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Title: PDE-Net: Learning PDEs from Data

Paper content:
Partial differential equations (PDEs)  play a prominent role in many disciplines such as applied mathematics, physics, chemistry, material science, computer science, etc.
PDEs are commonly derived based on physical laws or empirical observations.
However, the governing equations for many complex systems in modern applications are still not fully known.
With the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored.
Such vast quantity of data offers new opportunities for data-driven discovery of hidden physical laws.
Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models.
The basic idea of the proposed PDE-Net is to learn differential operators by learning convolution kernels (filters), and apply neural networks or other machine learning methods to approximate the unknown nonlinear responses.
Comparing with existing approaches, which either assume the form of the nonlinear response is known or fix certain finite difference approximations of differential operators, our approach has the most flexibility by learning both differential operators and the nonlinear responses.
A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network.
These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory).
We also discuss relations of the PDE-Net with some existing networks in computer vision such as Network-In-Network (NIN) and Residual Neural Network (ResNet).
Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.
Differential equations, especially partial differential equations(PDEs), play a prominent role in many disciplines to describe the governing physical laws underlying a given system of interest.
Traditionally, PDEs are derived based on simple physical principles such as conservation laws, minimum energy principles, or based on empirical observations.
Important examples include the NavierStokes equations in fluid dynamics, the Maxwell's equations for electromagnetic propagation, and the Schrödinger's equations in quantum mechanics.
However, many complex systems in modern applications (such as many problems in climate science, neuroscience, finance, etc.) still have eluded mechanisms, and the governing equations of these systems are only partially known.
With the rapid development of sensors, computational power, and data storage in the last decade, huge quantities of data can be easily collected and efficiently stored .
Such vast quantity of data offers new opportunities for data-driven discovery of potentially new physical laws.
Then, one may ask the following interesting and intriguing question: can we learn a PDE model (if there exists one) from a given data set and perform accurate and efficient predictions using the learned model?One
of earlier attempts on data-driven discovery of hidden physical laws is by BID0 and BID19 . Their
main idea is to compare numerical differentiations of the experimental data with analytic derivatives of candidate functions, and apply the symbolic regression and the evolutionary algorithm to determining the nonlinear dynamical system. Recently
, BID1 , BID18 , BID17 and BID21 propose an alternative approach using sparse regression. They construct
a dictionary of simple functions and partial derivatives that are likely to appear in the unknown governing equations. Then, they take
advantage of sparsity promoting techniques to select candidates that most accurately represent the data. When the form of
the nonlinear response of a PDE is known, except for some scalar parameters, presented a framework to learn these unknown parameters by introducing regularity between two consecutive time step using Gaussian process. More recently, introduced
a new class of universal function approximators called the physics informed neural networks which is capable of discovering nonlinear PDEs parameterized by scalars.These recent work greatly advanced the progress of the problem. However, symbolic regression
is expensive and does not scale very well to large systems. The sparse regression method
requires to fix certain numerical approximations of the spatial differentiations in the dictionary beforehand, which limits the expressive and predictive power of the dictionary. Although the framework presented
by ; is able to learn hidden physical laws using less data than the approach of sparse regression, the explicit form of the PDEs are assumed to be known except for a few scalar learnable parameters. Therefore, extracting governing
equations from data in a less restrictive setting remains a great challenge.The main objective of this paper is to accurately predict the dynamics of complex systems and to uncover the underlying hidden PDE models (should they exist) at the same time, with minimal prior knowledge on the systems. Our inspiration comes from the
latest development of deep learning techniques in computer vision. An interesting fact is that some
popular networks in computer vision, such as ResNet BID9 b) , have close relationship with PDEs BID4 BID6 BID8 BID20 BID13 . Furthermore, the deeper is the network
, the more expressive power the network possesses, which may enable us to learn more complex dynamics arose from fields other than computer vision. However, existing deep networks designed
in deep learning mostly emphasis on expressive power and prediction accuracy. These networks are not transparent enough
to be able to reveal the underlying PDE models, although they may perfectly fit the observed data and perform accurate predictions. Therefore, we need to carefully design the
network by combining knowledge from deep learning and applied mathematics so that we can learn the governing PDEs of the dynamics and make accurate predictions at the same time. Note that our work is closely related to BID4
where the authors designed their network based on discretization of quasilinear parabolic equations. However, it is not clear if the dynamics of image
denoising has to be governed by PDEs, nor did the authors attempt to recover the PDE (should there exists one).In this paper, we design a deep feed-forward network
, named PDE-Net, based on the following generic nonlinear evolution PDE DISPLAYFORM0 The objective of the PDE-Net is to learn the form of the nonlinear response F and to perform accurate predictions. Unlike the existing work, the proposed network only
requires minor knowledge on the form of the nonlinear response function F , and requires no knowledge on the involved differential operators (except for their maximum possible order) and their associated discrete approximations. The nonlinear response function F can be learned using
neural networks or other machine learning methods, while discrete approximations of the differential operators are learned using convolution kernels (i.e. filters) jointly with the learning of the response function F . If we have a prior knowledge on the form of the response
function F , we can easily adjust the network architecture by taking advantage of the additional information. This may simplify the training and improve the results.
We will also discuss relations of the PDE-Net to some existing
networks in computer vision such as Network-In-Network (NIN) and ResNet. Details are given in Section 2.In Section 3 and Section 4, we
conduct numerical experiments on a linear PDE (convection-diffusion equation) and a nonlinear PDE (convection-diffusion equation with a nonlinear source). We generate data set for each PDE using high precision numerical
methods and add Gaussian noise to mimic real situations. Our numerical results show that the PDE-Net can uncover the hidden
equations of the observed dynamics, and can predict the dynamical behavior for a relatively long time, even in a noisy environment.A particular novelty of our approach is that we impose appropriate constraints on the learnable filters in order to easily identify the governing PDE models while still maintaining the expressive and pre-dictive power of the network. This makes our approach different from existing deep convolutional
networks which mostly emphasis on the prediction accuracy of the networks, as well as all the existing approaches of learning PDEs from data which assume either the form of the response function is known or have fixed approximations of the differential operators. In other words, our proposed approach not only has vast flexibility
in fitting observed dynamics and is able to accurately predict its future behavior, but is also able to reveal the hidden equations driving the observed dynamics. The constraints on the filters are motivated by the earlier work of
BID2 ; where general relations between wavelet frame transforms and differential operators were established. In particular, it was observed in that we can relate filters and finite
difference approximation of differential operators by examining the orders of sum rules of the filters (an important concept in wavelet theory and closely related to vanishing moments of wavelet functions). These constraints on the filters may also be useful in network designs
for machine learning tasks in computer vision.2 PDE-NET: A FLEXIBLE DEEP ARCHTECTURE TO LEARN PDES FROM DATA Given a series
of measurements of some physical quantities {u(t, ·) : DISPLAYFORM1 : Ω → R, we want to discover the governing PDEs of the data. We assume that the observed data are associated with a PDE that takes the following
general form: DISPLAYFORM2 (1) Our objective is to design a feed-forward network, named the PDE-Net, that approximates the PDE (1) in the way that: 1) we can predict the dynamical behavior of the equation for as long time as possible
; 2) we are able to reveal the form of the response function F and the differential operators
involved. There are two main components of the PDE-Net that are combined together in the same network
: one is automatic determination on the differential operators involved in the PDE and their discrete approximations; the other is to approximate the nonlinear response function F . In this section, we start with discussions on the relation between convolutions and differentiations
in discrete setting.
This section presents numerical results of training the PDE-Net using the data set described in the previous subsection.
We will specifically observe how the learned PDE-Net performs in terms of prediction of dynamical behavior and identification of the underlying PDE model.
Furthermore, we will investigate the effects of some of the hyper-parameters (e.g. size of the filters, number of δt-blocks) on the learned PDE-Net.
This section presents numerical results of the trained PDE-Net using the data set described in Section 4.1.
We will observe how the trained PDE-Net performs in terms of prediction of dynamical behavior and identification of the underlying PDE model.
In this paper, we designed a deep feed-forward network, called the PDE-Net, to discover the hidden PDE model from the observed dynamics and to predict the dynamical behavior.
The PDE-Net consists of two major components which are jointly trained: to approximate differential operations by convolutions with properly constrained filters, and to approximate the nonlinear response by deep neural networks or other machine learning methods.
The PDE-Net is suitable for learning PDEs as general as in (1).
However, if we have a prior knowledge on the form of the response function F , we can easily adjust the network architecture by taking advantage of the additional information.
This may simplify the training and improve the results.
As an example, we considered a linear variable-coefficient convection-diffusion equation.
The results show that the PDE-Net can uncover the hidden equation of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.
Furthermore, having deep structure (i.e. multiple δt-blocks) and larger learnable filters can improve the PDE-Net in terms of stability and can prolong reliable predictions.
As part of the future work, we will try the proposed framework on real data sets.
One of the important directions is to uncover hidden variables which cannot be measured by sensors directly, such as in data assimilation.
Another interesting direction which is worth exploring is to learn stable and consistent numerical schemes for a given PDE model based on the architecture of the PDE-Net.

assistant: text: This paper proposes a new feed-forward network, call PDE-Net, to learn PDEs from data.