mattern
+import torch
+from functorch import vmap
+from hydra import compose, initialize
+from hydra.utils import get_class
+
+device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
+# display(device)
+
+
+def check_if_complete(sim_outputs, Nt=101):
+    try:
+        files = sorted(glob.glob(sim_outputs))
+        file = files[0]
+        with h5py.File(file, mode="r") as h5file:
+            data_file = h5file["tasks"]
+            keys = list(data_file.keys())
+            dims = data_file[keys[0]].dims
+            t = dims[0]["sim_time"][:]
+        if len(t) == Nt:
+            return True
+        else:
+            return False
+    except Exception:
+        return False
+
+
+if __name__ == "__main__":
+    import sys
+    
+    # Parse command line args before Hydra initialization
+    parser = argparse.ArgumentParser(add_help=False)
+    parser.add_argument('--Re', type=float, help='Reynolds number')
+    parser.add_argument('--N', type=int, help='Number of samples')
+    args, remaining_argv = parser.parse_known_args()
+    
+    # Initialize Hydra with remaining args
+    sys.argv = [sys.argv[0]] + remaining_argv
+    initialize(version_base=None, config_path=".", job_name="generate_mhd_field")
+    cfg = compose(config_name="mhd_field")
+
+    # Parameters - override with command line args if provided
+    Lx, Ly = cfg.Lx, cfg.Ly
+    Nx, Ny = cfg.Nx, cfg.Ny
+    Re = args.Re if args.Re is not None else cfg.Re  # Use CLI arg or default to config
+    Re = int(Re)
+    ReM = Re 
+    Schmidt = cfg.Schmidt  # 1
+    rho0 = cfg.rho0  # 1.0
+    dealias = cfg.dealias  # 3/2
+    stop_sim_time = cfg.tend
+    timestepper = get_class(cfg.timestepper)  # d3.RK443 #d3.RK222
+    Dt = cfg.Dt  #  1e-3
+    max_timestep = cfg.max_timestep  #  1e-2
+    output_dt = cfg.output_dt  #  1e-2 # 1e-1
+    log_iter = cfg.log_iter  # 10
+    dtype = get_class(cfg.dtype)  #  np.float64
+    max_writes = cfg.max_writes  #  None
+    logger = logging.getLogger(__name__)
+    output_dir = f"/Datasets/mhd_data/simulation_outputs_Re{Re}"
+    movie_dir = f"{output_dir}/movie"
+    use_cfl = cfg.use_cfl  # False
+    skip_exists = cfg.skip_exists  # False
+
+    ## ID Parameters
+    L = cfg.L  # 1
+    dim = 2
+    Nsamples = args.N if args.N is not None else cfg.N  # Use CLI arg or default to config
+    l_u = cfg.l_u  # 0.1
+    l_A = cfg.l_A  # 0.1
+    Nu = cfg.Nu  # None
+    sigma_u = cfg.sigma_u  # 0.1
+    sigma_A = cfg.sigma_A  # 5e-3
+
+    # Generate Random Initial Data
+    grf_u = GRF_Mattern(
+        dim,
+        Nx,
+        length=Lx,
+        nu=Nu,
+        l=l_u,
+        sigma=sigma_u,
+        boundary="periodic",
+        device=device,
+    )
+    grf_A = GRF_Mattern(
+        dim,
+        Nx,
+        length=Lx,
+        nu=Nu,
+        l=l_A,
+        sigma=sigma_A,
+        boundary="periodic",
+        device=device,
+    )
+
+    u0_pot = grf_u.sample(Nsamples).cpu().numpy().reshape(Nsamples, Nx, Ny)
+    A0 = grf_A.sample(Nsamples).cpu().numpy().reshape(Nsamples, Nx, Ny)
+    digits = int(math.log10(Nsamples)) + 1
+
+    # expected number of time steps
+    Nt = len(np.arange(0, stop_sim_time + Dt, output_dt))
+    indices = list(range(Nsamples))
+
+    if skip_exists:
+        completed_list = []
+        for j in range(Nsamples):
+            # print('hi')
+            sim_output_dir = os.path.join(output_dir, f"output-{j:0{digits}}")
+            sim_outputs = os.path.join(sim_output_dir, "*.h5")
+            # skip if the next output directory exists and if the output is complete
+            if os.path.exists(sim_output_dir):
+                completed = check_if_complete(sim_outputs, Nt=Nt)
+            else:
+                completed = False
+            completed_list.append(completed)
+        indices = [j for j, completed in enumerate(completed_list) if not completed]
+    print(indices)
+
+    def run_simulation(
+        i,
+        Lx=Lx,
+        Ly=Ly,
+        Nx=Nx,
+        Ny=Ny,
+        Re=Re,
+        ReM=ReM,
+        Schmidt=Schmidt,
+        rho0=rho0,
+        dealias=dealias,
+        stop_sim_time=stop_sim_time,
+        timestepper=timestepper,
+        Dt=Dt,
+        max_timestep=max_timestep,
+        output_dt=output_dt,
+        log_iter=log_iter,
+        dtype=dtype,
+        max_writes=max_writes,
+        logger=logger,
+        output_dir=output_dir,
+        use_cfl=use_cfl,
+        L=L,
+        dim=dim,
+        Nsamples=Nsamples,
+        l_u=l_u,
+        l_A=l_A,
+        Nu=Nu,
+        sigma_u=sigma_u,
+        sigma_A=sigma_A,
+        grf_u=grf_u,
+        grf_A=grf_A,
+        u0_pot=u0_pot,
+        A0=A0,
+        digits=digits,
+        Nt=Nt,
+    ):
+        sim_output_dir = os.path.join(output_dir, f"output-{i:0{digits}}")
+        sim_outputs = os.path.join(sim_output_dir, "*.h5")
+        print(
+            f"Running simulation {i:0{digits}} with outputs in {sim_output_dir}",
+            flush=True,
+        )
+        # Bases
+        coords = d3.CartesianCoordinates("x", "y")
+        dist = d3.Distributor(coords, dtype=dtype)
+        xbasis = d3.RealFourier(coords["x"], size=Nx, bounds=(0, Lx), dealias=dealias)
+        ybasis = d3.RealFourier(coords["y"], size=Ny, bounds=(0, Ly), dealias=dealias)
+
+        # Fields
+        p = dist.Field(name="p", bases=(xbasis, ybasis))
+        s = dist.Field(name="s", bases=(xbasis, ybasis))
+        u = dist.VectorField(coords, name="u", bases=(xbasis, ybasis))
+        B = dist.VectorField(coords, name="B", bases=(xbasis, ybasis))
+        A = dist.Field(name="A", bases=(xbasis, ybasis))
+        B2 = dist.Field(name="B2", bases=(xbasis, ybasis))
+        u_pot = dist.Field(name="u_pot", bases=(xbasis, ybasis))
+        Ax = dist.Field(name="Ax", bases=(xbasis, ybasis))
+        Ay = dist.Field(name="Ay", bases=(xbasis, ybasis))
+        Bx = dist.Field(name="Bx", bases=(xbasis, ybasis))
+        By = dist.Field(name="By", bases=(xbasis, ybasis))
+        u0 = dist.VectorField(coords, name="u0", bases=(xbasis, ybasis))
+        ux = dist.Field(name="ux", bases=(xbasis, ybasis))
+        uy = dist.Field(name="uy", bases=(xbasis, ybasis))
+        tau_p = dist.Field(name="tau_p")
+
+        # Substitutions
+        nu = 1 / Re
+        D = nu / Schmidt
+        eta = 1 / ReM
+        x, y = dist.local_grids(xbasis, ybasis)
+        X, Y = np.meshgrid(x, y, indexing="ij")
+        ex, ey = coords.unit_vector_fields(dist)
+        # ez = d3.CrossProduct(ex, ey)
+        curl2d_scalar = lambda x: -d3.skew(d3.grad(x))
+        curl2d_vector = lambda x: -d3.div(d3.skew(x))
+        B = curl2d_scalar(A)
+        B2 = d3.dot(B, B)
+        Bx = B @ ex
+        By = B @ ey
+        ux = u @ ex
+        uy = u @ ey
+
+        # Problem
+        problem = d3.IVP([u, p, A, tau_p, s], namespace=locals())
+        problem.add_equation(
+            "dt(u) + grad(p)/rho0 - nu*lap(u) = - 0.5*grad(B2)/rho0 - u@grad(u) + B@grad(B)/rho0"
+        )
+        problem.add_equation("dt(s) - D*lap(s) = - u@grad(s)")
+        problem.add_equation("dt(A) - eta*lap(A) = - u@grad(A)")
+        problem.add_equation("div(u) + tau_p = 0")
+        problem.add_equation("integ(p) = 0")  # Pressure gauge
+
+        # Solver
+        solver = problem.build_solver(timestepper)
+        # solver.stop_sim_time = stop_sim_time
+        solver.stop_sim_time = (
+            stop_sim_time + Dt
+        )  # Make sure we record the last timestep
+
+        # Initial conditions
+        u_pot["g"] = u0_pot[i]
+        u0 = curl2d_scalar(u_pot).evaluate()
+        u0.change_scales(1)
+        u["g"] = u0["g"]
+        ux = u @ ex
+        uy = u @ ey
+        B2 = d3.dot(B, B)
+        # s.set_global_data(u0_pot[i])
+        s["g"] = u0_pot[i]
+        # A.set_global_data(A0[i])
+        A["g"] = A0[i]
+
+        # Analysis (This overwrites existing files)
+        os.makedirs(sim_output_dir, exist_ok=True)
+        snapshots = solver.evaluator.add_file_handler(
+            sim_output_dir, sim_dt=output_dt, max_writes=max_writes
+        )
+
+        snapshots.add_task(s, name="tracer")
+        snapshots.add_task(A, name="vector potential")
+        snapshots.add_task(B, name="magnetic field")
+
+        snapshots.add_task(u, name="velocity")
+        snapshots.add_task(p, name="pressure")
+
+        # CFL (Don't actually use this.  Use constant timestep instead)
+        CFL = d3.CFL(
+            solver,
+            initial_dt=max_timestep,
+            cadence=10,
+            safety=0.2,
+            threshold=0.1,
+            max_change=1.5,
+            min_change=0.5,
+            max_dt=max_timestep,
+        )
+        CFL.add_velocity(u)
+
+        # Flow properties
+        flow = d3.GlobalFlowProperty(solver, cadence=10)
+        flow.add_property(d3.dot(u, u), name="w2")
+        flow.add_property(d3.dot(B, B), name="B2")
+        flow.add_property(d3.div(B), name="divB")
+
+        # Main loop
+        try:
+            logger.info("Starting main loop")
+            while solver.proceed:
+                if use_cfl:
+                    timestep = CFL.compute_timestep()
+                else:
+                    timestep = Dt
+                solver.step(timestep)
+                if (solver.iteration) % 10 == 0:
+                    max_w = np.sqrt(flow.max("w2"))
+                    max_B = np.sqrt(flow.max("B2"))
+                    max_divB = flow.max("divB")
+                    logger.info(
+                        f"Iteration={solver.iteration}, Time={solver.sim_time:#.3g}, dt={timestep:#.3g}, max(w)={max_w:#.3g}, max(B)={max_B:#.3g}, max(div_B)={max_divB:#.3g}"
+                    )
+            print(
+                f"Finished simulation {i:0{digits}} with outputs in {sim_output_dir}",
+                flush=True,
+            )
+        except:
+            logger.error("Exception raised, triggering end of main loop.")
+            raise
+        solver.log_stats()
+
+    # Run in parallel
+    with mp.Pool(mp.cpu_count() - 1) as pool:
+        pool.map(run_simulation, indices, chunksize=10)

mhd/generate_mhd_data/mhd_field.yaml

@@ -0,0 +1,39 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+Lx: 1.0                                                       # Length of domain in x direction
+Ly: 1.0                                                       # Length of domain in y direction
+Nx: 128                                                       # Number of points in x direction
+Ny: 128                                                       # Number of points in y direction
+Schmidt: 1.0                                                  # Schmit number
+rho0: 1.0                                                     # Density of fluid
+dealias: 1.5                                                  # Dealiasing factor
+tend: 1.0                                                     # End time of simulation
+Dt: 1.0e-3                                                    # Timestep size
+timestepper: dedalus.public.RK443                             # Timestepper type
+max_timestep: 1.0e-2                                          # Maximum timestep for CFL control
+output_dt: 1.0e-2                                             # Time between outputs
+log_iter: 10                                                  # Iterations between logging
+dtype: numpy.float64                                          # Datatype for simulation
+max_writes: null                                              # Maximum file writes
+L: 1.0                                                        # Length of domain for generating data
+l_u: 0.1                                                      # Length of typical spatial deviations for velocity potential
+l_A: 0.1                                                      # Length of typical spatial deviations for magnetic vector potential
+sigma_u: 0.1                                                  # Typical amplitude of velocity potential
+sigma_A: 0.5e-3                                               # Typical amplitude of magnetic vector potential
+Nu: null                                                      # Smoothness parameter for GRF
+use_cfl: false                                                # Whether to use timestep computed based on CFL
+skip_exists: true                                             # Skip existing output files

mhd/generate_mhd_data/my_random_fields.py

@@ -0,0 +1,229 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import torch
+import torch.nn.functional as F
+import math
+from math import pi, gamma, sqrt
+import numpy as np
+
+torch.manual_seed(0)
+
+
+class GRF_Mattern(object):
+    """Generate Random Fields"""
+
+    def __init__(
+        self,
+        dim,
+        size,
+        length=1.0,
+        nu=None,
+        l=0.1,
+        sigma=1.0,
+        boundary="periodic",
+        constant_eig=None,
+        device=None,
+    ):
+
+        self.dim = dim
+        self.device = device
+        self.bc = boundary
+
+        a = sqrt(2 / length)
+        if self.bc == "dirichlet":
+            constant_eig = None
+
+        if nu is not None:
+            kappa = sqrt(2 * nu) / l
+            alpha = nu + 0.5 * dim
+            self.eta2 = (
+                size**dim
+                * sigma
+                * (4.0 * pi) ** (0.5 * dim)
+                * gamma(alpha)
+                / (kappa**dim * gamma(nu))
+            )
+        else:
+            self.eta2 = size**dim * sigma * (sqrt(2.0 * pi) * l) ** dim
+
+        k_max = size // 2
+        if self.bc == "periodic":
+            const = (4.0 * (pi**2)) / (length**2)
+        else:
+            const = (pi**2) / (length**2)
+
+        if dim == 1:
+            k = torch.cat(
+                (
+                    torch.arange(start=0, end=k_max, step=1, device=device),
+                    torch.arange(start=-k_max, end=0, step=1, device=device),
+                ),
+                0,
+            )
+
+            k2 = k**2
+            if nu is not None:
+                eigs = 1.0 + (const / (kappa * length) ** 2 * k2)
+                self.sqrt_eig = self.eta2 / (length**dim) * eigs ** (-alpha / 2.0)
+            else:
+                self.sqrt_eig = (
+                    self.eta2
+                    / (length**dim)
+                    * torch.exp(-((l) ** 2) * const * k2 / 4.0)
+                )
+
+            if constant_eig is not None:
+                self.sqrt_eig[0] = constant_eig  # (size**dim)*sigma*(tau**(-alpha))
+            else:
+                self.sqrt_eig[0] = 0.0
+
+        elif dim == 2:
+            wavenumers = torch.cat(
+                (
+                    torch.arange(start=0, end=k_max, step=1, device=device),
+                    torch.arange(start=-k_max, end=0, step=1, device=device),
+                ),
+                0,
+            ).repeat(size, 1)
+
+            k_x = wavenumers.transpose(0, 1)
+            k_y = wavenumers
+
+            k2 = k_x**2 + k_y**2
+            if nu is not None:
+                eigs = 1.0 + (const / (kappa * length) ** 2 * k2)
+                self.sqrt_eig = self.eta2 / (length**dim) * eigs ** (-alpha / 2.0)
+            else:
+                self.sqrt_eig = (
+                    self.eta2
+                    / (length**dim)
+                    * torch.exp(-((l) ** 2) * const * k2 / 4.0)
+                )
+
+            if constant_eig is not None:
+                self.sqrt_eig[0, 0] = constant_eig  # (size**dim)*sigma*(tau**(-alpha))
+            else:
+                self.sqrt_eig[0, 0] = 0.0
+
+        elif dim == 3:
+            wavenumers = torch.cat(
+                (
+                    torch.arange(start=0, end=k_max, step=1, device=device),
+                    torch.arange(start=-k_max, end=0, step=1, device=device),
+                ),
+                0,
+            ).repeat(size, size, 1)
+
+            k_x = wavenumers.transpose(1, 2)
+            k_y = wavenumers
+            k_z = wavenumers.transpose(0, 2)
+
+            k2 = k_x**2 + k_y**2 + k_z**2
+            if nu is not None:
+                eigs = 1.0 + (const / (kappa * length) ** 2 * k2)
+                self.sqrt_eig = self.eta2 / (length**dim) * eigs ** (-alpha / 2.0)
+            else:
+                self.sqrt_eig = (
+                    self.eta2
+                    / (length**dim)
+                    * torch.exp(-((l) ** 2) * const * k2 / 4.0)
+                )
+
+            if constant_eig is not None:
+                self.sqrt_eig[
+                    0, 0, 0
+                ] = constant_eig  # (size**dim)*sigma*(tau**(-alpha))
+            else:
+                self.sqrt_eig[0, 0, 0] = 0.0
+
+        self.size = []
+        for j in range(self.dim):
+            self.size.append(size)
+
+        self.size = tuple(self.size)
+
+    def sample(self, N):
+
+        coeff = torch.randn(N, *self.size, dtype=torch.cfloat, device=self.device)
+        if self.bc == "dirichlet":
+            coeff.real[:] = 0
+        if self.bc == "neumann":
+            coeff.imag[:] = 0
+        coeff = self.sqrt_eig * coeff
+
+        u = torch.fft.irfftn(coeff, self.size, norm="backward")
+        return u
+
+
+if __name__ == "__main__":
+    from hydra import compose, initialize
+    import h5py
+    import os
+    import matplotlib.pyplot as plt
+
+    initialize(version_base=None, config_path=".", job_name="generate_random_field")
+    cfg = compose(config_name="example_field")
+
+    N = cfg.num_samples
+    n = cfg.num_points
+    dim = cfg.dim
+    L = cfg.length
+    device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
+    grf = GRF_Mattern(
+        dim=cfg.dim,
+        size=cfg.num_points,
+        length=cfg.length,
+        nu=cfg.nu,
+        l=cfg.length_scale,
+        sigma=cfg.sigma,
+        boundary=cfg.boundary_condition,
+        constant_eig=cfg.mean,
+        device=device,
+    )
+    U = grf.sample(N)
+    # convert to pad periodically
+    pad_width = [(0, 0)] + [(0, 1) for _ in range(dim)]
+
+    u = np.pad(U.cpu().numpy(), pad_width, mode="wrap")
+    x = np.linspace(0, L, n + 1)
+    digits = int(math.log10(N)) + 1
+    basefile = cfg.file
+    if basefile:
+        filedir, file = os.path.split(basefile)
+        if filedir:
+            os.makedirs(filedir, exist_ok=True)
+
+        for i, u0 in enumerate(u):
+            filename = f"{basefile}-{i:0{digits}d}.h5"
+            with h5py.File(filename, "w") as hf:
+                hf.create_dataset("u", data=u0)
+                for j in range(dim):
+                    coord_name = f"x{j+1}"
+                    hf.create_dataset(coord_name, data=x)
+
+    if cfg.plot:
+        # coords = [x for _ in dim]
+        # X = np.meshgrid(*coords, indexing='ij')
+        if dim == 2:
+            X, Y = np.meshgrid(x, x, indexing="ij")
+            plt.close("all")
+            fig = plt.figure()
+            pmesh = plt.pcolormesh(X, Y, u[0], cmap="jet", shading="gouraud")
+            plt.colorbar(pmesh)
+            plt.axis("square")
+            plt.title("Random Initial Data")
+            plt.show()

mhd/losses/__init__.py

@@ -0,0 +1,18 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from .losses import LpLoss
+from .loss_mhd_vec_pot_physicsnemo import LossMHDVecPot_PhysicsNeMo

mhd/losses/loss_mhd_vec_pot_physicsnemo.py

@@ -0,0 +1,441 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import torch
+import torch.nn.functional as F
+from physicsnemo.models.layers.spectral_layers import fourier_derivatives
+
+from .losses import (LpLoss, fourier_derivatives_lap, fourier_derivatives_ptot,
+                     fourier_derivatives_vec_pot)
+from .mhd_pde import MHD_PDE
+
+
+class LossMHDVecPot_PhysicsNeMo(object):
+    "Calculate loss for MHD equations with vector potential, using physicsnemo derivatives"
+
+    def __init__(
+        self,
+        nu=1e-4,
+        eta=1e-4,
+        rho0=1.0,
+        data_weight=1.0,
+        ic_weight=1.0,
+        pde_weight=1.0,
+        constraint_weight=1.0,
+        use_data_loss=True,
+        use_ic_loss=True,
+        use_pde_loss=True,
+        use_constraint_loss=True,
+        u_weight=1.0,
+        v_weight=1.0,
+        A_weight=1.0,
+        Du_weight=1.0,
+        Dv_weight=1.0,
+        DA_weight=1.0,
+        div_B_weight=1.0,
+        div_vel_weight=1.0,
+        Lx=1.0,
+        Ly=1.0,
+        tend=1.0,
+        use_weighted_mean=False,
+        **kwargs,
+    ):  # add **kwargs so that we ignore unexpected kwargs when passing a config dict):
+
+        self.nu = nu
+        self.eta = eta
+        self.rho0 = rho0
+        self.data_weight = data_weight
+        self.ic_weight = ic_weight
+        self.pde_weight = pde_weight
+        self.constraint_weight = constraint_weight
+        self.use_data_loss = use_data_loss
+        self.use_ic_loss = use_ic_loss
+        self.use_pde_loss = use_pde_loss
+        self.use_constraint_loss = use_constraint_loss
+        self.u_weight = u_weight
+        self.v_weight = v_weight
+        self.Du_weight = Du_weight
+        self.Dv_weight = Dv_weight
+        self.div_B_weight = div_B_weight
+        self.div_vel_weight = div_vel_weight
+        self.Lx = Lx
+        self.Ly = Ly
+        self.tend = tend
+        self.use_weighted_mean = use_weighted_mean
+        self.A_weight = A_weight
+        self.DA_weight = DA_weight
+        # Define 2D MHD PDEs
+        self.mhd_pde_eq = MHD_PDE(self.nu, self.eta, self.rho0)
+        self.mhd_pde_node = self.mhd_pde_eq.make_nodes()
+
+        if not self.use_data_loss:
+            self.data_weight = 0
+        if not self.use_ic_loss:
+            self.ic_weight = 0
+        if not self.use_pde_loss:
+            self.pde_weight = 0
+        if not self.use_constraint_loss:
+            self.constraint_weight = 0
+
+    def __call__(self, pred, true, inputs, return_loss_dict=False):
+        loss, loss_dict = self.compute_losses(pred, true, inputs)
+        return loss, loss_dict
+
+    def compute_loss(self, pred, true, inputs):
+        "Compute weighted loss"
+        pred = pred.reshape(true.shape)
+        u = pred[..., 0]
+        v = pred[..., 1]
+        A = pred[..., 2]
+
+        # Data
+        if self.use_data_loss:
+            loss_data = self.data_loss(pred, true)
+        else:
+            loss_data = 0
+        # IC
+        if self.use_ic_loss:
+            loss_ic = self.ic_loss(pred, inputs)
+        else:
+            loss_ic = 0
+
+        # PDE
+        if self.use_pde_loss:
+            Du, Dv, DA = self.mhd_pde(u, v, A)
+            loss_pde = self.mhd_pde_loss(Du, Dv, DA)
+        else:
+            loss_pde = 0
+
+        # Constraints
+        if self.use_constraint_loss:
+            div_vel, div_B = self.mhd_constraint(u, v, A)
+            loss_constraint = self.mhd_constraint_loss(div_vel, div_B)
+        else:
+            loss_constraint = 0
+
+        if self.use_weighted_mean:
+            weight_sum = (
+                self.data_weight
+                + self.ic_weight
+                + self.pde_weight
+                + self.constraint_weight
+            )
+        else:
+            weight_sum = 1.0
+
+        loss = (
+            self.data_weight * loss_data
+            + self.ic_weight * loss_ic
+            + self.pde_weight * loss_pde
+            + self.constraint_weight * loss_constraint
+        ) / weight_sum
+        return loss
+
+    def compute_losses(self, pred, true, inputs):
+        "Compute weighted loss and dictionary"
+        pred = pred.reshape(true.shape)
+        u = pred[..., 0]
+        v = pred[..., 1]
+        A = pred[..., 2]
+
+        loss_dict = {}
+
+        # Data
+        if self.use_data_loss:
+            loss_data, loss_u, loss_v, loss_A = self.data_loss(
+                pred, true, return_all_losses=True
+            )
+            loss_dict["loss_data"] = loss_data
+            loss_dict["loss_u"] = loss_u
+            loss_dict["loss_v"] = loss_v
+            loss_dict["loss_A"] = loss_A
+        else:
+            loss_data = 0
+        # IC
+        if self.use_ic_loss:
+            loss_ic, loss_u_ic, loss_v_ic, loss_A_ic = self.ic_loss(
+                pred, inputs, return_all_losses=True
+            )
+            loss_dict["loss_ic"] = loss_ic
+            loss_dict["loss_u_ic"] = loss_u_ic
+            loss_dict["loss_v_ic"] = loss_v_ic
+            loss_dict["loss_A_ic"] = loss_A_ic
+        else:
+            loss_ic = 0
+
+        # PDE
+        if self.use_pde_loss:
+            Du, Dv, DA = self.mhd_pde(u, v, A)
+            loss_pde, loss_Du, loss_Dv, loss_DA = self.mhd_pde_loss(
+                Du, Dv, DA, return_all_losses=True
+            )
+            loss_dict["loss_pde"] = loss_pde
+            loss_dict["loss_Du"] = loss_Du
+            loss_dict["loss_Dv"] = loss_Dv
+            loss_dict["loss_DA"] = loss_DA
+        else:
+            loss_pde = 0
+
+        # Constraints
+        if self.use_constraint_loss:
+            div_vel, div_B = self.mhd_constraint(u, v, A)
+            loss_constraint, loss_div_vel, loss_div_B = self.mhd_constraint_loss(
+                div_vel, div_B, return_all_losses=True
+            )
+            loss_dict["loss_constraint"] = loss_constraint
+            loss_dict["loss_div_vel"] = loss_div_vel
+            loss_dict["loss_div_B"] = loss_div_B
+        else:
+            loss_constraint = 0
+
+        if self.use_weighted_mean:
+            weight_sum = (
+                self.data_weight
+                + self.ic_weight
+                + self.pde_weight
+                + self.constraint_weight
+            )
+        else:
+            weight_sum = 1.0
+
+        loss = (
+            self.data_weight * loss_data
+            + self.ic_weight * loss_ic
+            + self.pde_weight * loss_pde
+            + self.constraint_weight * loss_constraint
+        ) / weight_sum
+        loss_dict["loss"] = loss
+        return loss, loss_dict
+
+    def data_loss(self, pred, true, return_all_losses=False):
+        "Compute data loss"
+        lploss = LpLoss(size_average=True)
+        u_pred = pred[..., 0]
+        v_pred = pred[..., 1]
+        A_pred = pred[..., 2]
+
+        u_true = true[..., 0]
+        v_true = true[..., 1]
+        A_true = true[..., 2]
+
+        loss_u = lploss(u_pred, u_true)
+        loss_v = lploss(v_pred, v_true)
+        loss_A = lploss(A_pred, A_true)
+
+        if self.use_weighted_mean:
+            weight_sum = self.u_weight + self.v_weight + self.A_weight
+        else:
+            weight_sum = 1.0
+
+        loss_data = (
+            self.u_weight * loss_u + self.v_weight * loss_v + self.A_weight * loss_A
+        ) / weight_sum
+
+        if return_all_losses:
+            return loss_data, loss_u, loss_v, loss_A
+        else:
+            return loss_data
+
+    def ic_loss(self, pred, input, return_all_losses=False):
+        "Compute initial condition loss"
+        lploss = LpLoss(size_average=True)
+        ic_pred = pred[:, 0]
+        ic_true = input[:, 0, ..., 3:]
+        u_ic_pred = ic_pred[..., 0]
+        v_ic_pred = ic_pred[..., 1]
+        A_ic_pred = ic_pred[..., 2]
+
+        u_ic_true = ic_true[..., 0]
+        v_ic_true = ic_true[..., 1]
+        A_ic_true = ic_true[..., 2]
+
+        loss_u_ic = lploss(u_ic_pred, u_ic_true)
+        loss_v_ic = lploss(v_ic_pred, v_ic_true)
+        loss_A_ic = lploss(A_ic_pred, A_ic_true)
+
+        if self.use_weighted_mean:
+            weight_sum = self.u_weight + self.v_weight + self.A_weight
+        else:
+            weight_sum = 1.0
+
+        loss_ic = (
+            self.u_weight * loss_u_ic
+            + self.v_weight * loss_v_ic
+            + self.A_weight * loss_A_ic
+        ) / weight_sum
+
+        if return_all_losses:
+            return loss_ic, loss_u_ic, loss_v_ic, loss_A_ic
+        else:
+            return loss_ic
+
+    def mhd_pde_loss(self, Du, Dv, DA, return_all_losses=None):
+        "Compute PDE loss"
+        Du_val = torch.zeros_like(Du)
+        Dv_val = torch.zeros_like(Dv)
+        DA_val = torch.zeros_like(DA)
+
+        loss_Du = F.mse_loss(Du, Du_val)
+        loss_Dv = F.mse_loss(Dv, Dv_val)
+        loss_DA = F.mse_loss(DA, DA_val)
+
+        if self.use_weighted_mean:
+            weight_sum = self.Du_weight + self.Dv_weight + self.DA_weight
+        else:
+            weight_sum = 1.0
+
+        loss_pde = (
+            self.Du_weight * loss_Du
+            + self.Dv_weight * loss_Dv
+            + self.DA_weight * loss_DA
+        ) / weight_sum
+
+        if return_all_losses:
+            return loss_pde, loss_Du, loss_Dv, loss_DA
+        else:
+            return loss_pde
+
+    def mhd_constraint(self, u, v, A):
+        "Compute constraints"
+        nt = u.size(1)
+        nx = u.size(2)
+        ny = u.size(3)
+
+        f_du, _ = fourier_derivatives(u, [self.Lx, self.Ly])
+        f_dv, _ = fourier_derivatives(v, [self.Lx, self.Ly])
+        f_dBx, f_dBy, _, _, _ = fourier_derivatives_vec_pot(A, [self.Lx, self.Ly])
+
+        u_x = f_du[:, 0:nt, :nx, :ny]
+        v_y = f_dv[:, nt : 2 * nt, :nx, :ny]
+        Bx_x = f_dBx[:, 0:nt, :nx, :ny]
+        By_y = f_dBy[:, nt : 2 * nt, :nx, :ny]
+
+        div_B = self.mhd_pde_node[12].evaluate({"Bx__x": Bx_x, "By__y": By_y})["div_B"]
+        div_vel = self.mhd_pde_node[13].evaluate({"u__x": u_x, "v__y": v_y})["div_vel"]
+
+        return div_vel, div_B
+    
+    def mhd_constraint_loss(self, div_vel, div_B, return_all_losses=False):
+        "Compute constraint loss"
+        div_vel_val = torch.zeros_like(div_vel)
+        div_B_val = torch.zeros_like(div_B)
+
+        loss_div_vel = F.mse_loss(div_vel, div_vel_val)
+        loss_div_B = F.mse_loss(div_B, div_B_val)
+
+        if self.use_weighted_mean:
+            weight_sum = self.div_vel_weight + self.div_B_weight
+        else:
+            weight_sum = 1.0
+
+        loss_constraint = (
+            self.div_vel_weight * loss_div_vel + self.div_B_weight * loss_div_B
+        ) / weight_sum
+
+        if return_all_losses:
+            return loss_constraint, loss_div_vel, loss_div_B
+        else:
+            return loss_constraint
+
+    def mhd_pde(self, u, v, A, p=None):
+        "Compute PDEs for MHD using vector potential"
+        nt = u.size(1)
+        nx = u.size(2)
+        ny = u.size(3)
+        dt = self.tend / (nt - 1)
+
+        # compute fourier derivatives
+        f_du, _ = fourier_derivatives(u, [self.Lx, self.Ly])
+        f_dv, _ = fourier_derivatives(v, [self.Lx, self.Ly])
+        f_dBx, f_dBy, f_dA, f_dB, B2_h = fourier_derivatives_vec_pot(
+            A, [self.Lx, self.Ly]
+        )
+
+        u_x = f_du[:, 0:nt, :nx, :ny]
+        u_y = f_du[:, nt : 2 * nt, :nx, :ny]
+        v_x = f_dv[:, 0:nt, :nx, :ny]
+        v_y = f_dv[:, nt : 2 * nt, :nx, :ny]
+        A_x = f_dA[:, 0:nt, :nx, :ny]
+        A_y = f_dA[:, nt : 2 * nt, :nx, :ny]
+
+        Bx = f_dB[:, 0:nt, :nx, :ny]
+        By = f_dB[:, nt : 2 * nt, :nx, :ny]
+        Bx_x = f_dBx[:, 0:nt, :nx, :ny]
+        Bx_y = f_dBx[:, nt : 2 * nt, :nx, :ny]
+        By_x = f_dBy[:, 0:nt, :nx, :ny]
+        By_y = f_dBy[:, nt : 2 * nt, :nx, :ny]
+
+        u_lap = fourier_derivatives_lap(u, [self.Lx, self.Ly])
+        v_lap = fourier_derivatives_lap(v, [self.Lx, self.Ly])
+        A_lap = fourier_derivatives_lap(A, [self.Lx, self.Ly])
+
+        # note that for pressure, the zero mode (the mean) cannot be zero for invertability so it is set to 1
+        div_vel_grad_vel = u_x**2 + 2 * u_y * v_x + v_y**2
+        div_B_grad_B = Bx_x**2 + 2 * Bx_y * By_x + By_y**2
+        f_dptot = fourier_derivatives_ptot(
+            p, div_vel_grad_vel, div_B_grad_B, B2_h, self.rho0, [self.Lx, self.Ly]
+        )
+        ptot_x = f_dptot[:, 0:nt, :nx, :ny]
+        ptot_y = f_dptot[:, nt : 2 * nt, :nx, :ny]
+
+        # Plug inputs into dictionary
+        all_inputs = {
+            "u": u,
+            "u__x": u_x,
+            "u__y": u_y,
+            "v": v,
+            "v__x": v_x,
+            "v__y": v_y,
+            "Bx": Bx,
+            "Bx__x": Bx_x,
+            "Bx__y": Bx_y,
+            "By": By,
+            "By__x": By_x,
+            "By__y": By_y,
+            "A__x": A_x,
+            "A__y": A_y,
+            "ptot__x": ptot_x,
+            "ptot__y": ptot_y,
+            "u__lap": u_lap,
+            "v__lap": v_lap,
+            "A__lap": A_lap,
+        }
+
+        # Substitute values into PDE equations
+        u_rhs = self.mhd_pde_node[14].evaluate(all_inputs)["u_rhs"]
+        v_rhs = self.mhd_pde_node[15].evaluate(all_inputs)["v_rhs"]
+        A_rhs = self.mhd_pde_node[23].evaluate(all_inputs)["A_rhs"]
+
+        u_t = self.Du_t(u, dt)
+        v_t = self.Du_t(v, dt)
+        A_t = self.Du_t(A, dt)
+
+        # Find difference
+        Du = self.mhd_pde_node[18].evaluate({"u__t": u_t, "u_rhs": u_rhs[:, 1:-1]})[
+            "Du"
+        ]
+        Dv = self.mhd_pde_node[19].evaluate({"v__t": v_t, "v_rhs": v_rhs[:, 1:-1]})[
+            "Dv"
+        ]
+        DA = self.mhd_pde_node[24].evaluate({"A__t": A_t, "A_rhs": A_rhs[:, 1:-1]})[
+            "DA"
+        ]
+        return Du, Dv, DA
+
+    def Du_t(self, u, dt):
+        "Compute time derivative"
+        u_t = (u[:, 2:] - u[:, :-2]) / (2 * dt)
+        return u_t

mhd/losses/losses.py

@@ -0,0 +1,267 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import numpy as np
+import torch
+import torch.nn.functional as F
+import math
+from torch import Tensor
+from typing import List
+
+
+class LpLoss(object):
+    """
+    loss function with rel/abs Lp loss
+    """
+
+    def __init__(self, d=2, p=2, size_average=True, reduction=True):
+        super(LpLoss, self).__init__()
+
+        # Dimension and Lp-norm type are postive
+        assert d > 0 and p > 0
+
+        self.d = d
+        self.p = p
+        self.reduction = reduction
+        self.size_average = size_average
+
+    def abs(self, x, y):
+        num_examples = x.size()[0]
+
+        # Assume uniform mesh
+        h = 1.0 / (x.size()[1] - 1.0)
+
+        all_norms = (h ** (self.d / self.p)) * torch.norm(
+            x.view(num_examples, -1) - y.view(num_examples, -1), self.p, 1
+        )
+
+        if self.reduction:
+            if self.size_average:
+                return torch.mean(all_norms)
+            else:
+                return torch.sum(all_norms)
+
+        return all_norms
+
+    def rel(self, x, y):
+        num_examples = x.size()[0]
+
+        diff_norms = torch.norm(
+            x.reshape(num_examples, -1) - y.reshape(num_examples, -1), self.p, 1
+        )
+        y_norms = torch.norm(y.reshape(num_examples, -1), self.p, 1)
+
+        if self.reduction:
+            if self.size_average:
+                return torch.mean(diff_norms / y_norms)
+            else:
+                return torch.sum(diff_norms / y_norms)
+
+        return diff_norms / y_norms
+
+    def __call__(self, x, y):
+        return self.rel(x, y)
+
+
+def fourier_derivatives_lap(x: Tensor, ell: List[float]) -> Tensor:
+    """
+    Fourier derivative laplacian function
+    """
+
+    # check that input shape maches domain length
+    if len(x.shape) - 2 != len(ell):
+        raise ValueError("input shape doesn't match domain dims")
+
+    # set pi from numpy
+    pi = float(np.pi)
+
+    # get needed dims
+    n = x.shape[2:]
+    dim = len(ell)
+
+    # get device
+    device = x.device
+
+    # compute fourier transform
+    x_h = torch.fft.fftn(x, dim=list(range(2, dim + 2)))
+
+    # make wavenumbers
+    k_x = []
+    for i, nx in enumerate(n):
+        k_x.append(
+            (2 * pi / ell[i])
+            * torch.cat(
+                (
+                    torch.arange(start=0, end=nx // 2, step=1, device=device),
+                    torch.arange(start=-nx // 2, end=0, step=1, device=device),
+                ),
+                0,
+            ).reshape((i + 2) * [1] + [nx] + (dim - i - 1) * [1])
+        )
+    lap = torch.zeros_like(k_x[0])
+    for i in k_x:
+        lap = lap - i**2
+
+    # compute laplacian in fourier space
+    wx_h = lap * x_h
+
+    # inverse fourier transform out
+    wx = torch.fft.ifftn(wx_h, dim=list(range(2, dim + 2))).real
+    return wx
+
+
+def fourier_derivatives_ptot(
+    p: Tensor,
+    div_vel_grad_vel: Tensor,
+    div_B_grad_B: Tensor,
+    B2_h: Tensor,
+    rho0: float,
+    ell: List[float],
+) -> List[Tensor]:
+    """
+    Fourier derivative function to calculate ptot in MHD equations
+    """
+
+    # check that input shape maches domain length
+    if len(div_vel_grad_vel.shape) - 2 != len(ell):
+        raise ValueError("input shape doesn't match domain dims")
+
+    # set pi from numpy
+    pi = float(np.pi)
+
+    # get needed dims
+    n = div_vel_grad_vel.shape[2:]
+    dim = len(ell)
+
+    # get device
+    device = div_vel_grad_vel.device
+
+    # make wavenumbers
+    k_x = []
+    for i, nx in enumerate(n):
+        k_x.append(
+            torch.cat(
+                (
+                    torch.arange(start=0, end=nx // 2, step=1, device=device),
+                    torch.arange(start=-nx // 2, end=0, step=1, device=device),
+                ),
+                0,
+            ).reshape((i + 2) * [1] + [nx] + (dim - i - 1) * [1])
+        )
+    # note that for pressure, the zero mode (the mean) cannot be zero for invertability so it is set to 1
+    lap = torch.zeros_like(k_x[0])
+    for i, k_x_i in enumerate(k_x):
+        lap = lap - ((2 * pi / ell[i]) * k_x_i) ** 2
+    lap[..., 0, 0] = -1.0
+
+    if p is None:
+        # compute fourier transform
+        div_vel_grad_vel_h = torch.fft.fftn(
+            div_vel_grad_vel, dim=list(range(2, dim + 2))
+        )
+        div_B_grad_B_h = torch.fft.fftn(div_B_grad_B, dim=list(range(2, dim + 2)))
+        ptot_h = (div_B_grad_B_h - rho0 * div_vel_grad_vel_h) / lap
+        ptot_h[..., 0, 0] = B2_h[..., 0, 0] / 2.0
+    else:
+        p_h = torch.fft.fftn(p, dim=list(range(2, dim + 2)))
+        ptot_h = p_h + B2_h / 2.0
+
+    # compute laplacian in fourier space
+    j = torch.complex(
+        torch.tensor([0.0], device=device), torch.tensor([1.0], device=device)
+    )  # Cuda graphs does not work here
+    wx_h = [j * k_x_i * ptot_h * (2 * pi / ell[i]) for i, k_x_i in enumerate(k_x)]
+
+    # inverse fourier transform out
+    wx = torch.cat(
+        [torch.fft.ifftn(wx_h_i, dim=list(range(2, dim + 2))).real for wx_h_i in wx_h],
+        dim=1,
+    )
+    return wx
+
+
+def fourier_derivatives_vec_pot(x: Tensor, ell: List[float]) -> List[Tensor]:
+    """
+    Fourier derivative function for vector potential
+    """
+
+    # check that input shape maches domain length
+    if len(x.shape) - 2 != len(ell):
+        raise ValueError("input shape doesn't match domain dims")
+
+    # set pi from numpy
+    pi = float(np.pi)
+
+    # get needed dims
+    n = x.shape[2:]
+    dim = len(ell)
+
+    # get device
+    device = x.device
+
+    # compute fourier transform
+    x_h = torch.fft.fftn(x, dim=list(range(2, dim + 2)))
+
+    # make wavenumbers
+    k_x = []
+    for i, nx in enumerate(n):
+        k_x.append(
+            torch.cat(
+                (
+                    torch.arange(start=0, end=nx // 2, step=1, device=device),
+                    torch.arange(start=-nx // 2, end=0, step=1, device=device),
+                ),
+                0,
+            ).reshape((i + 2) * [1] + [nx] + (dim - i - 1) * [1])
+        )
+
+    # compute laplacian in fourier space
+    j = torch.complex(
+        torch.tensor([0.0], device=device), torch.tensor([1.0], device=device)
+    )  # Cuda graphs does not work here
+    Ax_h = j * k_x[0] * x_h * (2 * pi / ell[0])
+    Ay_h = j * k_x[1] * x_h * (2 * pi / ell[1])
+
+    B2_h = (Ay_h) ** 2 + (-Ax_h) ** 2
+
+    Bx_h = [j * k_x_i * Ay_h * (2 * pi / ell[i]) for i, k_x_i in enumerate(k_x)]
+    By_h = [j * k_x_i * -Ax_h * (2 * pi / ell[i]) for i, k_x_i in enumerate(k_x)]
+
+    # inverse fourier transform out
+    wA = torch.cat(
+        [
+            torch.fft.ifftn(w_h_i, dim=list(range(2, dim + 2))).real
+            for w_h_i in [Ax_h, Ay_h]
+        ],
+        dim=1,
+    )
+    wB = torch.cat(
+        [
+            torch.fft.ifftn(w_h_i, dim=list(range(2, dim + 2))).real
+            for w_h_i in [Ay_h, -Ax_h]
+        ],
+        dim=1,
+    )
+    wx = torch.cat(
+        [torch.fft.ifftn(wx_h_i, dim=list(range(2, dim + 2))).real for wx_h_i in Bx_h],
+        dim=1,
+    )
+    wy = torch.cat(
+        [torch.fft.ifftn(wx_h_i, dim=list(range(2, dim + 2))).real for wx_h_i in By_h],
+        dim=1,
+    )
+
+    return wx, wy, wA, wB, B2_h

mhd/losses/mhd_pde.py

@@ -0,0 +1,104 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from physicsnemo.sym.eq.pde import PDE
+from sympy import Function, Number, Symbol
+
+
+class MHD_PDE(PDE):
+    """MHD PDEs using PhysicsNeMo Sym"""
+
+    name = "MHD_PDE"
+
+    def __init__(self, nu=1e-4, eta=1e-4, rho0=1.0):
+
+        # x, y, time
+        x, y, t, lap = Symbol("x"), Symbol("y"), Symbol("t"), Symbol("lap")
+
+        # make input variables
+        input_variables = {"x": x, "y": y, "t": t, "lap": lap}
+
+        # make functions
+        u = Function("u")(*input_variables)
+        v = Function("v")(*input_variables)
+        Bx = Function("Bx")(*input_variables)
+        By = Function("By")(*input_variables)
+        A = Function("A")(*input_variables)
+        ptot = Function("ptot")(*input_variables)
+
+        u_rhs = Function("u_rhs")(*input_variables)
+        v_rhs = Function("v_rhs")(*input_variables)
+        Bx_rhs = Function("Bx_rhs")(*input_variables)
+        By_rhs = Function("By_rhs")(*input_variables)
+        A_rhs = Function("A_rhs")(*input_variables)
+
+        # initialize constants
+        nu = Number(nu)
+        eta = Number(eta)
+        rho0 = Number(rho0)
+
+        # set equations
+        self.equations = {}
+
+        self.equations["vel_grad_u"] = u * u.diff(x) + v * u.diff(y)
+        self.equations["vel_grad_v"] = u * v.diff(x) + v * v.diff(y)
+
+        self.equations["B_grad_u"] = Bx * u.diff(x) + v * Bx.diff(y)
+        self.equations["B_grad_v"] = Bx * v.diff(x) + By * v.diff(y)
+
+        self.equations["vel_grad_Bx"] = u * Bx.diff(x) + v * Bx.diff(y)
+        self.equations["vel_grad_By"] = u * By.diff(x) + v * By.diff(y)
+
+        self.equations["B_grad_Bx"] = Bx * Bx.diff(x) + By * Bx.diff(y)
+        self.equations["B_grad_By"] = Bx * By.diff(x) + By * By.diff(y)
+
+        self.equations["uBy_x"] = u * By.diff(x) + By * u.diff(x)
+        self.equations["uBy_y"] = u * By.diff(y) + By * u.diff(y)
+        self.equations["vBx_x"] = v * Bx.diff(x) + Bx * v.diff(x)
+        self.equations["vBx_y"] = v * Bx.diff(y) + Bx * v.diff(y)
+
+        self.equations["div_B"] = Bx.diff(x) + By.diff(y)
+        self.equations["div_vel"] = u.diff(x) + v.diff(y)
+
+        # RHS of MHD equations
+        self.equations["u_rhs"] = (
+            -self.equations["vel_grad_u"]
+            - ptot.diff(x) / rho0
+            + self.equations["B_grad_Bx"] / rho0
+            + nu * u.diff(lap)
+        )
+        self.equations["v_rhs"] = (
+            -self.equations["vel_grad_v"]
+            - ptot.diff(y) / rho0
+            + self.equations["B_grad_By"] / rho0
+            + nu * v.diff(lap)
+        )
+        self.equations["Bx_rhs"] = (
+            self.equations["uBy_y"] - self.equations["vBx_y"] + eta * Bx.diff(lap)
+        )
+        self.equations["By_rhs"] = -(
+            self.equations["uBy_x"] - self.equations["vBx_x"]
+        ) + eta * By.diff(lap)
+        # Node 18, 19, 20, 21
+        self.equations["Du"] = u.diff(t) - u_rhs
+        self.equations["Dv"] = v.diff(t) - v_rhs
+        self.equations["DBx"] = Bx.diff(t) - Bx_rhs
+        self.equations["DBy"] = By.diff(t) - By_rhs
+        # Node 22, 23, 24
+        # Vec potential equations
+        self.equations["vel_grad_A"] = u * A.diff(x) + v * A.diff(y)
+        self.equations["A_rhs"] = -self.equations["vel_grad_A"] + +eta * A.diff(lap)
+        self.equations["DA"] = A.diff(t) - A_rhs

mhd/tfno/__init__.py

@@ -0,0 +1,23 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from .tfno import TFNO, TFNO1DEncoder, TFNO2DEncoder, TFNO3DEncoder, TFNO4DEncoder
+from .spectral_layers import (
+    FactorizedSpectralConv1d,
+    FactorizedSpectralConv2d,
+    FactorizedSpectralConv3d,
+    FactorizedSpectralConv4d,
+)

mhd/tfno/spectral_layers.py

@@ -0,0 +1,657 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from typing import List, Tuple
+
+import numpy as np
+import torch
+import torch.nn as nn
+import torch.nn.functional as F
+from torch import Tensor
+import tltorch
+
+
+class FactorizedSpectralConv1d(nn.Module):
+    """1D Factorized Fourier layer. It does FFT, linear transform, and Inverse FFT.
+
+    Parameters
+    ----------
+    in_channels : int
+        Number of input channels
+    out_channels : int
+        Number of output channels
+    modes1 : int
+        Number of Fourier modes to multiply, at most floor(N/2) + 1
+    rank : float
+        Rank of the decomposition
+    factorization : {'CP', 'TT', 'Tucker'}
+        Tensor factorization to use to decompose the tensor
+    fixed_rank_modes : List[int]
+        A list of modes for which the initial value is not modified
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict
+        Additional arguments to initialization of factorized tensors
+    """
+
+    def __init__(
+        self,
+        in_channels: int,
+        out_channels: int,
+        modes1: int,
+        rank: float,
+        factorization: str,
+        fixed_rank_modes: bool,
+        decomposition_kwargs: dict,
+    ):
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.out_channels = out_channels
+        self.modes1 = (
+            modes1  # Number of Fourier modes to multiply, at most floor(N/2) + 1
+        )
+
+        self.scale = 1 / (in_channels * out_channels)
+        self.weights1 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.reset_parameters()
+
+    def compl_mul1d(
+        self,
+        input: Tensor,
+        weights: Tensor,
+    ) -> Tensor:
+        """Complex multiplication
+
+        Parameters
+        ----------
+        input : Tensor
+            Input tensor
+        weights : Tensor
+            Weights tensor
+
+        Returns
+        -------
+        Tensor
+            Product of complex multiplication
+        """
+        # (batch, in_channel, x ), (in_channel, out_channel, x) -> (batch, out_channel, x)
+        cweights = torch.view_as_complex(weights.to_tensor().contiguous())
+        return torch.einsum("bix,iox->box", input, cweights)
+
+    def forward(self, x: Tensor) -> Tensor:
+        bsize = x.shape[0]
+        # Compute Fourier coeffcients up to factor of e^(- something constant)
+        x_ft = torch.fft.rfft(x)
+
+        # Multiply relevant Fourier modes
+        out_ft = torch.zeros(
+            bsize,
+            self.out_channels,
+            x.size(-1) // 2 + 1,
+            device=x.device,
+            dtype=torch.cfloat,
+        )
+        out_ft[:, :, : self.modes1] = self.compl_mul1d(
+            x_ft[:, :, : self.modes1],
+            self.weights1,
+        )
+
+        # Return to physical space
+        x = torch.fft.irfft(out_ft, n=x.size(-1))
+        return x
+
+    def reset_parameters(self):
+        """Reset spectral weights with distribution scale*N(0,1)"""
+        self.weights1.normal_(0, self.scale)
+
+
+class FactorizedSpectralConv2d(nn.Module):
+    """2D Factorized Fourier layer. It does FFT, linear transform, and Inverse FFT.
+
+    Parameters
+    ----------
+    in_channels : int
+        Number of input channels
+    out_channels : int
+        Number of output channels
+    modes1 : int
+        Number of Fourier modes to multiply in first dimension, at most floor(N/2) + 1
+    modes2 : int
+        Number of Fourier modes to multiply in second dimension, at most floor(N/2) + 1
+    rank : float
+        Rank of the decomposition
+    factorization : {'CP', 'TT', 'Tucker'}
+        Tensor factorization to use to decompose the tensor
+    fixed_rank_modes : List[int]
+        A list of modes for which the initial value is not modified
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict
+        Additional arguments to initialization of factorized tensors
+    """
+
+    def __init__(
+        self,
+        in_channels: int,
+        out_channels: int,
+        modes1: int,
+        modes2: int,
+        rank: float,
+        factorization: str,
+        fixed_rank_modes: bool,
+        decomposition_kwargs: dict,
+    ):
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.out_channels = out_channels
+        self.modes1 = (
+            modes1  # Number of Fourier modes to multiply, at most floor(N/2) + 1
+        )
+        self.modes2 = modes2
+
+        self.scale = 1 / (in_channels * out_channels)
+        self.weights1 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights2 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.reset_parameters()
+
+    def compl_mul2d(self, input: Tensor, weights: Tensor) -> Tensor:
+        """Complex multiplication
+
+        Parameters
+        ----------
+        input : Tensor
+            Input tensor
+        weights : Tensor
+            Weights tensor
+
+        Returns
+        -------
+        Tensor
+            Product of complex multiplication
+        """
+        # (batch, in_channel, x, y), (in_channel, out_channel, x, y) -> (batch, out_channel, x, y)
+        cweights = torch.view_as_complex(weights.to_tensor().contiguous())
+        return torch.einsum("bixy,ioxy->boxy", input, cweights)
+
+    def forward(self, x: Tensor) -> Tensor:
+        batchsize = x.shape[0]
+        # Compute Fourier coeffcients up to factor of e^(- something constant)
+        x_ft = torch.fft.rfft2(x)
+
+        # Multiply relevant Fourier modes
+        out_ft = torch.zeros(
+            batchsize,
+            self.out_channels,
+            x.size(-2),
+            x.size(-1) // 2 + 1,
+            dtype=torch.cfloat,
+            device=x.device,
+        )
+        out_ft[:, :, : self.modes1, : self.modes2] = self.compl_mul2d(
+            x_ft[:, :, : self.modes1, : self.modes2],
+            self.weights1,
+        )
+        out_ft[:, :, -self.modes1 :, : self.modes2] = self.compl_mul2d(
+            x_ft[:, :, -self.modes1 :, : self.modes2],
+            self.weights2,
+        )
+
+        # Return to physical space
+        x = torch.fft.irfft2(out_ft, s=(x.size(-2), x.size(-1)))
+        return x
+
+    def reset_parameters(self):
+        """Reset spectral weights with distribution scale*N(0,1)"""
+        self.weights1.normal_(0, self.scale)
+        self.weights2.normal_(0, self.scale)
+
+
+class FactorizedSpectralConv3d(nn.Module):
+    """3D Factorized Fourier layer. It does FFT, linear transform, and Inverse FFT.
+
+    Parameters
+    ----------
+    in_channels : int
+        Number of input channels
+    out_channels : int
+        Number of output channels
+    modes1 : int
+        Number of Fourier modes to multiply in first dimension, at most floor(N/2) + 1
+    modes2 : int
+        Number of Fourier modes to multiply in second dimension, at most floor(N/2) + 1
+    modes3 : int
+        Number of Fourier modes to multiply in third dimension, at most floor(N/2) + 1
+    rank : float
+        Rank of the decomposition
+    factorization : {'CP', 'TT', 'Tucker'}
+        Tensor factorization to use to decompose the tensor
+    fixed_rank_modes : List[int]
+        A list of modes for which the initial value is not modified
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict
+        Additional arguments to initialization of factorized tensors
+    """
+
+    def __init__(
+        self,
+        in_channels: int,
+        out_channels: int,
+        modes1: int,
+        modes2: int,
+        modes3: int,
+        rank: float,
+        factorization: str,
+        fixed_rank_modes: bool,
+        decomposition_kwargs: dict,
+    ):
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.out_channels = out_channels
+        self.modes1 = (
+            modes1  # Number of Fourier modes to multiply, at most floor(N/2) + 1
+        )
+        self.modes2 = modes2
+        self.modes3 = modes3
+
+        self.scale = 1 / (in_channels * out_channels)
+        self.weights1 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, self.modes3, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights2 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, self.modes3, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights3 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, self.modes3, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights4 = tltorch.FactorizedTensor.new(
+            (in_channels, out_channels, self.modes1, self.modes2, self.modes3, 2),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.reset_parameters()
+
+    def compl_mul3d(self, input: Tensor, weights: Tensor) -> Tensor:
+        """Complex multiplication
+
+        Parameters
+        ----------
+        input : Tensor
+            Input tensor
+        weights : Tensor
+            Weights tensor
+
+        Returns
+        -------
+        Tensor
+            Product of complex multiplication
+        """
+        # (batch, in_channel, x, y, z), (in_channel, out_channel, x, y, z) -> (batch, out_channel, x, y, z)
+        cweights = torch.view_as_complex(weights.to_tensor().contiguous())
+        return torch.einsum("bixyz,ioxyz->boxyz", input, cweights)
+
+    def forward(self, x: Tensor) -> Tensor:
+        batchsize = x.shape[0]
+        # Compute Fourier coeffcients up to factor of e^(- something constant)
+        x_ft = torch.fft.rfftn(x, dim=[-3, -2, -1])
+
+        # Multiply relevant Fourier modes
+        out_ft = torch.zeros(
+            batchsize,
+            self.out_channels,
+            x.size(-3),
+            x.size(-2),
+            x.size(-1) // 2 + 1,
+            dtype=torch.cfloat,
+            device=x.device,
+        )
+
+        out_ft[:, :, : self.modes1, : self.modes2, : self.modes3] = self.compl_mul3d(
+            x_ft[:, :, : self.modes1, : self.modes2, : self.modes3], self.weights1
+        )
+        out_ft[:, :, -self.modes1 :, : self.modes2, : self.modes3] = self.compl_mul3d(
+            x_ft[:, :, -self.modes1 :, : self.modes2, : self.modes3], self.weights2
+        )
+        out_ft[:, :, : self.modes1, -self.modes2 :, : self.modes3] = self.compl_mul3d(
+            x_ft[:, :, : self.modes1, -self.modes2 :, : self.modes3], self.weights3
+        )
+        out_ft[:, :, -self.modes1 :, -self.modes2 :, : self.modes3] = self.compl_mul3d(
+            x_ft[:, :, -self.modes1 :, -self.modes2 :, : self.modes3], self.weights4
+        )
+
+        # Return to physical space
+        x = torch.fft.irfftn(out_ft, s=(x.size(-3), x.size(-2), x.size(-1)))
+        return x
+
+    def reset_parameters(self):
+        """Reset spectral weights with distribution scale*U(0,1)"""
+        self.weights1.normal_(0, self.scale)
+        self.weights2.normal_(0, self.scale)
+        self.weights3.normal_(0, self.scale)
+        self.weights4.normal_(0, self.scale)
+
+
+class FactorizedSpectralConv4d(nn.Module):
+    """4D Factorized Fourier layer. It does FFT, linear transform, and Inverse FFT.
+
+    Parameters
+    ----------
+    in_channels : int
+        Number of input channels
+    out_channels : int
+        Number of output channels
+    modes1 : int
+        Number of Fourier modes to multiply in first dimension, at most floor(N/2) + 1
+    modes2 : int
+        Number of Fourier modes to multiply in second dimension, at most floor(N/2) + 1
+    modes3 : int
+        Number of Fourier modes to multiply in third dimension, at most floor(N/2) + 1
+    rank : float
+        Rank of the decomposition
+    factorization : {'CP', 'TT', 'Tucker'}
+        Tensor factorization to use to decompose the tensor
+    fixed_rank_modes : List[int]
+        A list of modes for which the initial value is not modified
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict
+        Additional arguments to initialization of factorized tensors
+    """
+
+    def __init__(
+        self,
+        in_channels: int,
+        out_channels: int,
+        modes1: int,
+        modes2: int,
+        modes3: int,
+        modes4: int,
+        rank: float,
+        factorization: str,
+        fixed_rank_modes: bool,
+        decomposition_kwargs: dict,
+    ):
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.out_channels = out_channels
+
+        # Number of Fourier modes to multiply, at most floor(N/2) + 1
+        self.modes1 = modes1
+        self.modes2 = modes2
+        self.modes3 = modes3
+        self.modes4 = modes4
+
+        self.scale = 1 / (in_channels * out_channels)
+        self.weights1 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights2 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights3 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights4 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights5 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights6 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights7 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.weights8 = tltorch.FactorizedTensor.new(
+            (
+                in_channels,
+                out_channels,
+                self.modes1,
+                self.modes2,
+                self.modes3,
+                self.modes4,
+                2,
+            ),
+            rank=rank,
+            factorization=factorization,
+            fixed_rank_modes=fixed_rank_modes,
+            **decomposition_kwargs
+        )
+        self.reset_parameters()
+
+    def compl_mul4d(
+        self,
+        input: Tensor,
+        weights: Tensor,
+    ) -> Tensor:
+        """Complex multiplication
+
+        Parameters
+        ----------
+        input : Tensor
+            Input tensor
+        weights : Tensor
+            Weights tensor
+
+        Returns
+        -------
+        Tensor
+            Product of complex multiplication
+        """
+        # (batch, in_channel, x, y, z), (in_channel, out_channel, x, y, z) -> (batch, out_channel, x, y, z)
+        cweights = torch.view_as_complex(weights.to_tensor().contiguous())
+        return torch.einsum("bixyzt,ioxyzt->boxyzt", input, cweights)
+
+    def forward(self, x: Tensor) -> Tensor:
+        batchsize = x.shape[0]
+        # Compute Fourier coeffcients up to factor of e^(- something constant)
+        x_ft = torch.fft.rfftn(x, dim=[-4, -3, -2, -1])
+
+        # Multiply relevant Fourier modes
+        out_ft = torch.zeros(
+            batchsize,
+            self.out_channels,
+            x.size(-4),
+            x.size(-3),
+            x.size(-2),
+            x.size(-1) // 2 + 1,
+            dtype=torch.cfloat,
+            device=x.device,
+        )
+
+        # print(f'mod: size x: {x_ft.size()}, out: {out_ft.size()}')
+        # print(f'mod: x_ft[weight4]: {x_ft[:, :, self.modes1 :, self.modes2 :, : -self.modes3, :self.modes4].size()} weight4: {self.weights4.size()}')
+
+        out_ft[
+            :, :, : self.modes1, : self.modes2, : self.modes3, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, : self.modes1, : self.modes2, : self.modes3, : self.modes4],
+            self.weights1,
+        )
+        out_ft[
+            :, :, -self.modes1 :, : self.modes2, : self.modes3, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, -self.modes1 :, : self.modes2, : self.modes3, : self.modes4],
+            self.weights2,
+        )
+        out_ft[
+            :, :, : self.modes1, -self.modes2 :, : self.modes3, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, : self.modes1, -self.modes2 :, : self.modes3, : self.modes4],
+            self.weights3,
+        )
+        out_ft[
+            :, :, : self.modes1, : self.modes2, -self.modes3 :, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, : self.modes1, : self.modes2, -self.modes3 :, : self.modes4],
+            self.weights4,
+        )
+        out_ft[
+            :, :, -self.modes1 :, -self.modes2 :, : self.modes3, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, -self.modes1 :, -self.modes2 :, : self.modes3, : self.modes4],
+            self.weights5,
+        )
+        out_ft[
+            :, :, -self.modes1 :, : self.modes2, -self.modes3 :, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, -self.modes1 :, : self.modes2, -self.modes3 :, : self.modes4],
+            self.weights6,
+        )
+        out_ft[
+            :, :, : self.modes1, -self.modes2 :, -self.modes3 :, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, : self.modes1, -self.modes2 :, -self.modes3 :, : self.modes4],
+            self.weights7,
+        )
+        out_ft[
+            :, :, -self.modes1 :, -self.modes2 :, -self.modes3 :, : self.modes4
+        ] = self.compl_mul4d(
+            x_ft[:, :, -self.modes1 :, -self.modes2 :, -self.modes3 :, : self.modes4],
+            self.weights8,
+        )
+
+        # Return to physical space
+        x = torch.fft.irfftn(out_ft, s=(x.size(-4), x.size(-3), x.size(-2), x.size(-1)))
+        return x
+
+    def reset_parameters(self):
+        """Reset spectral weights with distribution scale*N(0,1)"""
+        self.weights1.normal_(0, self.scale)
+        self.weights2.normal_(0, self.scale)
+        self.weights3.normal_(0, self.scale)
+        self.weights4.normal_(0, self.scale)
+        self.weights5.normal_(0, self.scale)
+        self.weights6.normal_(0, self.scale)
+        self.weights7.normal_(0, self.scale)
+        self.weights8.normal_(0, self.scale)

mhd/tfno/tfno.py

@@ -0,0 +1,1043 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from dataclasses import dataclass
+from typing import List, Tuple, Union
+
+import torch
+import torch.nn as nn
+import torch.nn.functional as F
+from torch import Tensor
+
+import physicsnemo  # noqa: F401 for docs
+import physicsnemo.models.layers as layers
+from .spectral_layers import (
+    FactorizedSpectralConv1d,
+    FactorizedSpectralConv2d,
+    FactorizedSpectralConv3d,
+    FactorizedSpectralConv4d,
+)
+
+from physicsnemo.models.meta import ModelMetaData
+from physicsnemo.models.mlp import FullyConnected
+from physicsnemo.models.module import Module
+
+# ===================================================================
+# ===================================================================
+# 1D TFNO
+# ===================================================================
+# ===================================================================
+
+
+class TFNO1DEncoder(nn.Module):
+    """1D Spectral encoder for TFNO
+
+    Parameters
+    ----------
+    in_channels : int, optional
+        Number of input channels, by default 1
+    num_fno_layers : int, optional
+        Number of spectral convolutional layers, by default 4
+    fno_layer_size : int, optional
+        Latent features size in spectral convolutions, by default 32
+    num_fno_modes : Union[int, List[int]], optional
+        Number of Fourier modes kept in spectral convolutions, by default 16
+    padding :  Union[int, List[int]], optional
+        Domain padding for spectral convolutions, by default 8
+    padding_type : str, optional
+        Type of padding for spectral convolutions, by default "constant"
+    activation_fn : nn.Module, optional
+        Activation function, by default nn.GELU
+    coord_features : bool, optional
+        Use coordinate grid as additional feature map, by default True
+    rank : float, optional
+        Rank of the decomposition, by default 1.0
+    factorization : {'CP', 'TT', 'Tucker'}, optional
+        Tensor factorization to use to decompose the tensor, by default 'CP'
+    fixed_rank_modes : List[int], optional
+        A list of modes for which the initial value is not modified, by default None
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict, optional
+        Additional arguments to initialization of factorized tensors, by default dict()
+    """
+
+    def __init__(
+        self,
+        in_channels: int = 1,
+        num_fno_layers: int = 4,
+        fno_layer_size: int = 32,
+        num_fno_modes: Union[int, List[int]] = 16,
+        padding: Union[int, List[int]] = 8,
+        padding_type: str = "constant",
+        activation_fn: nn.Module = nn.GELU(),
+        coord_features: bool = True,
+        rank: float = 1.0,
+        factorization: str = "cp",
+        fixed_rank_modes: List[int] = None,
+        decomposition_kwargs: dict = dict(),
+    ) -> None:
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.num_fno_layers = num_fno_layers
+        self.fno_width = fno_layer_size
+        self.activation_fn = activation_fn
+
+        # TensorLy arguments
+        self.rank = rank
+        self.factorization = factorization
+        self.fixed_rank_modes = fixed_rank_modes
+        self.decomposition_kwargs = decomposition_kwargs
+
+        # Add relative coordinate feature
+        self.coord_features = coord_features
+        if self.coord_features:
+            self.in_channels = self.in_channels + 1
+
+        # Padding values for spectral conv
+        if isinstance(padding, int):
+            padding = [padding]
+        self.pad = padding[:1]
+        self.ipad = [-pad if pad > 0 else None for pad in self.pad]
+        self.padding_type = padding_type
+
+        if isinstance(num_fno_modes, int):
+            num_fno_modes = [num_fno_modes]
+
+        # build lift
+        self.build_lift_network()
+        self.build_fno(num_fno_modes)
+
+    def build_lift_network(self) -> None:
+        """construct network for lifting variables to latent space."""
+        self.lift_network = torch.nn.Sequential()
+        self.lift_network.append(
+            layers.Conv1dFCLayer(self.in_channels, int(self.fno_width / 2))
+        )
+        self.lift_network.append(self.activation_fn)
+        self.lift_network.append(
+            layers.Conv1dFCLayer(int(self.fno_width / 2), self.fno_width)
+        )
+
+    def build_fno(self, num_fno_modes: List[int]) -> None:
+        """construct FNO block.
+        Parameters
+        ----------
+        num_fno_modes : List[int]
+            Number of Fourier modes kept in spectral convolutions
+
+        """
+        # Build Neural Fourier Operators
+        self.spconv_layers = nn.ModuleList()
+        self.conv_layers = nn.ModuleList()
+        for _ in range(self.num_fno_layers):
+            self.spconv_layers.append(
+                FactorizedSpectralConv1d(
+                    self.fno_width,
+                    self.fno_width,
+                    num_fno_modes[0],
+                    self.rank,
+                    self.factorization,
+                    self.fixed_rank_modes,
+                    self.decomposition_kwargs,
+                )
+            )
+            self.conv_layers.append(nn.Conv1d(self.fno_width, self.fno_width, 1))
+
+    def forward(self, x: Tensor) -> Tensor:
+        if self.coord_features:
+            coord_feat = self.meshgrid(list(x.shape), x.device)
+            x = torch.cat((x, coord_feat), dim=1)
+
+        x = self.lift_network(x)
+        # (left, right)
+        x = F.pad(x, (0, self.pad[0]), mode=self.padding_type)
+        # Spectral layers
+        for k, conv_w in enumerate(zip(self.conv_layers, self.spconv_layers)):
+            conv, w = conv_w
+            if k < len(self.conv_layers) - 1:
+                x = self.activation_fn(conv(x) + w(x))
+            else:
+                x = conv(x) + w(x)
+
+        x = x[..., : self.ipad[0]]
+        return x
+
+    def meshgrid(self, shape: List[int], device: torch.device) -> Tensor:
+        """Creates 1D meshgrid feature
+
+        Parameters
+        ----------
+        shape : List[int]
+            Tensor shape
+        device : torch.device
+            Device model is on
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        bsize, size_x = shape[0], shape[2]
+        grid_x = torch.linspace(0, 1, size_x, dtype=torch.float32, device=device)
+        grid_x = grid_x.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1)
+        return grid_x
+
+    def grid_to_points(self, value: Tensor) -> Tuple[Tensor, List[int]]:
+        """converting from grid based (image) to point based representation
+
+        Parameters
+        ----------
+        value : Meshgrid tensor
+
+        Returns
+        -------
+        Tuple
+            Tensor, meshgrid shape
+        """
+        y_shape = list(value.size())
+        output = torch.permute(value, (0, 2, 1))
+        return output.reshape(-1, output.size(-1)), y_shape
+
+    def points_to_grid(self, value: Tensor, shape: List[int]) -> Tensor:
+        """converting from point based to grid based (image) representation
+
+        Parameters
+        ----------
+        value : Tensor
+            Tensor
+        shape : List[int]
+            meshgrid shape
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        output = value.reshape(shape[0], shape[2], value.size(-1))
+        return torch.permute(output, (0, 2, 1))
+
+
+# ===================================================================
+# ===================================================================
+# 2D TFNO
+# ===================================================================
+# ===================================================================
+
+
+class TFNO2DEncoder(nn.Module):
+    """2D Spectral encoder for TFNO
+
+    Parameters
+    ----------
+    in_channels : int, optional
+        Number of input channels, by default 1
+    num_fno_layers : int, optional
+        Number of spectral convolutional layers, by default 4
+    fno_layer_size : int, optional
+        Latent features size in spectral convolutions, by default 32
+    num_fno_modes : Union[int, List[int]], optional
+        Number of Fourier modes kept in spectral convolutions, by default 16
+    padding :  Union[int, List[int]], optional
+        Domain padding for spectral convolutions, by default 8
+    padding_type : str, optional
+        Type of padding for spectral convolutions, by default "constant"
+    activation_fn : nn.Module, optional
+        Activation function, by default nn.GELU
+    coord_features : bool, optional
+        Use coordinate grid as additional feature map, by default True
+    rank : float, optional
+        Rank of the decomposition, by default 1.0
+    factorization : {'CP', 'TT', 'Tucker'}, optional
+        Tensor factorization to use to decompose the tensor, by default 'CP'
+    fixed_rank_modes : List[int], optional
+        A list of modes for which the initial value is not modified, by default None
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict, optional
+        Additional arguments to initialization of factorized tensors, by default dict()
+    """
+
+    def __init__(
+        self,
+        in_channels: int = 1,
+        num_fno_layers: int = 4,
+        fno_layer_size: int = 32,
+        num_fno_modes: Union[int, List[int]] = 16,
+        padding: Union[int, List[int]] = 8,
+        padding_type: str = "constant",
+        activation_fn: nn.Module = nn.GELU(),
+        coord_features: bool = True,
+        rank: float = 1.0,
+        factorization: str = "cp",
+        fixed_rank_modes: List[int] = None,
+        decomposition_kwargs: dict = dict(),
+    ) -> None:
+        super().__init__()
+        self.in_channels = in_channels
+        self.num_fno_layers = num_fno_layers
+        self.fno_width = fno_layer_size
+        self.coord_features = coord_features
+        self.activation_fn = activation_fn
+
+        # TensorLy arguments
+        self.rank = rank
+        self.factorization = factorization
+        self.fixed_rank_modes = fixed_rank_modes
+        self.decomposition_kwargs = decomposition_kwargs
+
+        # Add relative coordinate feature
+        if self.coord_features:
+            self.in_channels = self.in_channels + 2
+
+        # Padding values for spectral conv
+        if isinstance(padding, int):
+            padding = [padding, padding]
+        padding = padding + [0, 0]  # Pad with zeros for smaller lists
+        self.pad = padding[:2]
+        self.ipad = [-pad if pad > 0 else None for pad in self.pad]
+        self.padding_type = padding_type
+
+        if isinstance(num_fno_modes, int):
+            num_fno_modes = [num_fno_modes, num_fno_modes]
+
+        # build lift
+        self.build_lift_network()
+        self.build_fno(num_fno_modes)
+
+    def build_lift_network(self) -> None:
+        """construct network for lifting variables to latent space."""
+        # Initial lift network
+        self.lift_network = torch.nn.Sequential()
+        self.lift_network.append(
+            layers.Conv2dFCLayer(self.in_channels, int(self.fno_width / 2))
+        )
+        self.lift_network.append(self.activation_fn)
+        self.lift_network.append(
+            layers.Conv2dFCLayer(int(self.fno_width / 2), self.fno_width)
+        )
+
+    def build_fno(self, num_fno_modes: List[int]) -> None:
+        """construct TFNO block.
+        Parameters
+        ----------
+        num_fno_modes : List[int]
+            Number of Fourier modes kept in spectral convolutions
+
+        """
+        # Build Neural Fourier Operators
+        self.spconv_layers = nn.ModuleList()
+        self.conv_layers = nn.ModuleList()
+        for _ in range(self.num_fno_layers):
+            self.spconv_layers.append(
+                FactorizedSpectralConv2d(
+                    self.fno_width,
+                    self.fno_width,
+                    num_fno_modes[0],
+                    num_fno_modes[1],
+                    self.rank,
+                    self.factorization,
+                    self.fixed_rank_modes,
+                    self.decomposition_kwargs,
+                )
+            )
+            self.conv_layers.append(nn.Conv2d(self.fno_width, self.fno_width, 1))
+
+    def forward(self, x: Tensor) -> Tensor:
+        if x.dim() != 4:
+            raise ValueError(
+                "Only 4D tensors [batch, in_channels, grid_x, grid_y] accepted for 2D FNO"
+            )
+
+        if self.coord_features:
+            coord_feat = self.meshgrid(list(x.shape), x.device)
+            x = torch.cat((x, coord_feat), dim=1)
+
+        x = self.lift_network(x)
+        # (left, right, top, bottom)
+        x = F.pad(x, (0, self.pad[1], 0, self.pad[0]), mode=self.padding_type)
+        # Spectral layers
+        for k, conv_w in enumerate(zip(self.conv_layers, self.spconv_layers)):
+            conv, w = conv_w
+            if k < len(self.conv_layers) - 1:
+                x = self.activation_fn(conv(x) + w(x))
+            else:
+                x = conv(x) + w(x)
+
+        # remove padding
+        x = x[..., : self.ipad[0], : self.ipad[1]]
+
+        return x
+
+    def meshgrid(self, shape: List[int], device: torch.device) -> Tensor:
+        """Creates 2D meshgrid feature
+
+        Parameters
+        ----------
+        shape : List[int]
+            Tensor shape
+        device : torch.device
+            Device model is on
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        bsize, size_x, size_y = shape[0], shape[2], shape[3]
+        grid_x = torch.linspace(0, 1, size_x, dtype=torch.float32, device=device)
+        grid_y = torch.linspace(0, 1, size_y, dtype=torch.float32, device=device)
+        grid_x, grid_y = torch.meshgrid(grid_x, grid_y, indexing="ij")
+        grid_x = grid_x.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1)
+        grid_y = grid_y.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1)
+        return torch.cat((grid_x, grid_y), dim=1)
+
+    def grid_to_points(self, value: Tensor) -> Tuple[Tensor, List[int]]:
+        """converting from grid based (image) to point based representation
+
+        Parameters
+        ----------
+        value : Meshgrid tensor
+
+        Returns
+        -------
+        Tuple
+            Tensor, meshgrid shape
+        """
+        y_shape = list(value.size())
+        output = torch.permute(value, (0, 2, 3, 1))
+        return output.reshape(-1, output.size(-1)), y_shape
+
+    def points_to_grid(self, value: Tensor, shape: List[int]) -> Tensor:
+        """converting from point based to grid based (image) representation
+
+        Parameters
+        ----------
+        value : Tensor
+            Tensor
+        shape : List[int]
+            meshgrid shape
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        output = value.reshape(shape[0], shape[2], shape[3], value.size(-1))
+        return torch.permute(output, (0, 3, 1, 2))
+
+
+# ===================================================================
+# ===================================================================
+# 3D TFNO
+# ===================================================================
+# ===================================================================
+
+
+class TFNO3DEncoder(nn.Module):
+    """3D Spectral encoder for TFNO
+
+    Parameters
+    ----------
+    in_channels : int, optional
+        Number of input channels, by default 1
+    num_fno_layers : int, optional
+        Number of spectral convolutional layers, by default 4
+    fno_layer_size : int, optional
+        Latent features size in spectral convolutions, by default 32
+    num_fno_modes : Union[int, List[int]], optional
+        Number of Fourier modes kept in spectral convolutions, by default 16
+    padding :  Union[int, List[int]], optional
+        Domain padding for spectral convolutions, by default 8
+    padding_type : str, optional
+        Type of padding for spectral convolutions, by default "constant"
+    activation_fn : nn.Module, optional
+        Activation function, by default nn.GELU
+    coord_features : bool, optional
+        Use coordinate grid as additional feature map, by default True
+    rank : float, optional
+        Rank of the decomposition, by default 1.0
+    factorization : {'CP', 'TT', 'Tucker'}, optional
+        Tensor factorization to use to decompose the tensor, by default 'CP'
+    fixed_rank_modes : List[int], optional
+        A list of modes for which the initial value is not modified, by default None
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict, optional
+        Additional arguments to initialization of factorized tensors, by default dict()
+    """
+
+    def __init__(
+        self,
+        in_channels: int = 1,
+        num_fno_layers: int = 4,
+        fno_layer_size: int = 32,
+        num_fno_modes: Union[int, List[int]] = 16,
+        padding: Union[int, List[int]] = 8,
+        padding_type: str = "constant",
+        activation_fn: nn.Module = nn.GELU(),
+        coord_features: bool = True,
+        rank: float = 1.0,
+        factorization: str = "cp",
+        fixed_rank_modes: List[int] = None,
+        decomposition_kwargs: dict = dict(),
+    ) -> None:
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.num_fno_layers = num_fno_layers
+        self.fno_width = fno_layer_size
+        self.coord_features = coord_features
+        self.activation_fn = activation_fn
+
+        # TensorLy arguments
+        self.rank = rank
+        self.factorization = factorization
+        self.fixed_rank_modes = fixed_rank_modes
+        self.decomposition_kwargs = decomposition_kwargs
+
+        # Add relative coordinate feature
+        if self.coord_features:
+            self.in_channels = self.in_channels + 3
+
+        # Padding values for spectral conv
+        if isinstance(padding, int):
+            padding = [padding, padding, padding]
+        padding = padding + [0, 0, 0]  # Pad with zeros for smaller lists
+        self.pad = padding[:3]
+        self.ipad = [-pad if pad > 0 else None for pad in self.pad]
+        self.padding_type = padding_type
+
+        if isinstance(num_fno_modes, int):
+            num_fno_modes = [num_fno_modes, num_fno_modes, num_fno_modes]
+
+        # build lift
+        self.build_lift_network()
+        self.build_fno(num_fno_modes)
+
+    def build_lift_network(self) -> None:
+        """construct network for lifting variables to latent space."""
+        # Initial lift network
+        self.lift_network = torch.nn.Sequential()
+        self.lift_network.append(
+            layers.Conv3dFCLayer(self.in_channels, int(self.fno_width / 2))
+        )
+        self.lift_network.append(self.activation_fn)
+        self.lift_network.append(
+            layers.Conv3dFCLayer(int(self.fno_width / 2), self.fno_width)
+        )
+
+    def build_fno(self, num_fno_modes: List[int]) -> None:
+        """construct FNO block.
+        Parameters
+        ----------
+        num_fno_modes : List[int]
+            Number of Fourier modes kept in spectral convolutions
+
+        """
+        # Build Neural Fourier Operators
+        self.spconv_layers = nn.ModuleList()
+        self.conv_layers = nn.ModuleList()
+        for _ in range(self.num_fno_layers):
+            self.spconv_layers.append(
+                FactorizedSpectralConv3d(
+                    self.fno_width,
+                    self.fno_width,
+                    num_fno_modes[0],
+                    num_fno_modes[1],
+                    num_fno_modes[2],
+                    self.rank,
+                    self.factorization,
+                    self.fixed_rank_modes,
+                    self.decomposition_kwargs,
+                )
+            )
+            self.conv_layers.append(nn.Conv3d(self.fno_width, self.fno_width, 1))
+
+    def forward(self, x: Tensor) -> Tensor:
+        if self.coord_features:
+            coord_feat = self.meshgrid(list(x.shape), x.device)
+            x = torch.cat((x, coord_feat), dim=1)
+
+        x = self.lift_network(x)
+        # (left, right, top, bottom, front, back)
+        x = F.pad(
+            x,
+            (0, self.pad[2], 0, self.pad[1], 0, self.pad[0]),
+            mode=self.padding_type,
+        )
+        # Spectral layers
+        for k, conv_w in enumerate(zip(self.conv_layers, self.spconv_layers)):
+            conv, w = conv_w
+            if k < len(self.conv_layers) - 1:
+                x = self.activation_fn(conv(x) + w(x))
+            else:
+                x = conv(x) + w(x)
+
+        x = x[..., : self.ipad[0], : self.ipad[1], : self.ipad[2]]
+        return x
+
+    def meshgrid(self, shape: List[int], device: torch.device) -> Tensor:
+        """Creates 3D meshgrid feature
+
+        Parameters
+        ----------
+        shape : List[int]
+            Tensor shape
+        device : torch.device
+            Device model is on
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        bsize, size_x, size_y, size_z = shape[0], shape[2], shape[3], shape[4]
+        grid_x = torch.linspace(0, 1, size_x, dtype=torch.float32, device=device)
+        grid_y = torch.linspace(0, 1, size_y, dtype=torch.float32, device=device)
+        grid_z = torch.linspace(0, 1, size_z, dtype=torch.float32, device=device)
+        grid_x, grid_y, grid_z = torch.meshgrid(grid_x, grid_y, grid_z, indexing="ij")
+        grid_x = grid_x.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1)
+        grid_y = grid_y.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1)
+        grid_z = grid_z.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1)
+        return torch.cat((grid_x, grid_y, grid_z), dim=1)
+
+    def grid_to_points(self, value: Tensor) -> Tuple[Tensor, List[int]]:
+        """converting from grid based (image) to point based representation
+
+        Parameters
+        ----------
+        value : Meshgrid tensor
+
+        Returns
+        -------
+        Tuple
+            Tensor, meshgrid shape
+        """
+        y_shape = list(value.size())
+        output = torch.permute(value, (0, 2, 3, 4, 1))
+        return output.reshape(-1, output.size(-1)), y_shape
+
+    def points_to_grid(self, value: Tensor, shape: List[int]) -> Tensor:
+        """converting from point based to grid based (image) representation
+
+        Parameters
+        ----------
+        value : Tensor
+            Tensor
+        shape : List[int]
+            meshgrid shape
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        output = value.reshape(shape[0], shape[2], shape[3], shape[4], value.size(-1))
+        return torch.permute(output, (0, 4, 1, 2, 3))
+
+
+# ===================================================================
+# ===================================================================
+# 4D TFNO
+# ===================================================================
+# ===================================================================
+
+
+class TFNO4DEncoder(nn.Module):
+    """4D Spectral encoder for TFNO
+
+    Parameters
+    ----------
+    in_channels : int, optional
+        Number of input channels, by default 1
+    num_fno_layers : int, optional
+        Number of spectral convolutional layers, by default 4
+    fno_layer_size : int, optional
+        Latent features size in spectral convolutions, by default 32
+    num_fno_modes : Union[int, List[int]], optional
+        Number of Fourier modes kept in spectral convolutions, by default 16
+    padding :  Union[int, List[int]], optional
+        Domain padding for spectral convolutions, by default 8
+    padding_type : str, optional
+        Type of padding for spectral convolutions, by default "constant"
+    activation_fn : nn.Module, optional
+        Activation function, by default nn.GELU
+    coord_features : bool, optional
+        Use coordinate grid as additional feature map, by default True
+    rank : float, optional
+        Rank of the decomposition, by default 1.0
+    factorization : {'CP', 'TT', 'Tucker'}, optional
+        Tensor factorization to use to decompose the tensor, by default 'CP'
+    fixed_rank_modes : List[int], optional
+        A list of modes for which the initial value is not modified, by default None
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict, optional
+        Additional arguments to initialization of factorized tensors, by default dict()
+    """
+
+    def __init__(
+        self,
+        in_channels: int = 1,
+        num_fno_layers: int = 4,
+        fno_layer_size: int = 32,
+        num_fno_modes: Union[int, List[int]] = 16,
+        padding: Union[int, List[int]] = 8,
+        padding_type: str = "constant",
+        activation_fn: nn.Module = nn.GELU(),
+        coord_features: bool = True,
+        rank: float = 1.0,
+        factorization: str = "cp",
+        fixed_rank_modes: List[int] = None,
+        decomposition_kwargs: dict = dict(),
+    ) -> None:
+        super().__init__()
+
+        self.in_channels = in_channels
+        self.num_fno_layers = num_fno_layers
+        self.fno_width = fno_layer_size
+        self.coord_features = coord_features
+        self.activation_fn = activation_fn
+
+        # TensorLy arguments
+        self.rank = rank
+        self.factorization = factorization
+        self.fixed_rank_modes = fixed_rank_modes
+        self.decomposition_kwargs = decomposition_kwargs
+
+        # Add relative coordinate feature
+        if self.coord_features:
+            self.in_channels = self.in_channels + 4
+
+        # Padding values for spectral conv
+        if isinstance(padding, int):
+            padding = [padding, padding, padding, padding]
+        padding = padding + [0, 0, 0, 0]  # Pad with zeros for smaller lists
+        self.pad = padding[:4]
+        self.ipad = [-pad if pad > 0 else None for pad in self.pad]
+        self.padding_type = padding_type
+
+        if isinstance(num_fno_modes, int):
+            num_fno_modes = [num_fno_modes, num_fno_modes, num_fno_modes, num_fno_modes]
+
+        # build lift
+        self.build_lift_network()
+        self.build_fno(num_fno_modes)
+
+    def build_lift_network(self) -> None:
+        """construct network for lifting variables to latent space."""
+        # Initial lift network
+        self.lift_network = torch.nn.Sequential()
+        self.lift_network.append(
+            layers.ConvNdFCLayer(self.in_channels, int(self.fno_width / 2))
+        )
+        self.lift_network.append(self.activation_fn)
+        self.lift_network.append(
+            layers.ConvNdFCLayer(int(self.fno_width / 2), self.fno_width)
+        )
+
+    def build_fno(self, num_fno_modes: List[int]) -> None:
+        """construct TFNO block.
+        Parameters
+        ----------
+        num_fno_modes : List[int]
+            Number of Fourier modes kept in spectral convolutions
+
+        """
+        # Build Neural Fourier Operators
+        self.spconv_layers = nn.ModuleList()
+        self.conv_layers = nn.ModuleList()
+        for _ in range(self.num_fno_layers):
+            self.spconv_layers.append(
+                FactorizedSpectralConv4d(
+                    self.fno_width,
+                    self.fno_width,
+                    num_fno_modes[0],
+                    num_fno_modes[1],
+                    num_fno_modes[2],
+                    num_fno_modes[3],
+                    self.rank,
+                    self.factorization,
+                    self.fixed_rank_modes,
+                    self.decomposition_kwargs,
+                )
+            )
+            self.conv_layers.append(
+                layers.ConvNdKernel1Layer(self.fno_width, self.fno_width)
+            )
+
+    def forward(self, x: Tensor) -> Tensor:
+        if self.coord_features:
+            coord_feat = self.meshgrid(list(x.shape), x.device)
+            x = torch.cat((x, coord_feat), dim=1)
+
+        x = self.lift_network(x)
+        # (left, right, top, bottom, front, back, past, future)
+        x = F.pad(
+            x,
+            (0, self.pad[3], 0, self.pad[2], 0, self.pad[1], 0, self.pad[0]),
+            mode=self.padding_type,
+        )
+        # Spectral layers
+        for k, conv_w in enumerate(zip(self.conv_layers, self.spconv_layers)):
+            conv, w = conv_w
+            if k < len(self.conv_layers) - 1:
+                x = self.activation_fn(conv(x) + w(x))
+            else:
+                x = conv(x) + w(x)
+
+        x = x[..., : self.ipad[0], : self.ipad[1], : self.ipad[2], : self.ipad[3]]
+        return x
+
+    def meshgrid(self, shape: List[int], device: torch.device) -> Tensor:
+        """Creates 4D meshgrid feature
+
+        Parameters
+        ----------
+        shape : List[int]
+            Tensor shape
+        device : torch.device
+            Device model is on
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        bsize, size_x, size_y, size_z, size_t = (
+            shape[0],
+            shape[2],
+            shape[3],
+            shape[4],
+            shape[5],
+        )
+        grid_x = torch.linspace(0, 1, size_x, dtype=torch.float32, device=device)
+        grid_y = torch.linspace(0, 1, size_y, dtype=torch.float32, device=device)
+        grid_z = torch.linspace(0, 1, size_z, dtype=torch.float32, device=device)
+        grid_t = torch.linspace(0, 1, size_t, dtype=torch.float32, device=device)
+        grid_x, grid_y, grid_z, grid_t = torch.meshgrid(
+            grid_x, grid_y, grid_z, grid_t, indexing="ij"
+        )
+        grid_x = grid_x.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1, 1)
+        grid_y = grid_y.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1, 1)
+        grid_z = grid_z.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1, 1)
+        grid_t = grid_t.unsqueeze(0).unsqueeze(0).repeat(bsize, 1, 1, 1, 1, 1)
+        return torch.cat((grid_x, grid_y, grid_z, grid_t), dim=1)
+
+    def grid_to_points(self, value: Tensor) -> Tuple[Tensor, List[int]]:
+        """converting from grid based (image) to point based representation
+
+        Parameters
+        ----------
+        value : Meshgrid tensor
+
+        Returns
+        -------
+        Tuple
+            Tensor, meshgrid shape
+        """
+        y_shape = list(value.size())
+        output = torch.permute(value, (0, 2, 3, 4, 5, 1))
+        return output.reshape(-1, output.size(-1)), y_shape
+
+    def points_to_grid(self, value: Tensor, shape: List[int]) -> Tensor:
+        """converting from point based to grid based (image) representation
+
+        Parameters
+        ----------
+        value : Tensor
+            Tensor
+        shape : List[int]
+            meshgrid shape
+
+        Returns
+        -------
+        Tensor
+            Meshgrid tensor
+        """
+        output = value.reshape(
+            shape[0], shape[2], shape[3], shape[4], shape[5], value.size(-1)
+        )
+        return torch.permute(output, (0, 5, 1, 2, 3, 4))
+
+
+# ===================================================================
+# ===================================================================
+# General TFNO Model
+# ===================================================================
+# ===================================================================
+
+
+class TFNO(nn.Module):
+    """Tensor Factorized Fourier neural operator (FNO) model.
+
+    Note
+    ----
+    The TFNO architecture supports options for 1D, 2D, 3D and 4D fields which can
+    be controlled using the `dimension` parameter.
+
+    Parameters
+    ----------
+    in_channels : int
+        Number of input channels
+    out_channels : int
+        Number of output channels
+    decoder_layers : int, optional
+        Number of decoder layers, by default 1
+    decoder_layer_size : int, optional
+        Number of neurons in decoder layers, by default 32
+    decoder_activation_fn : str, optional
+        Activation function for decoder, by default "silu"
+    dimension : int
+        Model dimensionality (supports 1, 2, 3).
+    latent_channels : int, optional
+        Latent features size in spectral convolutions, by default 32
+    num_fno_layers : int, optional
+        Number of spectral convolutional layers, by default 4
+    num_fno_modes : Union[int, List[int]], optional
+        Number of Fourier modes kept in spectral convolutions, by default 16
+    padding : int, optional
+        Domain padding for spectral convolutions, by default 8
+    padding_type : str, optional
+        Type of padding for spectral convolutions, by default "constant"
+    activation_fn : str, optional
+        Activation function, by default "gelu"
+    coord_features : bool, optional
+        Use coordinate grid as additional feature map, by default True
+    rank : float, optional
+        Rank of the decomposition, by default 1.0
+    factorization : {'CP', 'TT', 'Tucker'}, optional
+        Tensor factorization to use to decompose the tensor, by default 'CP'
+    fixed_rank_modes : List[int], optional
+        A list of modes for which the initial value is not modified, by default None
+        The last mode cannot be fixed due to error computation.
+    decomposition_kwargs : dict, optional
+        Additional arguments to initialization of factorized tensors, by default dict()
+
+    Example
+    -------
+    >>> # define the 2d TFNO model
+    >>> model = physicsnemo.models.fno.TFNO(
+    ...     in_channels=4,
+    ...     out_channels=3,
+    ...     decoder_layers=2,
+    ...     decoder_layer_size=32,
+    ...     dimension=2,
+    ...     latent_channels=32,
+    ...     num_fno_layers=2,
+    ...     padding=0,
+    ... )
+    >>> input = torch.randn(32, 4, 32, 32) #(N, C, H, W)
+    >>> output = model(input)
+    >>> output.size()
+    torch.Size([32, 3, 32, 32])
+
+    Note
+    ----
+    Reference: Rosofsky, Shawn G. and Huerta, E. A. "Magnetohydrodynamics with
+    Physics Informed Neural Operators." arXiv preprint arXiv:2302.08332 (2023).
+    """
+
+    def __init__(
+        self,
+        in_channels: int,
+        out_channels: int,
+        decoder_layers: int = 1,
+        decoder_layer_size: int = 32,
+        decoder_activation_fn: str = "silu",
+        dimension: int = 2,
+        latent_channels: int = 32,
+        num_fno_layers: int = 4,
+        num_fno_modes: Union[int, List[int]] = 16,
+        padding: int = 8,
+        padding_type: str = "constant",
+        activation_fn: str = "gelu",
+        coord_features: bool = True,
+        rank: float = 1.0,
+        factorization: str = "cp",
+        fixed_rank_modes: List[int] = None,
+        decomposition_kwargs: dict = dict(),
+    ) -> None:
+        super().__init__()
+
+        self.num_fno_layers = num_fno_layers
+        self.num_fno_modes = num_fno_modes
+        self.padding = padding
+        self.padding_type = padding_type
+        self.activation_fn = layers.get_activation(activation_fn)
+        self.coord_features = coord_features
+        self.dimension = dimension
+
+        # TensorLy arguments
+        self.rank = rank
+        self.factorization = factorization
+        self.fixed_rank_modes = fixed_rank_modes
+        self.decomposition_kwargs = decomposition_kwargs
+
+        # decoder net
+        self.decoder_net = FullyConnected(
+            in_features=latent_channels,
+            layer_size=decoder_layer_size,
+            out_features=out_channels,
+            num_layers=decoder_layers,
+            activation_fn=decoder_activation_fn,
+        )
+
+        TFNOModel = self.getTFNOEncoder()
+
+        self.spec_encoder = TFNOModel(
+            in_channels,
+            num_fno_layers=self.num_fno_layers,
+            fno_layer_size=latent_channels,
+            num_fno_modes=self.num_fno_modes,
+            padding=self.padding,
+            padding_type=self.padding_type,
+            activation_fn=self.activation_fn,
+            coord_features=self.coord_features,
+            rank=self.rank,
+            factorization=self.factorization,
+            fixed_rank_modes=self.fixed_rank_modes,
+            decomposition_kwargs=self.decomposition_kwargs,
+        )
+
+    def getTFNOEncoder(self):
+        "Return correct TFNO ND Encoder"
+        if self.dimension == 1:
+            return TFNO1DEncoder
+        elif self.dimension == 2:
+            return TFNO2DEncoder
+        elif self.dimension == 3:
+            return TFNO3DEncoder
+        elif self.dimension == 4:
+            return TFNO4DEncoder
+        else:
+            raise NotImplementedError(
+                "Invalid dimensionality. Only 1D, 2D, 3D and 4D FNO implemented"
+            )
+
+    def forward(self, x: Tensor) -> Tensor:
+        # Fourier encoder
+        y_latent = self.spec_encoder(x)
+
+        # Reshape to pointwise inputs if not a conv FC model
+        y_shape = y_latent.shape
+        y_latent, y_shape = self.spec_encoder.grid_to_points(y_latent)
+
+        # Decoder
+        y = self.decoder_net(y_latent)
+
+        # Convert back into grid
+        y = self.spec_encoder.points_to_grid(y, y_shape)
+
+        return y

mhd/train_mhd_vec_pot_tfno.py

@@ -0,0 +1,269 @@
+import os
+
+import hydra
+from omegaconf import OmegaConf
+import torch
+from omegaconf import DictConfig
+from physicsnemo.distributed import DistributedManager
+from physicsnemo.launch.logging import LaunchLogger, PythonLogger
+from physicsnemo.launch.utils import load_checkpoint, save_checkpoint
+from physicsnemo.sym.hydra import to_absolute_path
+from torch.nn.parallel import DistributedDataParallel
+from torch.optim import AdamW
+
+from dataloaders import Dedalus2DDataset, MHDDataloaderVecPot
+from losses import LossMHDVecPot_PhysicsNeMo
+from tfno import TFNO
+from utils.plot_utils import plot_predictions_mhd, plot_predictions_mhd_plotly
+
+dtype = torch.float
+torch.set_default_dtype(dtype)
+
+
+@hydra.main(
+    version_base="1.3", config_path="config", config_name="train_mhd_vec_pot_tfno.yaml"
+)
+def main(cfg: DictConfig) -> None:
+    DistributedManager.initialize()  # Only call this once in the entire script!
+    dist = DistributedManager()  # call if required elsewhere
+    cfg = OmegaConf.to_container(cfg, resolve=True)
+
+    # initialize monitoring
+    log = PythonLogger(name="mhd_pino")
+    log.file_logging()
+
+    log_params = cfg["log_params"]
+
+    # Load config file parameters
+    model_params = cfg["model_params"]
+    dataset_params = cfg["dataset_params"]
+    train_loader_params = cfg["train_loader_params"]
+    val_loader_params = cfg["val_loader_params"]
+    loss_params = cfg["loss_params"]
+    optimizer_params = cfg["optimizer_params"]
+    train_params = cfg["train_params"]
+
+    load_ckpt = cfg["load_ckpt"]
+    output_dir = cfg["output_dir"]
+
+    output_dir = to_absolute_path(output_dir)
+    os.makedirs(output_dir, exist_ok=True)
+
+    data_dir = dataset_params["data_dir"]
+    ckpt_path = train_params["ckpt_path"]
+
+    # Construct dataloaders
+    dataset_train = Dedalus2DDataset(
+        dataset_params["data_dir"],
+        output_names=dataset_params["output_names"],
+        field_names=dataset_params["field_names"],
+        num_train=dataset_params["num_train"],
+        num_test=dataset_params["num_test"],
+        num=dataset_params["num"],
+        use_train=True,
+    )
+    dataset_val = Dedalus2DDataset(
+        data_dir,
+        output_names=dataset_params["output_names"],
+        field_names=dataset_params["field_names"],
+        num_train=dataset_params["num_train"],
+        num_test=dataset_params["num_test"],
+        num=dataset_params["num"],
+        use_train=False,
+    )
+
+    mhd_dataloader_train = MHDDataloaderVecPot(
+        dataset_train,
+        sub_x=dataset_params["sub_x"],
+        sub_t=dataset_params["sub_t"],
+        ind_x=dataset_params["ind_x"],
+        ind_t=dataset_params["ind_t"],
+    )
+    mhd_dataloader_val = MHDDataloaderVecPot(
+        dataset_val,
+        sub_x=dataset_params["sub_x"],
+        sub_t=dataset_params["sub_t"],
+        ind_x=dataset_params["ind_x"],
+        ind_t=dataset_params["ind_t"],
+    )
+
+    dataloader_train, sampler_train = mhd_dataloader_train.create_dataloader(
+        batch_size=train_loader_params["batch_size"],
+        shuffle=train_loader_params["shuffle"],
+        num_workers=train_loader_params["num_workers"],
+        pin_memory=train_loader_params["pin_memory"],
+        distributed=dist.distributed,
+    )
+    dataloader_val, sampler_val = mhd_dataloader_val.create_dataloader(
+        batch_size=val_loader_params["batch_size"],
+        shuffle=val_loader_params["shuffle"],
+        num_workers=val_loader_params["num_workers"],
+        pin_memory=val_loader_params["pin_memory"],
+        distributed=dist.distributed,
+    )
+
+    # define FNO model
+    model = TFNO(
+        in_channels=model_params["in_dim"],
+        out_channels=model_params["out_dim"],
+        decoder_layers=model_params["decoder_layers"],
+        decoder_layer_size=model_params["fc_dim"],
+        dimension=model_params["dimension"],
+        latent_channels=model_params["layers"],
+        num_fno_layers=model_params["num_fno_layers"],
+        num_fno_modes=model_params["modes"],
+        padding=[model_params["pad_z"], model_params["pad_y"], model_params["pad_x"]],
+        rank=model_params["rank"],
+        factorization=model_params["factorization"],
+        fixed_rank_modes=model_params["fixed_rank_modes"],
+        decomposition_kwargs=model_params["decomposition_kwargs"],
+    ).to(dist.device)
+    # Set up DistributedDataParallel if using more than a single process.
+    # The `distributed` property of DistributedManager can be used to
+    # check this.
+    if dist.distributed:
+        ddps = torch.cuda.Stream()
+        with torch.cuda.stream(ddps):
+            model = DistributedDataParallel(
+                model,
+                device_ids=[dist.local_rank],  # Set the device_id to be
+                # the local rank of this process on
+                # this node
+                output_device=dist.device,
+                broadcast_buffers=dist.broadcast_buffers,
+                find_unused_parameters=dist.find_unused_parameters,
+            )
+        torch.cuda.current_stream().wait_stream(ddps)
+
+    # Construct optimizer and scheduler
+    optimizer = AdamW(
+        model.parameters(),
+        betas=optimizer_params["betas"],
+        lr=optimizer_params["lr"],
+        weight_decay=0.1,
+    )
+
+    scheduler = torch.optim.lr_scheduler.MultiStepLR(
+        optimizer,
+        milestones=optimizer_params["milestones"],
+        gamma=optimizer_params["gamma"],
+    )
+
+    # Construct Loss class
+    mhd_loss = LossMHDVecPot_PhysicsNeMo(**loss_params)
+
+    # Load model from checkpoint (if exists)
+    loaded_epoch = 0
+    if load_ckpt:
+        loaded_epoch = load_checkpoint(
+            ckpt_path, model, optimizer, scheduler, device=dist.device
+        )
+
+    # Training Loop
+    epochs = train_params["epochs"]
+    ckpt_freq = train_params["ckpt_freq"]
+    names = dataset_params["fields"]
+    input_norm = torch.tensor(model_params["input_norm"]).to(dist.device)
+    output_norm = torch.tensor(model_params["output_norm"]).to(dist.device)
+    for epoch in range(max(1, loaded_epoch + 1), epochs + 1):
+        with LaunchLogger(
+            "train",
+            epoch=epoch,
+            num_mini_batch=len(dataloader_train),
+            epoch_alert_freq=1,
+        ) as log:
+            if dist.distributed:
+                sampler_train.set_epoch(epoch)
+
+            # Train Loop
+            model.train()
+
+            for i, (inputs, outputs) in enumerate(dataloader_train):
+                inputs = inputs.type(torch.FloatTensor).to(dist.device)
+                outputs = outputs.type(torch.FloatTensor).to(dist.device)
+                # Zero Gradients
+                optimizer.zero_grad()
+                # Compute Predictions
+                pred = (
+                    model((inputs / input_norm).permute(0, 4, 1, 2, 3)).permute(
+                        0, 2, 3, 4, 1
+                    )
+                    * output_norm
+                )
+                # Compute Loss
+                loss, loss_dict = mhd_loss(pred, outputs, inputs, return_loss_dict=True)
+                # Compute Gradients for Back Propagation
+                loss.backward()
+                # Update Weights
+                optimizer.step()
+
+                log.log_minibatch(loss_dict)
+
+            log.log_epoch({"Learning Rate": optimizer.param_groups[0]["lr"]})
+            scheduler.step()
+
+        with LaunchLogger("valid", epoch=epoch) as log:
+            # Val loop
+            model.eval()
+            plot_count = 0
+            with torch.no_grad():
+                for i, (inputs, outputs) in enumerate(dataloader_val):
+                    inputs = inputs.type(dtype).to(dist.device)
+                    outputs = outputs.type(dtype).to(dist.device)
+
+                    # Compute Predictions
+                    pred = (
+                        model((inputs / input_norm).permute(0, 4, 1, 2, 3)).permute(
+                            0, 2, 3, 4, 1
+                        )
+                        * output_norm
+                    )
+                    # Compute Loss
+                    loss, loss_dict = mhd_loss(
+                        pred, outputs, inputs, return_loss_dict=True
+                    )
+
+                    log.log_minibatch(loss_dict)
+
+                    # Get prediction plots to log
+                    # Do for number of batches specified in the config file
+                    if (i < log_params["log_num_plots"]) and (
+                        epoch % log_params["log_plot_freq"] == 0
+                    ):
+                        # Add all predictions in batch
+                        for j, _ in enumerate(pred):
+                            # Make plots for each field
+                            for index, name in enumerate(names):
+                                # Generate figure
+                                _ = plot_predictions_mhd_plotly(
+                                    pred[j].cpu(),
+                                    outputs[j].cpu(),
+                                    inputs[j].cpu(),
+                                    index=index,
+                                    name=name,
+                                )
+                            plot_count += 1
+
+                    # Get prediction plots and save images locally
+                    if (i < 2) and (epoch % log_params["log_plot_freq"] == 0):
+                        # Add all predictions in batch
+                        for j, _ in enumerate(pred):
+                            # Generate figure
+                            plot_predictions_mhd(
+                                pred[j].cpu(),
+                                outputs[j].cpu(),
+                                inputs[j].cpu(),
+                                names=names,
+                                save_path=os.path.join(
+                                    output_dir,
+                                    "MHD_physicsnemo" + "_" + str(dist.rank),
+                                ),
+                                save_suffix=i,
+                            )
+
+            if epoch % ckpt_freq == 0 and dist.rank == 0:
+                save_checkpoint(ckpt_path, model, optimizer, scheduler, epoch=epoch)
+
+
+if __name__ == "__main__":
+    main()

mhd/utils/plot_utils.py

@@ -0,0 +1,379 @@
+# SPDX-FileCopyrightText: Copyright (c) 2023 - 2024 NVIDIA CORPORATION & AFFILIATES.
+# SPDX-FileCopyrightText: All rights reserved.
+# SPDX-License-Identifier: Apache-2.0
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import os
+import math
+import numpy as np
+
+import torch
+import torch.nn.functional as F
+from tqdm import tqdm
+import traceback
+
+import matplotlib.pyplot as plt
+import matplotlib
+from mpl_toolkits.axes_grid1.axes_divider import make_axes_locatable
+import imageio
+import plotly
+import plotly.express as px
+from plotly.subplots import make_subplots
+import plotly.graph_objects as go
+from IPython.display import HTML, display
+
+
+def plot_spectra_mhd(
+    k,
+    pred_spectra_kin,
+    true_spectra_kin,
+    pred_spectra_mag,
+    true_spectra_mag,
+    index_t=-1,
+    name="Re100",
+    save_path=None,
+    save_suffix=None,
+    font_size=None,
+    sci_limits=None,
+    style_kin_pred="b-",
+    style_kin_true="k-",
+    style_mag_pred="b--",
+    style_mag_true="k--",
+    xmin=0,
+    xmax=200,
+    ymin=1e-10,
+    ymax=None,
+):
+    "Plots spectra of predicted and true outputs"
+    if font_size is not None:
+        plt.rcParams.update({"font.size": font_size})
+
+    if sci_limits is not None:
+        plt.rcParams.update({"axes.formatter.limits": sci_limits})
+
+    E_kin_pred = pred_spectra_kin[index_t]
+    E_mag_pred = pred_spectra_mag[index_t]
+
+    E_kin_true = true_spectra_kin[index_t]
+    E_mag_true = true_spectra_mag[index_t]
+
+    fig = plt.figure(figsize=(6, 5))
+
+    plt.loglog(k, E_kin_pred, style_kin_pred, label="$E_{kin}$ Pred")
+    plt.loglog(k, E_kin_true, style_kin_true, label="$E_{kin}$ True")
+    plt.loglog(k, E_mag_pred, style_mag_pred, label="$E_{mag}$ Pred")
+    plt.loglog(k, E_mag_true, style_mag_true, label="$E_{mag}$ True")
+
+    plt.xlabel("k")
+    plt.ylabel("E(k)")
+    plt.axis([xmin, xmax, ymin, ymax])
+
+    plt.title(f"Spectra ${name}$")
+    plt.legend(loc="upper right")
+
+    if save_path is not None:
+        if save_suffix is not None:
+            figure_path = f"{save_path}_spectra_{save_suffix}.png"
+        else:
+            figure_path = f"{save_path}_spectra.png"
+        plt.savefig(figure_path, bbox_inches="tight")
+
+    return fig
+
+
+def plot_predictions_mhd(
+    pred,
+    true,
+    inputs,
+    index_t=-1,
+    names=[],
+    save_path=None,
+    save_suffix=None,
+    font_size=None,
+    sci_limits=None,
+    shading="auto",
+    cmap="jet",
+):
+    "Plots images of predictions and absolute error"
+    if font_size is not None:
+        plt.rcParams.update({"font.size": font_size})
+
+    if sci_limits is not None:
+        plt.rcParams.update({"axes.formatter.limits": sci_limits})
+    # Plot
+    fig = plt.figure(figsize=(24, 5 * len(names)))
+
+    # Make plots for each field
+    for index, name in enumerate(names):
+        Nt, Nx, Ny, Nfields = pred.shape
+        u_pred = pred[index_t, ..., index]
+        u_true = true[index_t, ..., index]
+        u_err = u_pred - u_true
+
+        initial_data = inputs[0, ..., 3:]
+        u0 = initial_data[..., index]
+
+        x = inputs[0, :, 0, 1]
+        y = inputs[0, 0, :, 2]
+        X, Y = torch.meshgrid(x, y, indexing="ij")
+        t = inputs[index_t, 0, 0, 0]
+
+        plt.subplot(len(names), 4, index * 4 + 1)
+        plt.pcolormesh(X, Y, u0, cmap=cmap, shading=shading)
+        plt.colorbar()
+        plt.title(f"Intial Condition ${name}_0(x,y)$")
+        plt.tight_layout()
+        plt.axis("square")
+        plt.axis("off")
+
+        plt.subplot(len(names), 4, index * 4 + 2)
+        plt.pcolormesh(X, Y, u_true, cmap=cmap, shading=shading)
+        plt.colorbar()
+        plt.title(f"Exact ${name}(x,y,t={t:.2f})$")
+        plt.tight_layout()
+        plt.axis("square")
+        plt.axis("off")
+
+        plt.subplot(len(names), 4, index * 4 + 3)
+        plt.pcolormesh(X, Y, u_pred, cmap=cmap, shading=shading)
+        plt.colorbar()
+        plt.title(f"Predict ${name}(x,y,t={t:.2f})$")
+        plt.axis("square")
+        plt.tight_layout()
+        plt.axis("off")
+
+        plt.subplot(len(names), 4, index * 4 + 4)
+        plt.pcolormesh(X, Y, u_pred - u_true, cmap=cmap, shading=shading)
+        plt.colorbar()
+        plt.title(f"Absolute Error ${name}(x,y,t={t:.2f})$")
+        plt.tight_layout()
+        plt.axis("square")
+        plt.axis("off")
+
+    if save_path is not None:
+        if save_suffix is not None:
+            figure_path = f"{save_path}_{save_suffix}.png"
+        else:
+            figure_path = f"{save_path}.png"
+        plt.savefig(figure_path, bbox_inches="tight")
+    # plt.show()
+    # return fig
+    plt.close()
+
+
+def generate_movie_2D(
+    preds_y,
+    test_y,
+    test_x,
+    key=0,
+    plot_title="",
+    field=0,
+    val_cbar_index=-1,
+    err_cbar_index=-1,
+    val_clim=None,
+    err_clim=None,
+    font_size=None,
+    movie_dir="",
+    movie_name="movie.gif",
+    frame_basename="movie",
+    frame_ext="jpg",
+    cmap="jet",
+    shading="gouraud",
+    remove_frames=True,
+):
+    "Generates a movie of the exact, predicted, and absolute error fields"
+    frame_files = []
+
+    if movie_dir:
+        os.makedirs(movie_dir, exist_ok=True)
+
+    if font_size is not None:
+        plt.rcParams.update({"font.size": font_size})
+
+    pred = preds_y[key][..., field]
+    true = test_y[key][..., field]
+    inputs = test_x[key]
+    error = pred - true
+
+    Nt, Nx, Ny = pred.shape
+
+    t = inputs[:, 0, 0, 0]
+    x = inputs[0, :, 0, 1]
+    y = inputs[0, 0, :, 2]
+    X, Y = torch.meshgrid(x, y, indexing="ij")
+
+    fig, axs = plt.subplots(1, 3, figsize=(18, 5))
+    ax1 = axs[0]
+    ax2 = axs[1]
+    ax3 = axs[2]
+
+    pcm1 = ax1.pcolormesh(
+        X, Y, true[val_cbar_index], cmap=cmap, label="true", shading=shading
+    )
+    pcm2 = ax2.pcolormesh(
+        X, Y, pred[val_cbar_index], cmap=cmap, label="pred", shading=shading
+    )
+    pcm3 = ax3.pcolormesh(
+        X, Y, error[err_cbar_index], cmap=cmap, label="error", shading=shading
+    )
+
+    if val_clim is None:
+        val_clim = pcm1.get_clim()
+    if err_clim is None:
+        err_clim = pcm3.get_clim()
+
+    pcm1.set_clim(val_clim)
+    plt.colorbar(pcm1, ax=ax1)
+    ax1.axis("square")
+    ax1.set_axis_off()
+
+    pcm2.set_clim(val_clim)
+    plt.colorbar(pcm2, ax=ax2)
+    ax2.axis("square")
+    ax2.set_axis_off()
+
+    pcm3.set_clim(err_clim)
+    plt.colorbar(pcm3, ax=ax3)
+    ax3.axis("square")
+    ax3.set_axis_off()
+
+    plt.tight_layout()
+
+    for i in range(Nt):
+        # Exact
+        ax1.clear()
+        pcm1 = ax1.pcolormesh(X, Y, true[i], cmap=cmap, label="true", shading=shading)
+        pcm1.set_clim(val_clim)
+        ax1.set_title(f"Exact {plot_title}: $t={t[i]:.2f}$")
+        ax1.axis("square")
+        ax1.set_axis_off()
+
+        # Predictions
+        ax2.clear()
+        pcm2 = ax2.pcolormesh(X, Y, pred[i], cmap=cmap, label="pred", shading=shading)
+        pcm2.set_clim(val_clim)
+        ax2.set_title(f"Predict {plot_title}: $t={t[i]:.2f}$")
+        ax2.axis("square")
+        ax2.set_axis_off()
+
+        # Error
+        ax3.clear()
+        pcm3 = ax3.pcolormesh(X, Y, error[i], cmap=cmap, label="error", shading=shading)
+        pcm3.set_clim(err_clim)
+        ax3.set_title(f"Error {plot_title}: $t={t[i]:.2f}$")
+        ax3.axis("square")
+        ax3.set_axis_off()
+
+        #         plt.tight_layout()
+        fig.canvas.draw()
+
+        if movie_dir:
+            frame_path = os.path.join(movie_dir, f"{frame_basename}-{i:03}.{frame_ext}")
+            frame_files.append(frame_path)
+            plt.savefig(frame_path, bbox_inches="tight")
+
+    if movie_dir:
+        movie_path = os.path.join(movie_dir, movie_name)
+        with imageio.get_writer(movie_path, mode="I") as writer:
+            for frame in frame_files:
+                image = imageio.imread(frame)
+                writer.append_data(image)
+
+    if movie_dir and remove_frames:
+        for frame in frame_files:
+            try:
+                os.remove(frame)
+            except:
+                pass
+
+
+def plot_predictions_mhd_plotly(
+    pred,
+    true,
+    inputs,
+    index=0,
+    index_t=-1,
+    name="u",
+    save_path=None,
+    font_size=None,
+    shading="auto",
+    cmap="jet",
+):
+    "Plots images of predictions and absolute error to be saved to wandb"
+    Nt, Nx, Ny, Nfields = pred.shape
+    u_pred = pred[index_t, ..., index]
+    u_true = true[index_t, ..., index]
+
+    ic = inputs[0, ..., 3:]
+    u_ic = ic[..., index]
+    u_err = u_pred - u_true
+
+    x = inputs[0, :, 0, 1]
+    y = inputs[0, 0, :, 2]
+    X, Y = torch.meshgrid(x, y, indexing="ij")
+    t = inputs[index_t, 0, 0, 0]
+
+    zmin = u_true.min().item()
+    zmax = u_true.max().item()
+    labels = {"color": name}
+
+    # Initial Conditions
+    title_ic = f"{name}0"
+    fig_ic = px.imshow(
+        u_ic,
+        binary_string=False,
+        color_continuous_scale=cmap,
+        labels=labels,
+        title=title_ic,
+    )
+    fig_ic.update_xaxes(showticklabels=False)
+    fig_ic.update_yaxes(showticklabels=False)
+
+    # Predictions
+    title_pred = f"Predict {name}: t={t:.2f}"
+    fig_pred = px.imshow(
+        u_pred,
+        binary_string=False,
+        color_continuous_scale=cmap,
+        labels=labels,
+        title=title_pred,
+    )
+    fig_pred.update_xaxes(showticklabels=False)
+    fig_pred.update_yaxes(showticklabels=False)
+
+    # Ground Truth
+    title_true = f"Exact {name}: t={t:.2f}"
+    fig_true = px.imshow(
+        u_true,
+        binary_string=False,
+        color_continuous_scale=cmap,
+        labels=labels,
+        title=title_true,
+    )
+    fig_true.update_xaxes(showticklabels=False)
+    fig_true.update_yaxes(showticklabels=False)
+
+    # Ground Truth
+    title_err = f"Error {name}: t={t:.2f}"
+    fig_err = px.imshow(
+        u_err,
+        binary_string=False,
+        color_continuous_scale=cmap,
+        labels=labels,
+        title=title_err,
+    )
+    fig_err.update_xaxes(showticklabels=False)
+    fig_err.update_yaxes(showticklabels=False)
+
+    return fig_ic, fig_pred, fig_true, fig_err

on_startup.sh

@@ -0,0 +1,5 @@
+#!/bin/bash
+# Write some commands here that will run on root user before startup.
+# For example, to clone transformers and install it in dev mode:
+# git clone https://github.com/huggingface/transformers.git
+# cd transformers && pip install -e ".[dev]"

requirements.txt

@@ -0,0 +1,10 @@
+Cython==3.1.0
+hydra-core==1.3.2
+imageio==2.37.0
+ipython==9.2.0
+mlflow>=2.1.1
+plotly==6.0.1
+tensorly==0.9.0
+tensorly-torch==0.5.0
+termcolor
+wandb

start_server.sh

@@ -0,0 +1,19 @@
+#!/bin/bash
+JUPYTER_TOKEN="${JUPYTER_TOKEN:=huggingface}"
+
+NOTEBOOK_DIR="/mhd-demo"
+
+jupyter labextension disable "@jupyterlab/apputils-extension:announcements"
+
+jupyter-lab \
+    --ip 0.0.0.0 \
+    --port 7860 \
+    --no-browser \
+    --allow-root \
+    --ServerApp.token="$JUPYTER_TOKEN" \
+    --ServerApp.tornado_settings="{'headers': {'Content-Security-Policy': 'frame-ancestors *'}}" \
+    --ServerApp.cookie_options="{'SameSite': 'None', 'Secure': True}" \
+    --ServerApp.disable_check_xsrf=True \
+    --LabApp.news_url=None \
+    --LabApp.check_for_updates_class="jupyterlab.NeverCheckForUpdate" \
+    --notebook-dir=$NOTEBOOK_DIR