Learning Time-Aware Assistance Functions for Numerical Fluid Solvers
Authors: Kiwon Um
Yun (Raymond) Fei
Philipp Holl
Nils Thuerey
Venue:  (2020)
Keywords: PDEs
convolutional neural networks
numerical simulation
fluids

Abstract: Improving the accuracy of numerical methods remains a central challenge in many disciplines and is especially important for nonlinear simulation problems. A representative example of such problems is fluid flow, which has been thoroughly studied to arrive at efficient simulations of complex flow phenomena. This paper presents a data-driven approach that learns to improve the accuracy of numerical solvers. The proposed method utilizes an advanced numerical scheme with a fine simulation resolution to acquire reference data. We, then, employ a neural network that infers a correction to move a coarse thus quickly obtainable result closer to the reference data. We provide insights into the targeted learning problem with different learning approaches: fully supervised learning methods with a naive and an optimized data acquisition as well as an unsupervised learning method with a differentiable Navier-Stokes solver. While our approach is very general and applicable to arbitrary partial differential equation models, we specifically highlight gains in accuracy for fluid flow simulations.

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LEARNING TIME-AWARE ASSISTANCE FUNCTIONS FOR NUMERICAL FLUID SOLVERS
Anonymous authors Paper under double-blind review
ABSTRACT
Improving the accuracy of numerical methods remains central challenge in many disciplines and is especially important for nonlinear simulation problems. rep- resentative example of such problems is fluid flow, which has been thoroughly studied to arrive at efficient simulations complex flow phenomena. This paper presents data-driven approach that learns improve the accuracy of numerical solvers. The proposed method utilizes an advanced numerical scheme with fine simulation resolution to acquire reference data. We, then, employ neural network that infers a correction to move a coarse thus quickly obtainable result closer to the reference data. We provide insights into the targeted learning problem with different learning approaches: fully supervised learning methods with a naive and an optimized data acquisition well an unsupervised learning method with a differentiable Navier-Stokes solver. While our approach is very general and ap- plicable to arbitrary partial differential equation models, we specifically highlight gains in accuracy for fluid flow simulations.
INTRODUCTION
Numerical methods are central component of many disciplines and widely used for solving variety of linear and nonlinear problems. One of the long-standing targets is fluid flow, which is renowned for its great diversity and complexity in terms of dynamics. Studies computational fluid dynamics have focused numerical simulations for such problems and invested huge efforts in solving spatio-tempont partial differential equations (PDEs) such as the Navier-Stokes equations, which represent the well-established physical model for fluids.
Traditional methods typically improve accuracy with fine discretizations both in space and time. While the methods and computing power for numerical simulation have seen advances in recent years, there is still a pressing need for better efficiency and accuracy. For most practical applications of computer simulations, we are still far away from fully resolving all necessary scales of nature around us (Verma et al. 2018 Cummins et al. 2018).
To tackle this problem, we propose data-driven approach that "assists" given numerical method to improve its accuracy. To this end, we introduce a first learning-based approach that puts special emphasis on the time dimension. We demonstrate of fluids: supervised version with an optimization algorithm for acquisition of temporally con- strained correction data and an unsupervised version with a differentiable PDE solver that allows us to autonomously takes into account temporal information when training. We compare advantages and disadvantages of both approaches, and our experiments show that, using our trained models, the simulation accuracy of the given solver can be significantly improved. In all cases, our trained mod- els yield improved dynamics, and the learned assistance function lets a coarse simulation reproduce the behavior of the reference data more closely. In particular, demonstrate the improvements of our approach over ad-hoc learning approaches.
variants to achieve this goal in the context
RELATED WORK
Data-driven approaches were shown be highly effective for inferring unknown PDEs and coeffi- cients of corresponding terms (Brunton et al. (016). Their capabilities were investigated for differ- problems, among others the Navier-Stokes equations (Rudy et al. 2017 Schaeffer 2017). Deep

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learning methods were successfully applied for learning stencils of advection diffusion problems (Bar-Sinai et al., 2018), to discover PDE formulations (Long et al., (017)) and to analyze families of Poisson equations (Magill et al.] 2018). While identitying governing equations represents in- teresting and challenging task, we instead focus on a general method to improve the solutions of chosen solution spaces.
Due to the complexity and nonlinear nature of fluid dynamics, they remain a challenging problem, and accordingly data-driven approaches including deep learning have received a lot of attention to reach an efficient solution for such problems (Kutz) (0177) Particularly, a turbulence modeling has been a representative objective this context (Duraisamy et al. 2015 Maulik & San 2017 Beck iet al. 2018; Mohan et al.) 2019). Moreover, both steady (Guo et al.) (016) and unsteady (Morton et al.
2018 flows as well as multiphase (Gibou al. 2018) flows have been investigated with deep learning based approaches. Additionally, the convolutional neural network (CNN) based methods
were studied for airfoil flow problems (Thuerey et al. 2018 Zhang et al. 2018).
For fluid simulation, particularly in the field of computer graphics, data-driven approaches have been considered as an efficient alternative to replace computationally expensive steps of numerical processes (Ladicky et al. Tomppon et al. (017). Moreover, deep learning, particularly with generative adversarial models (Goodfellow) 2016). has been used for efficiently synthesizing de- tails within coarsely resolved simulations (Chu & Thuerey 2017: Xie et al., 2018). In addition to scalar transport and smoke flows, liquid droplets have been efficiently tackled by learning stochastic models (Um et al. 2018). Due their capabilities for transforming space-time time data into reduced latent spaces, fluid simulations were processed and re-simulated with neural network (NN) models (Kim et al. 2019: Prantl et al. 2019). Motivated by the success of previous studies regarding data- driven approaches in numerical simulations, we aim for enabling the learning of general correction functions. Our aim differs from methods that synthesize details for low-resolution input data. As such, our method is orthogonal to super-resolution methods and shares similarities with methods to identify discontinuities in finite difference solutions (Ray Hesthaven 2018). However, we inves- tigate the problem for challenging two-dimensional flow problems and propose new approach for learning the correction bby taking into account temporal dynamics.
CORRECTING NUMERICAL SIMULATIONS
Numerical methods yield approximations of smooth function fxx) in a discrete setting with an approximation error of the form O(h*) where h is the step size of the discretization. Naturally, higher order methods with larger is are preferable but difficult to arrive at in practice. In our setting, the motions of fluids can be represented by a series of vector fields v(x) where bold vectors will denote two- or three-dimensional vector-valued functions in the following. The accuracy of the fluid motion likewise depends on the discretization error for V. In the following, we will focus on Cartesian Eulerian representations, meaning axis aligned grids with square cells. Discretizing functions like on such a grid yields approximations that scale with the grid spacing h. While small h can yield accurate representations, the computational requirements typically increase super- linearly with the number of unknowns, and as such, all practical methods strive for keeping h as large as possible in order to compute solutions quickly.
Due to the nonlinear nature of flow equations and the inherent truncation errors of numerical schemes, varying resolutions often lead to very significant differences in solutions. Such differ- ences are not readily interchangeable across the different resolutions, as especially higher frequency content can be difficult to represent on a coarse grid. In order to improve the accuracy of a numeri- cal method, coarse grids and is representable by a NN. More specifically, we employ a CNN (Krizhessky et al.) 2012) as the correction should be translation invariant for generalization. We target time-dependent problems for which we infer a per time step correction. This is particularly challenging as approxi- mation errors accumulate over time. In this work, we will focus on finite difference approximations Strrkwerra 2004).
propose learn correction function that compensates for discretization errors on
A discretized representation s of a smooth target function f typically consists of set of discrete physical quantities. Together, we call these discrete quantities a state with which our correction approach can be formulated as follows:
1+1 F(s + c.
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where F denotes numerical solver for the target function f moving state s" forward in time to a new state s Here, c" denotes the correction function. Note that c" potentially updates only a part of the full state, e.g., the velocity field v.
We target solutions of the Navier- Stoikr-Stokes equations, which for incompressible Newtonian fluids takes the form
Ov
+ Vv - Vp vV tv+g subject to V V v 0,
(2)
0t
where P. p. v, and 9 denote density, pressure, viscosity, and external forces, respectively, The constraint, v (i.e., volume preserving) velocities.
= 0, is particularly important as it restricts motions to the space of divergence-free
3.1 CORRECTION FUNCTION
The natural follow-up question how compute and represent the potentially highly nonlinear correction function of Eq. [ Thanks to their flexibility and ability to represent nonlincarities, we use fully convolutional networks as our model for the correction function. A fine discretization, optionally with advanced numerical schemes, is employed to compute the reference data. Our goal is, then, find an optimal approximation of the correction function for the given reference data. We will first approach this problem in a supervised manner with human intervention and then explain an improved approach that enables a NN model to learn the target function an unsupervised manner.
As the velocity is the key quantity for fluids, we target corrections of this function in the follow- ing. However, our approach likewise could be applied to other functions. The correction is applied additively, i.e. c and should be computed such that this sum matches the reference velocity VR. ie., ideally v + c = Vv- In practice, we will not aim for an equality but rather match the reference closely as possible. As the reference is typically best defined in terms of fine discretization with reduced approximation errors, it might have an altogether different resolution. In this case, care is needed when transferring functions between different discretizations especially when additional constraints such as physical laws of conservation need to be taken into account. We address this problem via an optimization and aim for minimizing the distance between representations on both grids with respect to a suitable metric.
For two different dimensionalities {. E N with & < X. consider two vector spaces H RX and L E R that both conserve volume (following Eq. i.e., V = 0 for vh E H, and V 1 = 0 for 4I L. Having finer vector field cH, which contains the necessary information, we aim to find the closest vector field Cr. (E L) to Cи (E H). We first describe how to transfer functions between both spaces general before explaining how find those that are particularly amenable for learning. Consider an interpolation operator W that introduces new data points within a vector field CL (E L), i.., Wez E RX. We, then, strive to minimize the distance between We2 and си such that с₁ best represent the information of eи without loosing its volume conserving properties. Thus, we aim for computing C₁ with
argmin cp
Wel 12 subject cc
This represents a constrained optimization problem with equality constraints, which we can solve via Lagrange multipliers X as follows:
+
|ee - cH + (V cL) X.
(4)
This results a system of equations as follows:
V (WWTww [
W W
CL X
w WTc cH
(5)
Using the Schur complement, we can simplify this system to speed up calculations:
vT(W TT cH
(6)
er = ( (WTw)- (W CH -
(7)
A NN, then, infers the correction é, by minimizing the supervised loss, Luu = Ele, c, for pre-computed data set. While this method can efficiently yield divergence free changes of
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Solver g"
Model
c
Solver
Model
cht1
+
Figure I: Overview of the recurrent training, in which the NN model is coupled with differentiable PDE solver. Each unit, shown with dashed-lines lines, represents one time step, and multiple steps are unrolled so that the model can take into account temporal information when learning.
resolution, i.c., move a function such as the aforementioned correction from fine coarse grids, the solutions obtained in this way are not necessarily optimal for deep learning methods as we will evaluate more detail below. Hence, we first introduce an extension that takes into account the temporal evolution of the correction function.
3.2 TEMPORAL REGULARIZATION OF CORRECTIONS
The vector fields we target are obtained from a numerical simulation. Here, the underlying flow equations solved for finite number of steps from an initial condition. The simulation produces a series of states, which represent the spatio-tempore changes of fluid volume and velocity. Given the setup as described so far, we can acquire the correction of basic numerical simulation by running the same simulation with reduced step size in space and time to reduce approximation errors. addition, we can employ more accurate numerical schemes for the reference as we will demonstrate below. Having both the basic simulation velocity field v and the reference simulation velocity field v RR we can compute the correction vector field e₁ via Eq. 7 for си = VR - WvB-
As our aim is to find learned representation of the corrections, crucial aspect consider is the sensitivity (Murphy et al., 2004) of the targeted function (i.e., the correction) with respect to the data at hand, i.c., in our case, the state of coarse simulation. Here, observe that the correction vector fields vary strongly in time even for smooth changes of the basic simulation. That means the correction function has very nonlinear and difficult to learn relationship with the observable data in simulation.
In order to address this difficulty, here, we introduce supervision in data acquisition by employing temporal regularization that makes the correction function learnable without sacrificing its correct- ing properties. As we are dealing with continuous models, we know that the flow motion changes smoothly in time if it is resolved finely enough. As our solution changes in space well as time, learned function has be able represent spatial well as temporal changes of the target function correctly. We focus on the inference of per-time step corrections, and thus it is especially important obtain correction functions that change smoothly over time (spatial regularization could poten- tially also be incorporated during learning implicitly or explicitly). Consequently, we regularize our correction vector fields such that they change smoothly time by penalizing temporal change the correction vector field. These are given by de;/ dt,, which minimize together with the transfer from fine to coarse discretizations:
argmin (|wcz cH 12 B11
deL 1²)
subject to
V Cr = 0.
(8)
dt
Here, 33 is the regularization coefficient. This yields system of equations follows:
w WwW+ W 21
[T cn + 3
where At is the timestep size, I is the identity matrix, and the superscript n 1 of сı denotes the state at the previous step in time. We used the 3 = in the following experiments and found effective for both the correction accuracy and training.
3.3 UNSUPERVISED LEARNING WITH DIFFERENTIABLE PHYSICS
So far, we have described how acquire learnable data obtained with human support for in- troducing temporal regularization. The NN can, then, learn the corrections given the low-resolution input simulations, i.e., minimizes the emor of inferred corrections. This naturally requires special
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(a) Initial sctup
(b) Basic simulation
(c) Refereace simulation
(d) Conrected simulation
Figure 2: Rising smoke. (a) The initial size of a smoke volume is randomized in two simulations. After 1,000 steps, the same initial setup results in significant differences between (b) the basic simu- lation and (c) the reference simulation. Moreover, the small variations in the initial volume result in very different density configurations. (d) The basic version assisted by the ground truth correction from Eq. without NN inference.
care in terms of choosing right amount of temporal regularization, which in practice is strongly problem dependent. To reduce the reliance on human intervention, we propose a training approach with which the model can directly learn the target function by recurrently observing the temporal dynamics of the physical model as illustrated in Fig. [ The recurrent training can be achieved via differentiable physics solvers that allow for back paok-propagttion of gradients through the discretized physical model. In this way, we can let a NN learn the sought-after correction function an un- supervised manner. As will demonstrate below, the main advantage of this approach is that the NN experiences how corrections influence the evolution employ a differentiable Navier-Stoke
the dynamics and receive gradients in
order improve inference accuracy. In the following. solver from concurrent work (anonymous) 2020). This solver builds on the automatic differentiation of machine learning framework to compute analytic derivatives and augments them with custom derivatives where built-in derivatives would be inefficient. This setup allows for straightforward integration of solver functionality with machine learning models and enables end-to-end training in recurrent settings.
Denoting the discretized physical model Fm, we can define our loss as Lu E." [I (sfoti) - f112, where the notation of the NN inference for c₁ is omitted for brevity, M is a sam- pling operator to make the spatial discretizations consistent, and m denotes the number of recurrent steps. Above, each loss evaluation starts at time ty. Note that state foo+ depends on all - pre- vious states, and thus requires pack-propagation of the gradients through In contrast to the supervised approach, we can directly evaluate the difference between the corrected simulation result, provided by the unsupervised loss formulation, and the reference state sn. In practice, this can be achieved without explicitly preparing for learnable input-output pairs, which are acquired using our optimization for training. Once sequences of reference simulations are collected, we can directly use the sequences by selecting an initial frame to. which is downsampled to the ba- sic solver's resolution and fed to the architecture as an initial input. During each recurrent step, the intermediate results are evaluated via the differentiable solver and the current state of the correction model. These intermediate results are compared with the given consecutive reference frames, and the model can update its weights accordingly. With this setup, the model can directly learn from simulation results that have experienced inferred corrections via the unrolled recurrent steps, each of which incorporates the physics solver.
1 solver evaluations.
EXPERIMENTS
To acquire our data sets, we generate a set of simulation sequences, which are started from ran- domized initial setups according to the given parameters. These sequences are used for obtaining pairs of input and correction velocity fields to training our NN. Anonymized and time-stamped sup- plemental material for our submission can be downloaded at wttttsss//ww.dtrooooooom////// b12 bl22xtt222oot1//AAAAarssttt Po0p
PGlzfxka7dl=1
4.1 SUPERVISED CORRECTION OF RISING SMOKE
This example encompasses a volume of hot smoke rising from the bottom of an enclosed container. The motion of the smoke volume is driven by buoyancy forces computed from the density field
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Basic model
Ground truth (a) Test A
Our model
Basic model
Ground truch (b) Test B
Our model
Figure 3: Corrected rising smoke simulation. Our model is applied two test cases and B. After correcting 400 steps, the density fields are compared among the basic method, ground truth, and corrected method (our model). (See Fig. in Appx. A.2 for more comparisons.)
using the Boussinesq model. We randomize the initial size of the smoke volume. Fig. 2 shows two selected randomized initial setups at the left most. Additionally, two different results of the reference and basic simulations after 1,000 simulation steps are shown in the subsequent columns, respectively.
worth noting that the reference simulations are significantly different from the basic ones, moreover, the small variations the initial setup result significant differences even after short periods of time. We use a regular MacCormack scheme (Selle et al., (2088) for the basic intensive advection-reflee
simulation and the more accurate and 2018) for the reference. The grid resolution of the basic simulation is 32 x 64, and a four times finer grid used for the reference.
scheme (Zehnder al.,
In the following, we denote the basic version corrected by the full correction function computed with Eq. as the "ground truth" version. The coarse version naturally cannot exactly represent the high-resolution reference, and as such, we consider the corrected version without NN inference as the "ground truth" version we are aiming for with our model. The ground truth of our selected two simulations is shown at the right most in Fig. 2
We train our model using the data collected from the 20 simulations; each simulation performs 1,000 steps, thus we collected 20,000 samples in total. From each simulation, we extract pairs of imput features X and correction vector field Vz- For the imput features, we can use mul- tiple channels depending on the complexity of target function. From pilot test, exper- imentally found that nine channels, i.e,, three subsequent states give best results for our prob- lem (see Fig. 10 in Appx. (A.2). These consist of the velocity and density fields, i.e., Xr, }. We randomly split the data into a training data set of 95% and validation data set of 5%. We network model A (see Appx. and Fig. 9a) and train for 200
{v" v Vi, Pi. PL )-
epochs with batch size of 32. We use the Luup loss and an Adam optimizer
Kingma & Ba 2014)
with an adaptive learning rate starting at 0.001.
We test the trained model within two simulations that are not part of the training data set. selected frame for these tests is shown in Fig. 3] which visualizes the density fields. Our model produces significantly closer results to the ground truth than the basic simulation. Since the error accumulates over time, differences become more appar- ent in later frames. Nevertheless, large number of nonlinear simulation steps, the inferred correction field manages to the solution close to the desired state. The average errors in both density and ve- locity fields shown in Fig. 4 Here, we show relative errors with respect to ground truth correction. The errors show that our supervised model corrects the simulations very well: errors stay very close to the ground truth values steps of simulation.
Canc Je
Can Iepurteat
(a) Density (b) Velocity u Figure 4: Relative errors of the rising smoke correction model. The trained model corrects the velocity field for 500 steps. Relative errors in (a) density and (b) velocity with respect to reference are shown for Test A from Fig. 3
find that, despite the
and start to slightly deviate only after 400
4.2 NAIVE CORRECTION
We now evaluate the effect of the proposed optimization approach to compute the correction func- tion on the learning process. our context, a straightforward, i.c., "naive", way to get a correction to directly downsample the reference correction (i.e., CH) to the target domain. We find that this

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correction function produces comparable ground truth results for the rising smoke, and hence, can train a network in a supervised manner with this downsampled correction data. Fig. [a shows example. Here, directly apply regularized and "naive" correction function without involv- ing the learning step. This figure highlights that both versions yield almost identical results and, hence, the regularization also retains the important characteristies of the high-resolution reference. Apart from omitting our optimization step, i.c., providing an unregularized set of target data for the supervised training, all steps are performed as described above.
Figs. 5ddddpsdddllel show the results of train- ing with both versions. The "naive" NN model converges to much higher overall level of error, and the resulting function approximation is unusable in practice. The inference emrors quickly cause the basic simu- lation to deteriorate and become un- stable. This behavior contrasts with to approach where the temporal regularization makes it possible to accurately represent the correction function with the CNN. We also demonstrate that our approach generalizes to other problems in the context of fluid flows. See Appx. A.3 for a liquid stream example.
(a) Conecttd
(b) Training
(c) (d) Naive
Figure 5: Comparison between the naive and our model.
We have shown that the smooth changes of correction field with respect smoothly changing in- puts are crucial for successfully learning the correction function. Despite obtaining gains in accuracy from the correction function, it requires manual input to make the correction function amenable to learning via suitable temporal regularization. In the following, we will show how our unsupervised learning approach alleviates this difficulty.
4.3 LEARNING WITH A DIFFERENTIABLE PDE SOLVER
We start with test for the reference data that presents relatively smooth motions close to the basic simulation. Hence, we first test the correction from a low-resolution to a high-resolution simulations with the same standard numerical scheme for both versions. this test, we use a semi- Lami-Lagrangian advection (Stam 1999), and the reference data is generated for 100 steps from 20 randomized initial sizes of the smoke volume, i.e., 2,000 samples in total. worth noting that the actual amount of samples the NN model sees is much larger, because the intermediate output states from the recurrent iterations are likewise seen by the model. These states vary over the course the training steps since the applied correction function changes.
Fig. 6/sows the evaluation of our trained model. is apparent that our model with 48 recurrent steps yields model that successfully removes the majority of the errors shown the left of Fig. 6 The graphs of Fig. (aa show velocity errors relative to the ground truth version for each three models trained with different numbers of recurrent steps. Here, steps, the model has a higher chance to learn the correction function for long term accuracy. Fig. 6b shows errors of inferences for the given 100 ground truth steps, where the ground truth correction is evaluated via the difference between the solver result and the downsampled reference data. These graphs demonstrate that the model with 48 recurrent steps yields the best long term accuracy despite
see that, with more recurrent
tbat rap
Basic model
Reference
rUC. steps
(a) Velocity
(b) Correction
Figure 6: Analysis of the corrected simulations for rising smoke. Example frames after 80 timesteps for the following versions are shown: basic, reference, and our model. Our model is trained with 48 recurrent steps in Fig. The trained model, then, corrects the velocity field for 100 steps. The La errors in (a) velocity and (b) correction are measured from the ground truth at every step. (See Fig. 12 in Appx. A.2 for more results.)

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containing larger per step differences with respect to ground truth corrections. This indicates that, by looking ahead more steps from the solver, the model learns to anticipate the future dynamics and corrects the target function accordingly instead of relying the seemingly ideal corrections for each time step.
Next, test model in more complex settings us- ing the reference data used in Sec. 4.1 Fig. g shows the corrected results of our unsupervised model, which again trained with 48 recurrent steps, and relative errors are given in Fig. 7 It worth noting that, within the first ca. 300 steps, the unsupervised model learns even more accurate corrections than the target ground truth version. This illustrates that the model can learn from the gradients provided over the course of the many timesteps by the differentiable solver to find correction function that per- forms better than the carefully precomputed supervised version. On the other hand, also that the gains disappear over the course of very long sequences (more than ca. 500 steps). Presumably, the un- supervised model would need to be trained with more recurrent steps learn anticipate very long-term changes.
(a) Density (b) Velocity u Figure 7: Relative errors of both super- vised and unsupervised models. Rela- tive errors in (a) density and (b) velocity with respect to reference are shown for Test A from Fig.
5 DISCUSSION AND CONCLUSION
key benefit of our approach is the gain performance resulting from our trained models. A correction velocity field for given input is inferred only the basic simulation grid, i.e., the low- resolution grid. This inference happens for each solving step to assist the underlying numerical solver and only requires fixed (n)) cost for n degrees of freedom. For example, for our rising smoke test, a simulation involving the trained NN model took ca. 20 seconds for 1,000 steps whereas its corresponding high-resol resolution counterpart took ca. 104 seconds compute. See Table [ in Appx. A.4 for more details.
At training time, the unsupervised learning approach leads significantly longer training times since each recurrent step the architecture can require evaluating a complex numerical procedure. potential remedy and interesting topic for future work would be to use larger timestep size in the solver such that the model could directly learn the correction for longer horizons. As the excellent performance of the unsupervised model for the initial stages of our simulations suggests, this is very promising avenue.
To summarize, we introduced a novel approach that assists numerical methods by learning a correc- tion function to improve the accuracy of the solution. We demonstrated that taking into account the temporal information is crucial for our goal and an optimization step can ensure that sequences of corrections can be learned accurately by NN model. The model successfully improves the accu- racy for previously unseen PDE solves. Additionally, an unsupervised training via differentiable solver can be employed to further improve the learned correction for time spans that do not strongly exceed the number of steps seen during training. Despite focusing on fluids, we envision that our approach can be applied to variety of other application domains that involve numerical methods for opatio-tempora problems, from plasma physics Lewis & Miller 1984) to climate modeling Randall et al. (2007).
Ground
Supervised (a) Test
Unsupervised
Reference
Ground
Supervised
Unsupervised
Reference
(b) Test
Figure 8: Corrected rising smoke simulations. Both supervised and unsupervised models applied to two test cases A and B. After correcting 200 steps, the corrected density fields are compared with respect the ground truth and the reference.
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Under review as a conference paper at ICLR 2020
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Figure 9: NN architectures. Two models that are used in our experiments are presented. The blue and magenta colors indicate the ReLU and LeakyReLU activation functions, respectively. The curved lines indicate skip connections; the dotted lines show an additional I x convolution, which makes the number of channels same when adding.
A
APPENDIX
A.1 NEURAL NETWORK MODEL
For the NN architecture to represent the correction function of Eq. we use fully convolutional network with residual blocks (He et al. 2015). The two network models, which are used for our ex- periments, are shown in Fig. The models A and consist of 405K and 265K training parameters, respectively. Hence, models are implemented using the TensorFlow framework (Abadi et al. 2015).
the model B for setups with smaller amount of training data. Our
A.2 RISING SMOKE EXPERIMENTS
In order to decide an effective input feature for NN model, we compare two models trained with different numbers of input channel. Fig. 10 shows the errors of inferred correction between two models and indicates that the model trained with three subsequent states (i.e., k = 3) performs slightly better particularly for longer steps. Nevertheless, we observe that the difference is negligible.
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Velocity
Figure 10: Comparison of errors inferred correction between two supervised models with different numbers of input channels. Here, & denotes the number of subsequent states used for the input.
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Under review as a conference paper at ICLR 2020
Basic model
Ground truih
Our model
Basic model
Ground truth (b) After 400 steps
Our model
Basic model
Ground truth
Our model
(a) After 200 steps
(c) After 800 steps
Figure 11: Corrected rising smoke simulation. Our supervised model is applied to two test cases shown each row. At three selected steps, the density fields are visualized among the basic method, ground truth, and corrected method (our model).
Basic model
roc. steps
16 rec. steps 48 rec. steps
Reference
Basic model
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16 rec. steps 48 rec. steps
Reference
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Test B
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Test A
Test B
Error velocity
Error in inferred correction
Figure 12: Two test simulations for the rising smoke example. After individual training with differ- numbers (i.e., 8, 16, and 48) recurrent steps, each model is applied two test setups. test A and B. The images at the top row show the density visualization after 80 steps. The graphs in (a) compare the corrected velocity fields, and those (b) show the errors of inferences for the given input steps.
A.3 LIQUID STREAM WITH STEP OBSTACLES
As an extension of our method to other types of fluid simulations, we target liquid stream flowing over backward facing step. To make the setup slightly more complex, we introduce second step on the right side of the domain. Height and width of both steps are randomized to generate data sets. We note that Lagrangian approaches, i.e., particle-based discretizations, form an important class of fluid-related problems and our approach also generalizes to such approaches via auxiliary grid. This example demonstrates how to use our method in conjunction with the methods such as the fluid implicit particle algorithm (Brackbill et al. 1988). In this method, the flow motion is transferred to a helper grid in every simulation step and follow the remaining steps as explained before. Fig. [3] shows this setup for two different initial conditions. Here, the color indicates the direction of velocity such that it highlights the locations of vortex structures the flow. All simulations adopt the APIC method without surface tension but instead with gravitational force and fixed inflow velocity.
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Under review as a conference paper at ICLR 2020
Inirial setup
Ground
Initial setup
Figure 13: Stream flow starting from randomly initialized setup for the width and height of two obstacles. The color indicates the direction of velocity, in each case, for basic, reference, and ground truth versions.
Basic model
Ground Iruhh
Our model
Busic model
Ground truth
Our model
Test A (after 300 steps)
Test B (after 500 steps)
Figure 14: Corrected stream flow simulations. Our model is applied two test cases A and B. The density fields are compared at the selected steps. The color indicates the direction of velocity.
The grid resolution for the basic and corrected simulations is 32x 16, and the reference grid is four times finer.
Comparing the reference and basic simulations, we can find that the basic simulation has difficulties resolving the rotating motions that form behind the step geometries. We use the network model B (see Appx. J. and Fig. 9b) and train the model with the data from ten simulations. Fig. 14 shows the results of model applied manages to reintroduce the vortex above the right step that was lost in the basic version. The shape of the left vortex also improves match the ground truth version.
test cases. A notable improvement of model that
A.4 PERFORMANCE
Here, give details about the performance of our method to illustrate the gains that be achieved by evaluating our trained model. Table shows the timings measured from different simulations of each example. All experiments were performed 128 GB memory. The trained NN models were evaluated with the CUDA support and TensorFlow NVIDIA GeForce GTX 960 GPU with 4 GB video memory. The reference is full simulation with four times higher resolution evaluated with an optimized CPU-based solver (the same one using generate the input for our method).
Intel Xeon E5-1620 3.70 GHz processor with
Table 1: Performance comparison. Simulation timings (in seconds) are measured for 1,000 simula- tion steps. Here, un. and sup. denote the unsupervised and supervised learning models, respectively.
Example
Test
Basic
Reference
Ours
Rising smoke (Fig.
A
9.40
104,46
17.56 (un.) 19.91 (sup.)
B
9,87
117.21
17.12 (un.)
18.96 (sup.)
Liquid stream (Fig.
14
A
18,00
85,49
29.86 (sup.)
19.58
84.10
31.82 (sup.)
14


Reviews:
[{"review_id": "rJgClbLZqr", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Numerical solvers for partial differential equations take a lot of time to get high resolution results since they have to explore high dimensional grid in function domain. Thus, it is important to interpolate between grids to get high resolution results. In this paper, the authors propose the model that assists PDE solver by correcting residuals in a data-driven way. Specifically, they try to approximate NN to correction function in supervised and unsupervised manners. They also propose a temporal regularization method that smooths behavior of fluid between times. As a result, proposed method can generate high resolution results in efficient way with smoke rising simulation dataset.\n\nSignificance\n- The field that the paper target is too narrow so that it may not be significant to general machine learning community.\n\nNovelty\n- I think this approach is somewhat novel since it introduces new direction in the field where the model assists PDE solver to improve performance with learned correction function.\n\nClarity\n- I think this paper is well written in most places, but the paper is not intended to make it easy for general machine learning researchers to understand and follow the problem thoroughly.\n\nPros\n- The proposed method is orthogonal to other methods such as super-resolution and may be complemented to other existing approaches.\n\nCons\nMy main concern is that the evaluation of proposed method is quite limited in the sense that- The method is evaluated with only single dataset.- Only basic model is used to compare to proposed model. What about other baseline models use similar deep learning approaches?\n- There should be various PDE problems other than Navier-Stokes to evaluate the general effectiveness of the proposed model.\n- I think time comparison/analysis should be provided since the paper targets to get efficient approximation (using NN) in numerical analysis field.\n\nQuestions and comments\n- I wonder the affect of hyper-parameter in temporal regularization.\n- I wonder performance comparison to existing NN based approaches.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_experiments": "I did not assess the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}, {"review_id": "H1ldAtK6KB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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 proposes learning NN to correct for inaccuracies in numerical solvers of PDEs, with experimental focus on fluid flow simulation. It lists two approaches: (1) compute correction in high resolution simulation from reference states, convert to low-resolution correction, and train NN to predict low-res correction (optionally with temporal regularization, and (2) directly simulate forward using correction prediction and differentiable PDE solver and optimize to match the given reference states. It shows empirical results on better approximating fluid flow simulation. \n\nThe paper tackles an important problem of using NN to speed up expensive simulation computations. The main limitation appears to be the significance of machine learning approaches. Approach (1) is a naive prediction using NN, and approach (2) involves interesting differentiation through PDEs, but that’s directly borrowed from another concurrent anonymized submission. This paper alone does not seem to have enough novel ML contributions for acceptance. \n\nThe writing can benefit from more clarity and better structure. For example,\n- Write prediction as \\hat{c}_L(s) to show what is input for the NN\n- Put definition for c_H in Section 3.1. It seems to be based on v_R and v_B but would benefit from being explicit. \n- If beta=0 is used, why do you need Section 3.2? Is it a typo?\n- In Section 3.3, I am unsure if it should be called unsupervised, as you are given reference state s_H. How is the assumption different from Sections 3.1 and 3.2? Also, a relevant reference [1]\n\n[1] Bengio, Samy, et al. \"Scheduled sampling for sequence prediction with recurrent neural networks.\" Advances in Neural Information Processing Systems. 2015.\n\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "H1efFRwPFr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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 authors aim at improving the accuracy of numerical solvers (e.g. for simulations of partial differential equations) by training a neural network on simulated reference data. The neural network is used to correct the numerical solver. For different tasks they set up an approximation scheme via minimizing a square loss plus a task specific regularization (e.g. volume preservation in the Navier-Stokes equation example). This is then trained in a supervised manner. They also explore an unsupervised version by back-progagating through a differentiable numberical solver.\nThe proposed method seems straight forward, but effective as the simulations seem to indicate.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper provides a data-driven approach that learns to improve the accuracy of numerical solvers. It solves an important problem and provides some promising direction. However, the presented paper is not novel in terms of ML methodology. The presentation can be significantly improved for ML audience (e.g., it would be preferred to explicitly state the problem setting in the beginning of Section 3).
Decision: Reject