Learning Incompressible Fluid Dynamics from Scratch - Towards Fast, Differentiable Fluid Models that Generalize
Authors: Nils Wandel
Michael Weinmann
Reinhard Klein
Venue: ICLR 2021 Spotlight (2021)
Keywords: Unsupervised Learning
Fluid Dynamics
U-Net

Abstract: Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer-generated imagery to computer-aided design in research and development. However, solving the partial differential equations of incompressible fluids is a challenging task and traditional numerical approximation schemes come at high computational costs. Recent deep learning based approaches promise vast speed-ups but do not generalize to new fluid domains, require fluid simulation data for training, or rely on complex pipelines that outsource major parts of the fluid simulation to traditional methods.

In this work, we propose a novel physics-constrained training approach that generalizes to new fluid domains, requires no fluid simulation data, and allows convolutional neural networks to map a fluid state from time-point t to a subsequent state at time t+dt in a single forward pass. This simplifies the pipeline to train and evaluate neural fluid models. After training, the framework yields models that are capable of fast fluid simulations and can handle various fluid phenomena including the Magnus effect and Kármán vortex streets. We present an interactive real-time demo to show the speed and generalization capabilities of our trained models. Moreover, the trained neural networks are efficient differentiable fluid solvers as they offer a differentiable update step to advance the fluid simulation in time. We exploit this fact in a proof-of-concept optimal control experiment. Our models significantly outperform a recent differentiable fluid solver in terms of computational speed and accuracy.

Published as a conference paper at ICLR 2021
LEARNING INCOMPRESSIBLE FLUID DYNAMICS FROM SCRATCH TOWARDS FAST, DIFFERENTIABLE FLUID MODELS THAT GENERALIZE
Nils Wandel
Michael Weinmann Department of Computer Science
Department of Computer Science
University of Bonn
University of Bonn mw@cs uni uni-bone -bonn. de
wandelnêcs, uni-boni mai-bonn.co de
Reinhard Klein
Department of Computer Science University of Bonn rk@cs. uni -bonn. de
ABSTRACT
Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer-generatee imagery to computer-aided design in research and development. However, solving the partial differential equations incompressible fluids is a challenging task and traditional numerical approximation schemes come high computational costs. Recent deep learning based approaches promise vast speed-ups but do not generalize to new fluid domains, require fluid simulation data for training, rely on complex pipelines that outsource major parts of the fluid simulation to traditional methods.
In this work, propose novel physics- constriied training approach that generalizes to new fluid domains, requires no fluid simulation data, and allows convolutional neural networks to map fluid state from time-point t to subsequent state at time + dt in a single forward pass. This simplifies the pipeline train and evaluate neural fluid models. After training, the framework yields models that are capable of fast fluid simulations and can handle various fluid phenomena including the Magnus effect and Kármán vortex streets. We present an interactive real-time demo to show the speed and generalization capabilities of our trained models. Moreover, the trained neural networks are efficient differentiable fluid solvers as they offer differentiable update step to advance the fluid simulation in time. We exploit this fact in a proof-of-concee optimal control experiment. Our models significantly outperform recent differentiable fluid solver terms of computational speed and accuracy.
INTRODUCTION
Simulating the behavior of fluids by solving the incompressible Navier-Stokes equations of great importance for a wide range of applications and accurate well as fast fluid simulations are long-standing research goal. top of simulating the behavior of fluids, several applications such as sensitivity analysis of fluids gradient-based control algorithms rely on differentiable fluid simulators that allow to propagate gradients throughout the simulation (Holl al. (2020)).
Recent advances in deep learning aim for fast and accurate fluid simulations but rely on vast datasets and or do not generalize to new fluid domains. Kim et al. (2019) present framework to learn parameterized fluid simulations and allow to interpolate efficiently in between such simulations. However, their work does not generalize to new domain geometries that lay outside the training data. Kim & Lee (2020) train RNN-GAN that produces turbulent flow fields within pipe domain, but do not show generalization results beyond pipe domains. Xie et al. (2018) introduce a tempoGAN perform temporally consistent superresolution of smoke simulations. This allows to

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produce plausible high-resolution smoke-density fields for arbitrary low-resolution inputs, but our fluid model should output a complete fluid state deseription consisting of a velocity and a pressure field. Tompson al. (017)) present how Helmholtz projection step be learned to accelerate Fulerian fluid simulations. This method generalizes to new domain geometries, but a particle tracer is needed to deal with the advection term of the Navier-Stokes equations. Furthermore, as Fulerian fluids do not model viscosity, effects like e.g. the Magnus effect or Kármáá vortex streets cannot be simulated. [eneva & Zabaras (2020) propose physics-informed framework to learn the entire update step for the Burgers equations in 1D and 2D, but no generalization results for new domain geometries are demonstrated. All of the aforementioned methods rely on the availability of vast amounts of data from fluid-solvers such as FEniCS, OpenFOAM methods do not generalize well or outsource a major part of the fluid simulation to traditional methods such as low-resolution fluid solvers or particle tracer.
Mantaflow. Most of these
In this work, we propose a novel unsupervised training framework to learn incompressible fluid dynamics from scratch. It does not require any simulated fluid-data (neither as ground truth data, nor to train an adversarial network, nor to initialize frames for a physics-constrained loss) and generalizes to fluid domains unseen during training. It allows CNNs to learn the entire update-step of mapping a fluid domain from time-point to t + dt without having to rely on low resolution fluid-solvers or particle-tracer. In fact, we will demonstrate that physics- constrained loss function combined with a simple strategy to recycle fluid-data generated by the neural network at training time suffices to teach CNNs fluid dynamics on increasingly realistic statistics of fluid states. This drastically simplifies the training pipeline. Fluid simulations get efficiently unrolled in time by recurrently applying the trained model on fluid state. Furthermore, the fluid models include viscous friction and handle effects such as the Magnus effect and Kármán vortex streets. On top of that, we show by a gradient- based optimal control example how backpropagation through time can be used to differentiate the fluid simulation. Code and pretrained models are publicly available at htttt::////gtt t/gbbhab..........//.../...... commas com/aschethor/ Unsupervised Deep Learning of Incompressible Fluid Dynamics/
2 RELATED WORK
In literature, several different approaches can be found that aim to approximate the dynamics of PDEs in general and fluids in particular with efficient, leasning-based surrogate models.
Lagrangian methods such as smoothed particle hydrodynamcs (SPH) Gingold & Monaghan 1977) handle fluids from the perspective of many individual particles that move with the velocity field.
Following this approach, learning-based methods using regression forests by Ladický et al. 2015), graph neural networks by MMrwwaa et al.] (2018); Li et al. (2019) and continuous convolutions by Ummenhofer et al. (2020) have been developed. In addition, Smooth Particle Networks (St-Nets) Schenck&& & Fox 2018) allow for differentiable fluid simulations within the Lagrangian frame of reference. These Lagrangian methods are particularly suitable when a fluid domain exhibits large, dynamic surfaces (e.g. waves or droplets). However, to simulate the dynamics within a fluid domain accurately, Eulerian methods, that treat the Navier-Stoke equations in fixed frame of reference, are usually better suited.
Continuous Eulerian methods allow for mesh-free solutions by mapping domain coordinates (c.g. J.y..) directly onto field values (e.g. velocity 5 / pressure p) (Sirignano & Spiliopoulos (2018): Grohs et al.] (2018); Khoo et al. (2019). Recent applications focused on flow through porous media (7hu & Zabaras (2018): Zhu et al. (2019): "Tripathy & Bilionns] (2018). fluid modeling (Yang et al. (016)) Raissi et al. (2018)), turbulence modeling (Geneva & Zabaras (2019): Ling ct al. (2016) and modeling of molecular dynamics (Schöberl et al. (2019)). Training is usually loss functions that penalize residuals of the underlying PDEs. Similar to approach, RRaissi et al. (2019) uses vector potentials to obtain continuous divergence-free velocity fields to approximate the incompressible Navier-Stokes equations. Continuous methods return smooth, accurate results and can overcome the curse of dimensionality of discrete techniques in -igh--imensional PDEs (Grohs et al. (2018)). However, these networks are trained on a specific domain and cannot generalize to new environments or be used in interactive scenarios.
based on physics-constraing
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Discrete Eulerian methods, on the other hand, aim to solve the underlying PDEs a grid and early work dates back to Harlow & Welch ( (1965) and Stam (1999). Accelerating such traditional works major field of research and all of the methods mentioned in the
with deep learning techniques introduction fall into this category. Further methods include the approach by (Thuerey et al. (2019) to learn solutions of the Reynolds-averaged Savier-Stokes equations for airfoil flows, but requires large amounts of training data and does not generalize beyond airfoil flows. In the work by Um al. (2020), correction step learned that brings solutions of low-resolution differentiable fluid solver closer to solutions of a high-resolution fluid simulation. However, generalization results for new domain geometries were not presented. The works of Mohan et al. (2020) and Kim et al. (2019) show that vector potentials are suitable to enforce the inconpressibility constraint in fluids but do not generalize to new fluid domains beyond their training data.
METHOD
In this section, we briefly review the incompressible Navier-Stokes equations. which are be solved by the neural network. Then, we explain how the Helmholtz decomposition can be exploited to ensure incompressibility within the fluid domain. Furthermore, we provide details of our discrete spatio-temporal fluid representation and introduce the fluid model. Afterwards, we formulate physics- physics-constrained loss function based on residuals of the Navier-Stoker equations and introduce pressure regularization term for very high Reynolds numbers. Finally, we explain the unsupervised training strategy.
3.1 INCOMPRESSIBLE AAVIER-STOKES EQUATIONS
Most fluids be modeled with the incompressible Navier-Stokes equations set of non-linear equations that describe the interplay of a velocity field 7 and a pressure field within fluid domain Ω:
V E = 0
incompressibility on Ω
(1)
pē p p ( 00
( Ot + (0 V Vp MA++ f
conservation of momentum on Ω
Here, P describes the fluid density and μ the viscosity. Equation T states that the fluid incompressible and thus ö is divergence- free. Equation 2 states that the change in momentum of fluid partieles must correspond to the sum of forces that arise from the pressure gradient, viscous friction and external forces. Here, external forces the fluid (such as e.g. gravity) be neglected, so we f
These incompressible Navier-Stokes equations shall be solved by CNN given initial conditions 5" and pd at the beginning of the simulation and Dirichlet boundary conditions which constrain the velocity field at the domain boundary ØΩ:
Ud
Dirichlet boundary condition on an
3.2 HELMHOLTZ DECOMPOSITION
A common method to ensure incompressibilltyy of fluid (see Equation 1 is to project the flow field onto the divergence-free part of its Helmholtz decomposition. The Helmholtz theorem states that every vector field if can be decomposed into a curl-free part (Vq) and a divergence-free part (V x a):
Vg + x
(4)
Note, that V (Vq) = Ö and (V x σ) = 0. The Helmholtz projection consists of solving the Poisson problem V J Aq for q. followed by substracting Vq from the original flow field. However, solving the Poisson equation on arbitrary domains comes at high computational costs for classical methods and one has rely e.g. conjugate gradient methods to approximate its solution.
Here, we propose a different approach and directly try to learn a vector potential ă with 전 -= V x ā. This ensures that the network outputs fivergence-free velocity field within the domain Ω and
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automatically solves Equation I
In this work, consider 2D fluid simulations, only the z- component of i as, is interest since vs and all derivatives with respect to the z-axis are zero:
0,a; 0_ay
0ya= 0za=
Ur Byy
V x ā
0.ag 0, Og dyas)
(5)
3.3 DISCRETE SPATIO-TEMPORAL FLUID REPRESENTATION
Marker-And-Cell (MAC) grid To solve the Navier-Stokes equations, represent the relation
between dz Up, Uyy on a 2D staggered marker-and-cell (MAC) grid (see Figure lak. Therefore, we
discretise time and space follows:
(we) p(a, y. (v))
á(z, y. =
o(., v. =
ph.j
(6)
(az)
Obtaining gradient, divergence, Laplace and curl operations on this grid with finite differences is straight forward and can be efficiently implemented with convolutions (see appendix A).
x
Qt+dt va ++t
U-Net
6
(U+1)
features
P
P
14
curl
(+UJ)
[1+1-1)
a²
a a++tt
p
-++t
(a) Layout of Staggered Marker- And-Cell (MAC) grid in 2D.
(b) Diagram of the fluid model. By recurrently applying the model on the fluid state (p" and a²), we can unroll the fluid simulation in time.
Figure 1: MAC grid and diagram of the fluid model.
Explicit, Implicit, EupllccttExxplici (IMEX) time integration methods time domain is needed to deal with the time-derivative of the velocity field becomes:
The discretization the
0u in Equation 2 which Øt
T
++
+ (IT
V
Vpttdt
DAT +f
(7)
dt
The goal to take as large as possible timesteps while maintaining stable and accurate solutions. Stability and accuracy largely depend on the definition of v"". In literature, choosing ed = of is often referred to explicit integration methods and frequently leads to unstable behavior. Choosing ot dl usually associated with implicit integration methods and gives stable solutions the cost of numerical dissipation. Implicit-Explee (IMEX) methods, which set v² (v' w'+dt)/2 are a compromise between both methods and considered to be more accurate but less stable than implicit methods.
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3.4 FLUID MODEL
We represent the fluid dynamics by a recurrent model that maps the fluid state p', a² for timestep I and the domain description 2 gf+dt to the fluid state p ah of the next timestep. Here, consider
p' describes the pressure field and at describes the vector potential of th For 1 initial states yd 0 and a Ô. however, other initial conditions could be considered well. Q is binary mask that contains the domain geometry and is for the fluid domain and everywhere else. For the boundary of the domain, simply take the inverse of Ω2: 8Ω = Ω. t represents the Dirichlet boundary conditions and contains velocity field that must be matched by oh+ at the domain boundaries. Figure Ib shows diagram of the fluid model. First, (p', d², D4+ds Ud txtd)) are taken derive slightly more meaningful feature representation that comprises (p', a', V a", f2* a [2 V 12 pp. 00 sttt)) These features be very efficiently computed with convolutions and are then fed into U-Net (Ronneberger et al. (2015) with a reduced number of channels (the exact network configuration be found in appendix B). The mean of the U-Net output set to () in order keep y and Ø well defined and prevent drifting offset values. Finally, the output is added to p and a' to obtain the updated fluid state p'+-dt and a
3.5 PHYSICS PHYSICS-CONSTRANEE CONSTRAINED LOSS FUNCTION
Using the residuals of the Navier-Stoker equations (Equations and can formulate the following loss terms on and 0?:
L V
divergence loss on Ω
(8)
+ (T V + Vp pAύ- - F
Lp
momentum loss on Q
(9)
0L
La 1 2 lll
boundary loss on OQ
(10)
Combining the described loss terms, we obtain the following loss function:
I. aL + BI.p + T.b
(11)
where a, 8, 7 are hyperparameters that weight the contributions of the different loss terms. Note that if we use a vector potential ग = V x ā, Lu = 0 is automatically fulfilled and we can set = 0. This loss function can be computed very efficiently with convolutions in O(N) (where N number of grid cells), whereas solving the Savier-Stoker equations explicitly would be computationally lot more expensive. For detailed descriptions regarding the fully discretized foss-function, we refer to appendix
3.6 PRESSURE REGULARIZATION
For very high Reynolds numbers (see Equation [3) viscous friction cannot dissipate enough energy out of the system. This leads unrealistic gradients i and p. For such cases, we introduce an additional regularization term for the loss function (1] that can be traded off with Lp to stabilize training:
and inviscid flows, training becomes unstable as
L, = |
(12)
The intuition behind this regularization term is, that we want to penalize unrealistically high energies in the pressure field.
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3.7 TRAINING STRATEGY
Training starts with initializing a pool {22, (va) (a=)), of randomized domains Dg and boundary conditions (va) as well as initial conditions for the vector potential and pressure fields that we both set to zero ((a2)} = 0 and p.g = 0). The resolution of our training domains is 100x300 grid cells and example-domain of the training pool are shown in appendix C Note that our training pool does not rely on any previously simulated fluid-data.
At each training step, a random mini-batch ( (va)k> (az) ph. Emiiibbaih)) is drawn from the pool and fed into the neural network which is designed to predict the velocity (ix+tt V x dx att and pressure (pt+tt fields of the next time step. Based physics-constrained foss-function (Equation I. we update the weights of the network using the Adam optimizer (Kingma & Ba (2015). At the end of each training step, the pool is updated by replacing the old vector potential and pressure fields .P. dt This recycling strategy fills the training pool with more and more realistic fluid states as the model becomes better at simulating fluid dynamics.
(ag)te pk by the newly predicted ones (az))+d d+
From time to time, old environments of the training pool are replaced by new randomized environments and the vector potential as well as the pressure fields are reset to O. This increases the variance of the training pool and helps the neural network to learn "cold starts" from O-velocity and O-pressure fields.
Besides the fluid model described above, which we denote as ä-Net in the following, we also trained an ablation model, ö-Net, that directly learns to predict the velocity field without a vector potential. For the implementation of both models, we used the popular machine learning framework Pytorch and trained the models on a NVidia GeForce RTX 2080 Ti. Training converged after about day. The hyperparameters in the loss-function for the 7-Net were B = and for choosing a higher weight for the loss term La than for Lp was the observation, that errors in La can lead to unrealistic flows leaking through boundaries. For the ablation study (ū-Net), we used a = 100, B = 1, = 0.001. Here, we had to choose a very high weight for La to ensure incompressibilityy of the fluid, otherwise unrealistic source and sink effects start to appear. For Lt, on the other hand, we used a very low weight as the boundary conditions can be trivially learned by the T-Net. We used these parameter settings for all experiments.
20. The reason
4 RESULTS
To evaluate the potential of our method, assess its ability reproduce physical effects such as Kármán vortex streets and the Magnus effect. In addition, we demonstrate its generalization capability and real-time performance. Finally, we test the fluid models quantitatively.
4.1 QUALITATIVE EVALUATION
Qualitative analysis of wake dynamies Qualitative effects in fluid dynamics such as the wake dynamics behind an obstacle are closely related to the Reynolds number. It is dimensionless quantity defined by:
P || D
Re
(13)
1
Here, is the fluid density, ||²|| is the fluid speed, D is the diameter of the obstacle, and is the viscosity. (We use the units of the grid).
We retrained models for different values of μ and to compare the fluid behavior for wide range of Reynolds numbers. Figure[]shows that the trained models are able to predict the wake dynamics behind an obstacle in good accordance with qualitative expectations from fluid dynamics. As rule of thumb, for Re < 1, the flow becomes time-reversible. This can be noticed in Figure 2a by the symmetry of the flow before and after the obstacle and the nearly constant pressure gradient within the pipe. Starting from Re z 10, the flow is still laminar but a static wake is forming behind the obstacle (see Figure 2bp. For Reynolds numbers Re > 90, Kármán vortex streets start appear (see Figure 2c). A Kärmán vortex street consists of clock and counterclockwis. spinning vortices that generated at the obstacle and then start moving in a regularly oscillating pattern with the

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flow. For very large Reynolds numbers inviscid flows, the flow field becomes turbulent, which can be recognized by the irregular patterns behind the obstacle in Fig 2d
(a) Re=0.6, 5.p 0.2
(b) Re=30, 1s = 0.5,p
(c) Re=600, 0.1, p
(d) Re-+ oc, 14 0,p (obtained with regularization on Vp)
Figure 2: After training, models are able to show correct wake flow dynamics for wide range of different Reynolds numbers. (D 30, || 0.5). Streamlines indicate flow direction, linewidth indicates speed and colors represent the pressure field (blue: low pressure yellow: high pressure).
Magnus effect The Magnus effect appears when a flow interacts with rotating body, It is widely known e-g. sports such as soccer or tennis where spin used to deflect the path ball. The reason for the deflection stems from a low pressure field where the surface of the object moves along flow direction and high pressure field where the object surface moves against the flow. Figure Ba shows, that our models are able to reproduce the Magnus effect around rotating cylinder.
(a) Magnus effect clock-wise turning cylinder.(b) Generalization example: Note that the fluid model never been confronted with wing- profiles during training.
Figure 3: Our models feature the Magnus effect and generalize new fluid domains. Further examples presented appendix and the video.
Analysis of generalization capability We tested the networks capability to generalize to objects seen during training. Figure[b]shoov the networks capability to meet boundary conditions of an airfoil and return plausible pressure field that produces lift (see low pressure on top of wing). Note that contrast the approach by Thuerey et al. (2019), which learns simplified, time-averaged solutions of the Navier-Stoke equations, Navier-Stokes equations for an airfoil without relying on any ground truth data or having seen airfoil- geometries during training. In fact, the network was only trained on simple randomized domains as highlighted in appendix and Figure Possible reasons for the networks generalization capabilities are:
method able simulate the full incompressible
During training, the network gets confronted with infinite number of different flow- fields and randomized domain configurations because the training pool gets updated at every training step. This prevents the network from over-fitting.

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The dynamics of fluid-particle mostly determined by its local neighborhood / surrounding particles. This means, the update step for a certain cell on the MAC grid is mostly determined by close neighboring MAC-grid cells. Since more complicated shapes can be seen locally as a composition of basic shapes (e.g. the front of the wing can be locally regarded as cylinder), it suffices to train basic shapes that provide the network with enough examples to generalize to more complicated shapes.
Further generalization examples are provided in appendix D
Real-time capability The fluid simulation be easily parallelized and takes low computational costs as one time-integration step consists just of a single forward pass through a convolutional neural network. This enables for example interactive real-time simulations. We implemented demo that allows to interact with a fluid by moving obstacles, rotating spheres and changing the flow speed within a pipe (see video in supplementary material and source code). Our method runs at 250 timesteps per second on a 100x300 grid. In the respective experiments, we used N NVidia GeForce RTX 2080 Ti consuming about 860 MB of GPU memory.
4.2 QUANTITATIVE EVALUATION
We compare method (Ø-Net) quantitatively with PhiFlow by Holl et al. (2020). Phiflow is recent, open source, differentiable fluid simulator based on a MAC grid data structure. Furthermore, provide ablation study (ū-Net) that does not make use of the Helmholtz decomposition but directly works on the velocity field T.
Quantitative comparison of different fluid solvers is challenging, their performance is highly dependent on factors like the geometry of the domain, fluid parameters such as viscosity or density, flow speed the timestep of the integrator. As benchmarks for fluid simulations MAC grids not yet available, we built a simple toy domain on a 100 x 100 grid which simulates a flow around an obstacle within pipe (more details are provided in appendix Eb.
First, we compared the computational speed on CPU and GPU by comparing the integration time-steps per second (see Table I). The ö-Net as well as the ä-Net significantly faster than PhiFlow (1 Ix on CPU and 40x on GPU) as they do not rely on an iterative conjugate gradient solver but instead use single forward pass through a convolutional neural network that can be easily parallelized on a GPU. To provide a fair comparison on La. we set the velocity field at the boundaries equal to Va. This enables us to compute La for the 7-Net architecture on the domain boundaries which would otherwise have zero divergence everywhere. This way, T.a can be interpreted metric on how well the orthogonal components of the Dirichlet boundary conditions are met (i.c. no flow leaks through the boundaries). For dt 4, outperformed Phiflow by several orders of magnitude. For both, La and Lp, the ä-Net architecture significantly outperformed the more naive j-Net approach.
Furthermore, we investigated stability by evaluating the evolution of L, and La for the a-Net over time (see Figure 4). As the fluid state is initialized with a, and p 0. the ö-Net has to perform a cold-start which is the reason for high Tup and La during the first circa 70 steps. Afterwards, the ä-Net continues an accurate and stable fluid simulation.
10
L
Method
CPU TTPSI
GPU [TPS
La
L,
10-4
Phiilow
6.2e-4
17 Net (ours)
311
8.66e 7
4.87e.5
10-5
7-Net (ours)
311
5.44c-7
1.56e-5
10-4
Table 1: Quantitative comparison of timesteps per second (TPS) on CPU/ / GPU as well as divergence loss and momentum loss differentiable fluid solvers on a 100x100 grid for viscosity = 0.1, density = 4 and timesteps of size dt =
0
100
200
300
400
500
timestep
Figure 4: Loong term stability of fluid
simulations performed by the ë-Net
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4.3 OPTIMAL CONTROL OF VORTEX SHEDDING FREQUENCY
In this section, we present a proof-of-coneept experiment that aims at controlling the shedding frequency of a Kármán vortex street behind obstacle by changing the flow speed (see Figure Sak. To this end, we exploit our previously trained differentiable fluid models.
speed: 0.852
10
0.75
frequency
target
0.25
target
0.00
100
frequency
Iteration
(a) control setup (domain size: 200x 100 grid cells)
(b) frequency distribution before reaching convergence
optimization
Figure 5: The frequency of vortex streets can be controlled using our differentiable fluid models.
First, we measure the y-component the velocity field v, (t) behind an obstacle (see white box in Figure 5a over 200 time steps. Then, we compute the frequency spectrum Vy( of (ty() using the fast Fourier transform (see Figure 5bb. Now, we want adjust the inflow outflow boundary conditions in Vh such that F = f. Here, f the target frequency. To optimize va. we define loss function L (BIIV f)² and compute the gradients 3e, through time. This is possible since all parts of the loss function including the fluid simulation that is performed by our trained neural fluid model as well as the fast Fourier transform are differentiable. Computing the gradients with standard automatic differentiation library (Pytorch) took 3.5 seconds for all 200 time steps on our 200x 100 domain setup. This is considerably faster than the current tate-ooftthearr differentiable fluid solver by Takahashi et al. (2021) which takes 5.42 seconds for only 30 time steps on a smaller 128x 128 grid. The update steps of da are done using the ADAM- optimizer and converge after approximately 70 iterations (see Figure (5c]. We want emphasize that differentiable fluid simulations are limited to scenarios with low Reynolds numbers as in the presence of turbulences, chaotic behavior will lead exploding gradients.
at with backpropagation
DISCUSSION AND OUTLOOK
In this work, we present unsupervised learning scheme for the incompressible Navier-Stokes equations and introduce fluid model that uses vector potential to output divergence-free velocity fields. Qualitative results of our trained fluid models fluid dynamics for wide range of Reynolds numbers and generalize to unknown fluid domains. Quantitative assessment showed superior performance terms of accuracy and speed compared to Phiflow and an ablation study that directly predicts the velocity field. We present a real-time demo and demonstrate how differentiability can be used a proof-of-concept fluid control scenario. believe that our fluid models can significantly speed up more sophisticated fluid control pipelines such as described by Holl et al. (2020).
good accordance with expectations from
First experiments of extending this approach to 3D deliver encouraging results and are topic of future research. Furthermore, on top of Dirichlet boundary conditions, Neumann boundary conditions and multi-phase domains could be incorporated in future fluid models as well.

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Arvind T. Mohan, Nicholas Lubbers, Daniel Livescu, and Michael Chertkov. Embedding hard physical constraints in neural network coarse-graining of 3d turbulence, 2020.
Damian Mrowca, Chengxu Zhuang, Elias Wang. Nick Haber, Li Fei-Fei, Joshua B. Tenenbaum, and Daniel L. K. Yamins. Flexible neural representation for physics prediction. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS' 18. pp- 8813-8824, Red Hook, NY, USA, 2018. Curran Associates Inc
Maziar Raissi, Alireza Yazdani, and George Em Karniadakis. Hidden fluid mechanics: A navier- stokes informed deep learning framework for assimilating flow visualization data. arXiv preprint arXiv: 1808 04327, 2018
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Maziar Raissi, P. Perdikaris, and George Em Karniadakis. Physics-informe neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686 707, 2019. ISSN 0021- 9991. doi:
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scci /ttpc////ttiilleee/////001199999

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Published as a conference paper at ICLR 2021
A PHYSICS-CONSTRAINE LOSS ON A MAC GRID
As mentioned in Section 3.3 of the paper, our method relies on a staggered marker-and-cell grid representation for the vector potential well the velocity and pressure fields. In the following, provide further details on how to apply this representation to learn incompressible fluid dynamics.
To calculate the velocity field き V of vector potential ă MAC grid in 2D, have to compute the curl as follows:
(Ux)i.s (0z)i+1.j (az)is
(14)
(My)as (az)i. (az)ig+1
this vector potential is inserted into the divergence operator on a MAC grid, we can show that V Ueyy 0 is indeed fulfilled:
V U. = (Va)is+1 - (U2)4s + (By)e41, - (Vg)sss
(15)
((a=)=+1511(aa))) - (az)is+1) ((02)i+13 (a=)as) + ((a))++1) - (a.)+415+1) - ((as)is- - -a22)15+1)
(16)
(17)
Thus, for the ä-Net, the incompressibility equation is automatically fulfilled and no further training on the divergence loss La is required. However, for the ö-Net, the residuals of the divergence are still of importance:
(Ra)
b+
+ (= 0 for ä-Net)
(18)
The residuals of the momentum equation in æ-direction can be computed as follows:
(vo)Es (va)is + (ow)i. (U2) (ua)
(Rp, 1i =p
dt
2
(wa)l's st(owhis (ww))ss (v=) + (wilin ytoWhns ((1)) (1:=)
2
2
+ (pitt
-P1--1) P A(Ua))
(19)
Here, we use the following isotropic Laplace operator:
Asi.j
(1 * $;-1j-1 8;-1.j 8;-1,141
(20)
+2 -
12 si + 8t.j+1
+1*81411j11+ 8i+1. 8411541)
The derivation of the advection term for Rg. bit more complex since on MAC grid, v. and By displaced by half pixel in x-direction and y-direction. To obtain the residuals of the momentum equation in y-direction, Rij.. one has to take (Rp))i. and swap x and y and the indices respectively.
Now, the discretized loss terms can be written as follows:
dt
Σ s ((Ra)
(21)
L d+
Σ
dtd
(((Rpa) )add) + ((Rpy) 22
(22)
dt
Σ
de batdet 2
(23)
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Published as a conference paper at ICLR 2021
Note, that all mentioned operations be efficiently implemented with convolutions. To obtain the final velocities on a square grid, we project the velocity fields the MAC grid back onto the a-grid using linear interpolation:
(Uz) (. + (Uz) Ú = 2 TTy k-j1 (Uy)i.)
(24)
B NETWORK ARCHITECTURE
Our fluid model is based on the U-Net architecture (Ronneberger et al. (2015) with fewer channels (see Figure 6). As the pressure field and vector potential can have an arbitrary offset, we always normalize the mean of the pressure (Ap) and vector potential (Aag) to 0 to keep these fields well- defined and prevent drifting offset values.
features
Ap, Aa
80
1660 80
conv 3x3, ReLu
copy and crop
160 160
920
160
|
max pool 2x2
up-conv 2x2
conv 1x1 normalize
Figure 6: U-Net architecture with fewer channels.
C EXAMPLES OF TRAINING DOMAINS
The domains we used for training consist of 100 x 300 grids. We used 3 different randomized domains exemplary depicted in Figure First, we have boxes with randomized height and width that float on randomized paths inspired by Brownian motion a pipe with randomized flow speed. Second, we have the same setup but replaced the boxes by cylinders with randomized radii and angular velocities order to learn the Magnus effect. Finally, we have a folded pipe system with randomized flow speed, that is randomly flipped along the r-axis.
D FURTHER EXAMPLES OF GENERALIZATION
Note that the network only trained on simple domain geometries presented in appendix C Still, as can be seen in Figure g the network capable of generalizing to far more complicated domain geometries (e.g. shark, car). Figure 8c shows that it can generalize to multiple objects in the scene, although the training set contained at most one object per scene. And Figure shows that can alter the outer boundary conditions well. For real-time simulations, please have a look at our source code and the supplementary video.
E QUANTITATIVE ANALYSIS: THE BENCHMARK PROBLEM
Figure 9 shows the domain Ω and ut on 100 100 grid which was used as the benchmark problem
for quantitative analysis. The flow speed for the inlet and outlet was set 0.5. The timestep of the integrator was set to dt = 4 and the viscosity and fluid density were set to μ = 0.1 and respectively.
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Published as a conference paper at ICLR 2021
(a)
(b)
(c)
(d)
Figure 7: The left column shows S2 (in white) / 00 (in black) and the right column shows T. for three examples of training domains. (Colors indicate the direction and magnitude of th as depicted in Figure Pa
(a) Shark
(b) Car
(c) Smiley
(d) Smiley in cave
Figure 8: Our models generalize to various domain geometries, although being trained only on simple shapes (see Figure 7
QUALITATIVE COMPARISON OF 7-NET AND Ü-NET
We give a qualitative example show the benefits of using vector potential. Figure 10 demonstrates that the 7-Net finds plausible solutions for a folded pipe domain while the J-Net looses most of the flow in the center of the domain. This is in good accordance with quantitative results shown in section [] The folded pipe domain is particularly difficult to learn as the flow field contains long range dependencies to the inlet and outlet (as shown in the bottom row in Figure 77).
G TRAINING WITHOUT RESETTING ENVIRONMENTS
We performed an ablation study to investigate what happens if we do not reset old environments from time to time and, thus, do not continuously present the fluid model with cold starts during training. [Tgure[[[shove that in this case, large error spikes appear in the validation curve. These error spikes appear since the model has troubles perform cold start be in Figure 11b compared
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Published as a conference paper at ICLR 2021
max
(a)
(b)
(c)
Figure a) shows legend for da;; b) shows 12 (in white) 8n (in black) for the benchmark problem; c) shows be the benchmark problem. (Colors indicate direction of vig as depicted a)
(a) ü-Net
ö-Net
Figure 10: Qualitative comparison of a-Net and U-Net in folded pipe domain
to properly trained model (see Figure 4 the model takes longer to perform cold start (ca 100 steps) and converges to solution with high Lyp and La- losses. By resetting the environments from time to time during training, prevent these error spikes shown in Figure 11c
Lp
Lp
10-3
-4
10-4
Loss
10-5
100
10-6
M
10-5
100
200
epoch
timestep
epoch
(a)
(b)
(c)
Figure a) ablation study without resetting environments: validation curve shows large error spikes during training: b) error spike: the fluid model takes longer to perform a cold start and converges to solution with high losses: c) original training with resetting environments: validation curve is stable
15


Reviews:
[{"review_id": "fUKvePOSokq", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper is generally a good contribution especially in the field of machine learning for physics. However, I do have some questions about the novelty of this work and what makes it different from other physics informed approaches. Below are the pros and cons and the I list out a set of comments and additional results that I believe would make the contribution of the paper much more clear. \n\nPros:\n-- A physics informed neural architecture that optimizes the governing PDEs to perform forward integration of the system. Generally a great approach since it does not require high-res numerical simulations\n-- Generalizes to new geometries\n\nCons\n-- I might be missing something, but I generally feel that there is a lack of novelty. Approach is very similar to certain papers I point out in the comments. That however, does not take anything away from the approach proposed and it is still a good contribution once certain concerns (in the comments) are resolved\n-- The authors claim that their network generalizes but I do not see an explanation for it. Even an intuitive explanation would be good.\n\n1) The introduction section of the paper does talk about different approaches to physics informed neural networks especially the Raissi et al., papers and mention that Raissi's approach (https://arxiv.org/pdf/1711.10566.pdf and https://arxiv.org/abs/1711.10561) does not generalize to new domains. Can the authors explain why they believe that their network would automatically generalize to new geometries? Does the introduction of $\\Omega$ and $\\partial \\Omega$ in their input features make their framework generalizable?\n\n2) The use of vector potentials to ensure divergence free by construction inside neural architectures using fixed convolution operations has been shown before By Mohan et al., (https://arxiv.org/pdf/2002.00021.pdf). The authors should cite this paper. \n\n3) In Figure 2, I'm concerned about how well the network satisfies BCs. If possible, I would like to see a plot showing velocity with time during inference near the edges of the obstacle.\n\n4) There is a complete absence of baselines. Sure, the a-net outperforms v-net when it comes to the loss. However, the loss term is $10^{-3}$ for v-net and $10^{-5}$ for a-net. Both are pretty low. How much improvement in inference would one expect?  It is evident that constraining divergence by construction definitely helps in reducing the loss and this has been shown in https://arxiv.org/pdf/2002.00021.pdf. \n\n5) I would like to see a comparison between the a-net prediction and a numerical simulation and possibly a comparison with general PINNs (https://arxiv.org/pdf/1711.10566.pdf)  for the rectangular obstacle. It should clearly show that their approach is more accurate in terms of RMSE error between a-net and numerical simulation as compared to PINN and numerical simulation. What would be more interesting to see is how well the authors satisfy BCs as compared to PINN. Moreover according to the authors PINNs would not generalize to the airfoil shaped obstacle while their network would . This would clearly show an advantage of the proposed mechanism. Without a baseline however, there is not much to prove that this is a better approach.  \n\n6)  The problem solved in this paper is a very simple problem. Even numerical solvers are not expensive when it comes to solving this problem. This physics informed approach is very promising because it can reduce the computational cost immensely (during inference). Would it be possible for the authors to show this for a *real* turbulent flow, e.g. the system considered in the Mohan et al., paper (https://arxiv.org/pdf/2002.00021.pdf) or the Kolmogorov system as shown in this paper (https://advances.sciencemag.org/content/advances/3/9/e1701533.full.pdf) or even the 2D Kraichnan system shown in https://arxiv.org/abs/1808.02983. Unless these approaches generalize to fully turbulent flow, the use for such architectures are limited for real applications\n\nThis is still an important contribution in the field of deep learning for computational physics, specifically in fluid dynamics. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "Wgm7RHh4U6L", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "- Summary\n\nThis paper presents a \"physics-informed\" deep learning model of fluid dynamics. The underlying deep learning architecture employed is a somewhat standard u-net, but one of the proposed method's distinguishing features is that it enforces its adherence to physical behavior at its loss terms, by penalizing predictions that are not incompressible or do not conserve momentum. Notably, this approach allows it to be trained unsupervisedly, without requiring the generation of ground-truth simulations.\n\n- Pros\n\nThe enforcement of the physical behavior as a core feature of the architecture (e.g., through the \"a\" vector and the conservation losses) makes the network to generate stable simulations.\n\nThis also brings with it the interesting benefit of being able to learn without the need for ground truth data, lowering the cost of training the model.\n\nThe proposed model performs favorably, both in terms of speed and error, to phiflow, the current \"reference\" differentiable fluid simulator. Since the proposed method is still deep learning based, it is also differentiable.\n\n\n\n- Cons\n\nMost of the experiments serve only to validate the model's accuracy and its ability to learn a proper fluid simulation. \nThe only practical application demonstrated is in a simple control task, which is not explored too deeply. \nPhiflow (Holl et al., 2020), for example, performs more extensive and diverse evaluation of control settings.\n\n\n- Reasons for score\n\nOverall, given the \"pros\" described above, notably the distinct loss formulation that allows the model to learn unsupervisedly to perform efficient, differentiable fluid simulations, I recommend this paper for acceptance. \n \n\n- Additional comments\n\nIs training improved if actual simulations are used for data, instead of \"cold starts\"? \nIf so how do these compare with training with cold starts (as presented in the paper), both in terms of training speed and in terms of final (test) results?\nDoes starting from more realistic starting points fix the issue of the initial error \"spike\" in Fig 4?\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "IV_P8MF9mk", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes to learn the dynamics of an incompressible fluid via a physics informed loss formulation using an unsupervised training framework. It employs a custom solver that is executed at training time to learn a Navier-Stokes residual with a incompressible (curl of a stream function) formulation. This setup is demonstrated for two dimensional karman vortex streets, and a control example for the magnus effect. \n\nThe paper definitely targets an interesting and relevant direction for machine learning, but it is a (fairly identical) resubmission of a paper that was rejected at NeurIPS. It is of course possible to resubmit papers, but unfortunately in this case it was a fairly clear rejection, and while I cannot compare versions side by side, I think the submitted content is largely the same. As both NeurIPS and ICLR are currently mostly on par in terms of aims, content, and expectations for accepted submissions, I cannot recommend accepting this paper in its current form to ICLR.\n\nFor the previous submission, I believe the following points were the most important issues, as raised in the reviews: Most importantly, the benefits of the proposed method over existing ones (ie supervised from regular solver) are not made clear enough. Then, there is the insufficient quantitative evaluation, that the stream function formulation does not properly extend to 3D, and that the direction was not fully clear (performance vs control applications).\n\nI'm aware that it must be frustrating for authors to be rejected for very similar reasons, but it is likewise not nice for reviewers to repeatedly look at papers searching for differences, and discovering that authors have not taken comments from previous submissions into account. I would be curious to hear in the rebuttal how or whether the authors have updated their submission after the NeurIPS reviews. As already mentioned in the previous cycle: there is merit to the direction of the paper, but I think it is important to make a clear step forward for one or more of the issues raised with the submission (as outlined in the previous paragraph). With such extensions, there should be the potential for an improved evaluation. However, as it stands I don't think this submission is suitable for ICLR.\n\nA minor note, but the title seems somewhat inapproprate to me - physics-informed typically implies that derivatives are computed via autodiff from a network representing the solution. Here, the authors instead discretize the solution on a grid, while a network yields the solution for the next step, and the physical model is evaluated with FDs on the grid. \"Physics-constrained\" (as in the section title) would be a better choice for the title.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "2sJKNxcavnM", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### Summary of my understanding\n\nThe authors propose an unsupervised (or rather, I would say self-supervised) method to build a simulator of incompressible fluid flows based on neural networks. Their neural network is given the pressure and velocity (or its potential) fields at time $t$ and is trained so that it outputs the fields at time $t+dt$ with the \"physics-constrained loss functions.\" The proposed loss functions are designed to make the outputs of the neural net fulfill the Navier-Stokes equation and boundary conditions. They train the neural net on randomly-generated domains and boundary conditions. They show that the learned model outputs qualitatively plausible flows, even if the domain is not exactly handled in the training phase.\n\n### Evaluation\n\nThe paper is easy to follow. The proposed method is technically reasonable. The related work section is comprehensive. While the technical novelty of each component of the proposed method looks moderate, the overall framework would be valuable as a fast differentiable fluid dynamics solver, which the authors claim. The experiment section is convincing to some extent, but it lacks comparison to classical solvers (i.e., somewhat inherently-faithful simulations), which limits our capability to assess the soundness of the results in Section 4.1 and Appendix D. Also, the paper lacks information on the range of hyperparameter search.\n\n### Questions\n\n[Q1]\nHow did you create the randomized boundary conditions $(v_d)_k^0$? I could not find it in the paper.\n\n[Q2]\nThe training strategy described in Section 3.7, especially the pool's update and random renew, seems essential to the performance of the proposed method. Did you performed some ablation study in this regard, or do you have some notes on what happened if the presented strategy was not adopted partially? Such a description would help a reader's understanding much.\n\n[Q3]\nCan you provide the range of hyperparameter search and information on how intensively the search was done? Such information is important for assessing training computational cost.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper introduces a learning framework for solving incompressible Navier-Stokes fluid using a physics informed loss formulation. The PDE is solved on a grid, and the model, implemented via convolutions and a U-Net, is trained to minimize the NS residual. The model is trained on a variety of randomized contexts, in a way that allows training to explore a large number of configurations. The paper presents original contributions compared to previous Physics informed framework (discrete formulation, conditioning on the domain conditions, …). All the reviewers agree that the detailed rebuttal provides answers to their questions and that the contribution is significant, they all have a positive assessment of the paper.
Decision: Accept (Spotlight)