A composable machine-learning approach for steady-state simulations on high-resolution grids
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Abstract
In this paper we show that our Machine Learning (ML) approach, CoMLSim( Co mposable M achine L earning Sim ulator), can simulate PDEs on highly-resolvedgrids with higher accuracy and generalization to out-of-distribution source termsand geometries than traditional ML baselines.
Our unique approach combineskey principles of traditional PDE solvers with local-learning and low-dimensionalmanifold techniques to iteratively simulate PDEs on large computational domains.
The proposed approach is validated on more than 5 steady-state PDEs acrossdifferent PDE conditions on highly-resolved grids and comparisons are made withthe commercial solver,
Ansys Fluent as well 4 other state-of-the-art ML methods.
The numerical experiments show that our approach outperforms ML baselines interms of 1) accuracy across quantitative metrics and 2) generalization to out-ofdistribution conditions as well as mesh resolutions.
Additionally, we provide resultsof conducting a large number of ablations experiments to highlight componentsof our approach that strongly influence the results.
We conclude that our locallearning and iterative-inferencing approach reduces the challenge of generalizationthat most ML models face.
Introduction
Engineering simulations utilize solutions of partial differential equations (PDEs) to model variouscomplex physical processes like weather prediction, combustion in a car engine, thermal cooling onelectronic chips etc., where the complexity in physics is driven by several factors such as Reynoldsnumber, heat source, geometry etc.
Traditional PDE solvers use discretization techniques to approximate PDEs on discrete computational domains and combine them with linear and non-linearequation solvers to compute PDE solutions.
The PDE solutions are dependent on several conditionsimposed on the PDE, such as geometry and boundary conditions of the computational domain as wellas source terms such as heat generation, buoyancy etc.
Moreover, many applications in engineeringrequire highly-resolved computational domains to accurately approximate the PDEs solutions aswell as to capture the intricacies of geometry and source terms imposed on the PDEs.
As a result,traditional PDE solvers, although accurate and generalizable across various PDE conditions, can becomputationally slow, especially in the presence of complicated physics and large computationaldomains.
In our work, we specifically aim to increase the speed of engineering simulations using MLtechniques on complex PDE conditions used for enterprise simulations.
The idea of using Machine Learning (ML) with PDEs is not unique and has been explored for severaldecades [1, 2].
ML approaches are computationally fast, but they fall short in terms of accuracy,generalization to a wide range and out-of-distribution PDE conditions and in their ability to scale tohighly-resolved computational grids, when compared to traditional PDE solvers.
A few shortcomingsof the current methods are outlined below and subsequently are motivations for this work.
ML approaches use static inferencing to predict PDE solutions as a function of the PDEconditions.
In many cases, solutions and conditions have high-dimensional and sparse representations, which are challenging to generalize with such black-boxed static inferencing.2.
Most engineering applications require highly-resolved computational grids to capture detailed solution features, such as hot spots on an electronic chip surface.
This adverselyimpacts training in terms of GPU memory, computational time and data requirements.
Most ML approaches fail to use powerful information from traditional PDE solvers relatedto solver methods, numerical discretization etc.
In this work, we introduce a novel ML approach, Co mposable M achine L earning Sim ulator (CoMLSim) to work with high-resolution grids without compromising accuracy.
To achieve that, our methoddecomposes highly-resolved computational grids into smaller local subdomains with Cartesian grids.
PDE solutions and conditions on each subdomain are represented by lower-dimensional latent vectorsdetermined using autoencoders.
The latent vectors of solutions corresponding to conditions arepredicted in each subdomain and stitched together to maintain local consistency, analogous to fluxconservation in traditional solvers.
In this paper, we describe our method (shown in fig. 2) forbuilding CoMLSim for different PDEs, showcase different experiments to compare accuracy andgeneralizability with different ML methods and provide an extensive ablation study to understand theimpact of CoMLSim’s different components on the results.
Significant contributions of this work:
The CoMLSim approach combines traditional PDE solver strategies such as domain dis-
cretization, flux conservation, solution methodologies etc.
with ML techniques to accurately
model numerical simulations.
Our approach operates on local subdomains and solves PDEs in a low
-dimensional space.
This enables generalizing to out-of-distribution PDE conditions and scaling to higherdimensional grids.
The iterative inferencing algorithm is unsupervised and allows for coupling with traditional
PDE solvers.
Related Works: The use of ML for solving PDEs has gained tremendous traction in the past fewyears.
Much of the research has focused on improving neural network architectures and optimizationtechniques to enable generalizable and accurate learning of PDE solutions.
More recently, therehas been a lot of focus on learning PDE operators with neural networks (NNs) [3, 4, 5, 6, 7, 8, 9].
The neural operators are trained on high-fidelity solutions generated by traditional PDE solvers on acomputational grid of specific resolution and do not require any knowledge of the PDE.
However,accuracy and out-of-distribution generalizability deteriorates as the resolution of computational gridincreases (for example, 2048 2 in 2-D or 256 3 in 3-D).
Furthermore, these methods are limited by anupper cap on GPU memory.
A different research direction focuses on training neural networks with physics constrained optimization [10, 11].
These method use Automatic Differentiation (AD) [12] to compute PDE derivatives.
The physics-based approaches have been extended to solve complicated PDEs representing complexphysics [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
More recently, alternateapproaches that use discretization techniques using higher order derivatives and specialize numericalschemes to compute derivatives have shown to provide better regularization for faster convergence[31, 32, 33, 34].
However, the use of optimization techniques to solve PDEs, although accurate,has proved to be extremely slow as compared to traditional solvers and hence, non-scalable to highresolution meshes.
Domain decomposition techniques have been successfully used in the context of ML applied to PDEs[35].
These techniques enable local learning from smaller restricted domains and have proved toaccelerate learning of neural networks.
In [36, 35], the authors proposed a hybrid method combining
ML with domain decomposition to reduce the computational cost of finite element solvers.
Similarly,domain decomposition has been extensively used in the context of physics informed neural networksto reduce training costs by enabling distributed training on multiple GPUs [37, 38].
[39] and [40]learn on localized domains but infer on larger computational domains using a stitching process.
[41]and [42] learn coefficients of numerical discretization schemes from high fidelity data, which issub-sampled on coarse grids.
[43] learn surrogate models for smaller components to allow for cheapersimulations.
[44] learn local prolongation operators from discretization matrices, to improve the rateof convergence of the multigrid linear solvers.
Alternatively, other methods such as [45, 46, 47] makeus of graph-based models to improve generalization by learning on local elements.
In this work, weuse a Cartesian grid to discretize each subdomain.
However, GNNs [48, 49, 50, 51, 52] as well as
FEM use other discretizations such as triangle or polygonal meshes.
In future, these ideas can be usedto handle unstructured meshes in subdomains.
Additionally, other ML methods improve the learningof ML models by compressing PDE solutions on to lower-dimensional manifolds.
This has shown toimprove accuracy and generalization capability of neural networks [53, 54, 55, 56, 57, 58].
Although,here we propose an iterative inferencing approach in the context of solving PDEs, it is inspired fromthe implicit deep learning approach used in a variety of machine learning tasks [59, 60, 61, 62].
Inthis work, domain decomposition is combined with latent space learning is employed to accuratelyrepresent solutions on local subdomains and allow scalability to high-resolution computational grids.
CoMLSim approach
2.1 Similarities with traditional PDE solvers
Consider a set of coupled steady-state PDEs with n solution variables.
For the sake of notationsimplicity, we take n = 2 , such that u ( x, y, z ) and v ( x, y, z ) are defined on a computational domainΩ with boundary conditions specified on the boundary of the computational domain, Ω b .
The coupled
PDEs are defined as follows:
[Equation]
where, L 1 , L 2 denote PDE operators, F 1 , F 2 represent PDE source terms and u = u b , v = v b forΩ = Ω b .
The PDE operators can vary for different PDEs.
For example, in a non-linear PDE such asthe unsteady, incompressible Navier-Stokes equation the operator, L = ⃗a.⃗ ∇ − ⃗ ∇ .⃗
∇
Traditional FVM/FDM based PDE solvers solve PDEs in Eq. 1 by computing solutions variables,u, v , and their linear/non-linear derivatives on a discretized computational domain.
Iterative solution algorithms are used to conserve fluxes between neighboring computational elements and determineconsistent PDE solutions over the entire domain at convergence.
The CoMLSim approach is designedto perform similar operations but at the level of subdomains ( n 3 elements) using ML techniques.
2.2 Solution algorithm for steady-state PDEs
The solution algorithm of CoMLSim approach is shown in Fig.
2 and described in detail in Alg.
Similar to traditional solvers, the CoMLSim approach discretizes the computational domain intosmaller subdomains contain n p computational elements, where p refers to spatial dimensionalityand n is predetermined.
Each subdomain has a constant physical size and represents both PDEsolutions and conditions.
For example, a subdomain cutting across the cylinder in Fig.
2 representsthe geometry and boundary conditions of the part-cylinder as well as the corresponding solution.
Ina steady-state problem, the solutions on the domain are initialized either with uniform solutions orthey are generated from coarse-grid PDE solvers.
The initial solution ⃗p =
( u ( ⃗x ) , v ( ⃗x )) is shownwith randomly oriented flow vectors in Fig.
Pretrained encoders e ⃗p , e g , e b , e s are used to encodethe initial solutions as well as user-specified PDE conditions into lower-dimensional latent vectors,η ⃗p corresponding to PDE solutions and η g , η b , η s corresponding to geometry, boundary conditionsand source terms, respectively.
Solution and condition latent vectors on groups of neighboringsubdomains are concatenated together evaluated using a pre-trained flux conservation autoencoder( Θ ) to get a new set of solution latent vectors ( η ′ ⃗p ) on each subdomain, which are more locallyconsistent than the original vectors.
The new solution latent vectors are iteratively passed throughΘ to improve their local consistency and the iteration is stopped when the L 2 norm of change insolution latent vectors meets a specified tolerance, otherwise the iteration continues with the updatedlatent vectors.
The latent vectors of the PDE conditions are not updated and help in steering thesolution latent vectors to an equilibrium state that is decoded to PDE solutions using pretraineddecoders ( g ⃗p ) on the computational domain.
The converged solution in Fig.
2 is represented withflow vectors that are locally consistent with neighboring subdomains.
The iterative procedure usedin the CoMLSim approach can be implemented using several linear or non-linear equation solvers,such as Fixed point iterations [59], Gauss Seidel, Newton’s method etc., that are used in commercial
PDE solvers.
Out of those we have explored Point Jacobi and Gauss Seidel, which are describedbelow.
The use of autoencoders for compressed solution and condition representation in tandem withiterative inferencing solution algorithm is inspired from [58, 31] but the main difference in this workis that the solution procedure is carried out on local subdomains as opposed to entire computationaldomains to align with principles used in traditional PDE solvers.
Domain Decomposition: Computational domain Ω → Subdomains Ω c
⃗x ∈
Stability: The iterative inferencing approach proposed in this work is similar to traditional approachesused in solving system of linear equations.
Our algorithm uses fixed point iterations to solve Eq.
[Equation]
where, Θ corresponds to the weights of flux conservation network, η () refers to the latent vectors andnb refers to neighboring subdomains defined below in Eq.
[Equation]
Similar to linear systems, the stability of our iterative inferencing approach is governed by thecondition number [63] defined in Eq.
[Equation]
Solution algorithms: Point Jacobi vs Gauss Seidel: Since the algorithm loops over all the subdomains in a specified order, while updating the solution encoding of a subdomain c , there aresubdomains in the neighborhood that already have an updated solution encoding.
The Gauss Seidelmethod uses these updated solution encodings ( η ′ ⃗p ) on neighboring subdomains to update the solutionencoding on subdomain, c .
On the other hand, Point Jacobi method does not make use of thisand hence can be easily vectorized for significantly faster computation.
The implementation has aremarkable similarity with how traditional solvers use these methods [64] and leaves the door open touse other non-linear optimizers with physics-based constraints.
2.3 Neural network components in CoMLSim
Our algorithm employs two types of autoencoders, CNN autoencoders to establish a lowerdimensional representation of PDE solutions and conditions from local subdomains grids and FCNN
Autoencoders for flux conservation, where the goal is to learn a reduced representation of solutionand condition latent vectors in a local neighborhood.
Let us consider an example of solving the
Laplace equation, ∇ 2 u = 0 in 2D for arbitrarily shaped computational domains, Ω g .
In this example,the CoMLSim algorithm will require 3 autoencoders, described in Eq.
5, to learn lower-dimensionalrepresentations on local subdomains.
[Equation]
where, u, η u , S refers to the solution, its latent vector and the weights of the autoencoder, respectively,g, η g , G refers to a representation of geometry, such as Signed Distance Fields (SDF) [65], its latentvector and the weights of the autoencoder, Θ refers to the weights of flux conservation autoencoder andη nb represents a set of concatenated latent vectors, η u and η g on a group of neighboring subdomains.
All the autoencoders are trained with samples of PDE solutions generated for the same use case.
Inthe case of coupled PDEs, a single autoencoder is trained for all solution variables.
Additional detailsrelated to the autoencoder networks are provided in the supplementary materials.
Why Autoencoders?:
Solutions to classical PDEs such as the Laplace equation can be representedby homogeneous solutions as follows:
[Equation]
where,
⃗A = a 0 , a 1 , a 2 , ..., a n are constant coefficients that can be used to reconstruct the PDE solutionon any local subdomain.
⃗A can be considered as a compressed encoding of the Laplace solutions.
Since, it is not possible to explicitly derive such compressed encodings for other high dimensional andnon-linear PDEs, the CoMLSim approach relies on autoencoders to compute them.
It is known thatnon-linear autoencoders with good compression ratios can learn powerful non-linear generalizations[66, 67, 68].
Autoencoders also have great denoising abilities, which improve robustness and stability,when used in iterative settings [58].
In their paper, Park and Lee [69] demonstrate that the latentmanifold established by a trained autoencoder is stable for varying intensities of Gaussian noise.
Thisquality of autoencoders is useful for the convergence of the iterative inferencing approach.
Experiments
In this section as well as in the supplementary materials, we consider a number of use cases withvarying degrees of difficulty resulting from the PDE formulation as well as source terms, geometryand boundary conditions.
The PDEs have applications in fluid mechanics, structural mechanics andsemiconductor simulations.
The source code demonstrating an experiment and the datasets used inour experiments and analysis can be made available upon acceptance.
Details of experiments: Here we provide some details about the experiments.
Additional details andexperiments on Laplace and Darcy equation may be found in the supplementary materials.
2D Poisson equation: The Poisson’s equation, shown in Eq.
7, is very popular in engineering simulations, for example, chip temperature prediction, pressure equation in fluids etc.
[Equation]
where, u is the solution variable and f is the source term.
This PDE is solved on a 1024 x 1024grid to resolve the high-frequency features of the source term.
The source, f , is sampledfrom a Gaussian mixture model shown in Eq. 8.
[Equation]
where, x, y correspond to the grid coordinates.
A k randomly assumes either 0 or 1 tovary the number of active Gaussians in the model.
µ x , µ y and σ x , σ y are the mean andstandard deviations of Gaussians in x and y directions, respectively.
The means and standarddeviations vary randomly between 0 to 1 and 0 .
001 to 0 .
01 , respectively.
The smallermagnitude of standard deviation results in hot spots that require highly-resolved grids.
Inthis case, 256 training and 100 testing solutions are generated using Ansys Fluent.2.
2D non-linear coupled Poisson equation: The coupled non-linear Poisson’s equation isshown below in Eq.
u, v are the solution variables and f is the source term, similar to thedescription in Eq. 8.
This PDE has applications in reactive flow simulations.
[Equation]
The data generation is similar to Experiment 2 with a slight difference that the source termis less stiff with a standard deviation that varies between 0 .
005 to 0 .
3D Reynolds-Averaged Navier-Stokes external flow: This use case consists of a 3-Dchannel flow with resolution 304 x 64 x 64 at high Reynolds number over arbitrarily shapedobjects.
The corresponding PDEs are presented in supplementary materials or may bereferred from [70].
The characteristics of flow generated on the downstream has importantapplications in the design of automobiles and airplanes.
In this case, we use 5 primitive 3-Dgeometries namely, cylinder, cuboid, trapezoid, airfoil wing and wedge and their randomcombinations and rotational augmentations to create 150 training geometries and 50 testinggeometries, which are solved with Ansys Fluent to generate the data.
Out-of-distributiontesting is carried out on simplistic automobile geometries as shown in Figure 1.4.
3D chip cooling with Natural convection: In this case, we extend the complexity of
Experiments 1 , 2 and 3 to a industrial 3-D case of chip cooling with natural convection.
Inthis problem, the computational grid ( 128 3 resolution) consists of a chip that is subjected tothe powermap specified by Eq. 8.
The heating of the chip results in generation of velocity inthe fluid and at steady-state, a balance is achieved.
We use Ansys Fluent to generate 300training and 100 testing solutions for velocity, pressure, temperature for arbitrary sources.
Results and Ablation studies
In this section, we provide a variety of results for the experiments outlined in Section 3.
Additionally,we conduct thorough ablation studies to understand the different components of the CoMLSimapproach in more detail.
We use the 2-D Linear Poisson’s equation experiment for these studiesunless otherwise specified.
Details regarding the CoMLSim set up for each experiment, baselinenetwork architectures as well as additional results corresponding to contour and line plots andcomparisons across other metrics are provided in supplementary materials.
4.1 Comparison with Ansys Fluent and other ML baselines
The CoMLSim is compared with other ML baselines namely, UNet [71], FNO [8] and DeepONet[7] for the experiments outlined in Section 3.
We compare the mean absolute error with respect to
Ansys Fluent and averaged over the unseen testing cases and all solution variables.
In the case of 3-Dchip cooling we report the L ∞ norm of temperature as this metric is more suited to this industrialapplication.
It may be observed from Table 1 that the CoMLSim approach performs better than other
ML baselines.
All the ML methods perform better for the non-linear Poisson’s case as compared tothe linear Poisson’s because the source term is less stiff but the CoMLSim approach does a better jobin modeling the non-linearity arising due solution coupling.
It must be noted that CoMLSim as wellas the baselines are trained with reasonably training samples.
The baseline performance can havedifferent outcomes with increasing the size of the training data.
[Table]
4.2 Assessment of generalizability
In this section, we assess the generalizability of our method specifically with respect to out-ofdistribution PDE conditions and scaling to higher resolution grids.
The results are compared with
Ansys Fluent as well as other ML baselines discussed in section 4.1.
Out-of-distribution source term: The source term for linear andnon-linear Poisson equations are sampled from a Gaussian mixturemodel described in Eq.
8, where the number of Gaussian is randomly chosen between 0 and 30 .
In this experiment, we evaluatethe generalizability of our approach for source terms with exactly30 , 40 , 50 and 60 number of Gaussians, corresponding to out-ofdistribution for the training data distributions.
For each case, theresults averaged over 10 testing samples are compared with Ansys Fluent.
It may be observed from 3A and B, that the accuracyof all ML approaches decreases as you move further away fromthe source term distribution.
However, the CoMLSim approachperforms significantly better than other ML baselines and the accuracy is reasonable even for the case with 60 Gaussian mixturemodel, which is substantially different from the training distribution.
Increasing mesh resolution: Next, we evaluate the performance of
CoMLSim for higher grid resolutions and compare it with other ML baselines.
In this experiment,we compute solutions for the 100 source term conditions in the test set but on 4 different meshresolutions, 1024 2 , 2048 2 , 4096 2 , 8192 2 , respectively.
The mean and standard deviation of Gaussiansin the testing set are proportionally increased with the mesh resolution to ensure that the problemdefinition does not change.
It is important to note that none of the networks are retrained for higherresolution grids.
It maybe observed from table 3 that the CoMLSim continues to scale to higherresolution with similar accuracy as the original mesh resolution.
The slight drop in accuracy due toincrease in mesh resolution is attributed to accumulation of error caused by the flux conservationnetworks resulting from the increase in number of subdomains.
Other ML baselines such as UNetand FNO scale up to a grid resolution of 2048 2 but cannot evaluate beyond that due to GPU memory constraints.
DeepONet has a point-wise inference and can also scale to higher-resolution grids but itsaccuracy is lower than CoMLSim.
Out-of-distribution geometries: Next, we evaluate the performance of CoMLSim on 2 unseen car models presented at different3-D angles of rotation to the external flow.
It should be noted thatthese geometries are more complicated than the primitive objectsconsidered for training our approach 3.
The mean absolute error with respect to Ansys Fluent across all solution variables are0.0145, 0.06924, 0.04175, 0.91 and 0.13725 for CoMLSim, UNet, FNO, DeepONet and FCNN respectively.
It may be observed that CoMLSim significantly outperforms these baselines.
Additionalresults and analysis are provided in supplementary materials.
4.3 Analysis of Subdomain size
In this experiment, we analyze the effect of subdomain resolutionon accuracy and computational cost of the CoMLSim approach.
We train 3 instances at resolutions of 32 2 , 64 2 and 128 2 .
Thecompression ratio in autoencoders is kept the same for the different subdomain resolutions.
The mean absolute error on testingset is 0 .
011 and 0 . 029 , respectively.
Additionally, the iterations required to convergence are 420 , 162 and 89 , respectively.
The convergence history is shown in Fig. 5A.
The 128 2 subdomain resolution has a lower accuracy because it is challenging totrain accurate autoencoders as bigger subdomains capture a largeamount of information.
On the other hand, the computationalcost is the lowest for the 128 2 resolution because the number ofsubdomains in the entire computational domain are significantlyless and hence, the solution algorithm converges faster.
4.4 Flux conservation bottleneck layer size
The flux conservation autoencoder as it is the primary workhorseof the CoMLSim solution algorithm.
As a result, we evaluatethe effect of the bottleneck layer size on the accuracy and computational speed.
Each instance of CoMLSim with differentbottleneck size is tested on 100 unseen test cases.
We verify thatall models satisfy the stability criterion specified in Eq. 3.
It may be observed from Figure 3D that as the compression ratio of the flux conservation autoencoder decreases, it begins tooverfit and the testing error as well as the number of convergenceiterations and computational time significantly increase.
On theother hand, if the compression ratio is too large the testing errorincreases because the autoencoders underfit.
In alignment withthe collective intuition about autoencoders, there exists an optimum bottleneck size compression ratio where the best testingerror is obtained for small computational times.
4.5 Robustness and stability
A long standing challenge in the field of numerical simulationis to guarantee the stability and convergence of non-linear PDEsolvers.
However, we believe that the denoising capability ofautoencoders [72, 66, 73, 74, 75] used in our iterative solutionalgorithm presents a unique benefit, irrespective of the choice ofinitial conditions.
Here, we empirically demonstrate the robustness and stability of our approach.
In scenario A , we randomly sample 25 initial solutions from auniform distribution and in scenario B we sample from 6 different distributions, namely Uniform, Gumbel, Laplace, Gamma,
Normal, Logistic etc.
The mean convergence trajectory is plottedin Fig.
5C and 5D show that the log of L 2 norm of the convergence error falls below an acceptable tolerance for all cases anddemonstrates the stability of our approach.
4.6 CoMLSim solution methods: Point Jacobi vs Gauss Seidel
Next, we compare the performance of the Point Jacobi and Gauss Seidel implementations in the
CoMLSim approach across 3 metrics, mean absolute error, computational cost and iterations toconverge.
Both methods have a similar prediction accuracy with a mean absolute error of 0 .
013 and0 .
017 , respectively
but the Point Jacobi method is computationally cheaper requiring 4 .
3 secondsto converge as compared to 72 seconds of Gauss Seidel.
It may be observed from Fig.
5E that
the
Gauss Seidel method takes significantly fewer convergence iterations, but the Point Jacobi method isfaster because it can be efficiently vectorized on a GPU.
4.7 Coupling with traditional PDE solvers and super-resolution
In this experiment, we analyze the potential of coupling the CoMLSim approach with a traditional
PDE solver such as Ansys Fluent and subsequently demonstrate how our approach inherentlypossesses the capability to perform super-resolution.
We mainly carry out 5 experiments, wherewe start from a coarse grid solution for the 100 test samples of linear Poisson’s equation solved by
Ansys Fluent on a 64 x 64 grid ( 256 x smaller than the fine resolution), add random noise to the coarseinitial solution with varying amplitudes ( 10% , 25% and 50% ), and use that as an initial solution inthe CoMLSim approach.
The convergence trajectory in Fig.
5B shows that the case with coarse gridinitialization converges the fastest, followed by initialization with noise and zero initialization.
Allsolutions converge to the same accuracy.
4.8 Computational speed
We observe that the CoMLSim approach is about 40 - 50 x faster as compared to commercial steadystate PDE solvers such as Ansys Fluent for the same mesh resolution in all the experiments presentedin this work.
In comparison to the ML baselines, our approach is expected to be slightly slowerbecause it adopts an iterative inferencing approach.
But, this is compensated by solution accuracy, generalizability and robustness on high-resolution grids.
A detailed analysis is provided insupplementary materials.
Conclusion
In this work, we introduce the CoMLSim approach, which is an unsupervised, low-dimensional andlocal machine learning approach for solving PDE on highly-resolved grids and generalizing acrossa wide range of PDE conditions.
Our approach is inspired from strategies employed in traditional
PDE solvers and adopts iterative inferencing.
The proposed approach is demonstrated to predictaccurate solutions for a range of PDEs, generalize reasonably across geometries, source terms and
BCs.
Moreover, it scales to high resolution domains.
Broader impact, future work & limitations Although the proposed ML-model can generalize toout-of-distribution geometries, source terms and BCs, but like other ML approaches, extrapolationto any and all PDE conditions that are significantly different from the original distribution stillremains a challenge.
However, this work takes a big step towards laying down the framework onhow truly generalizable ML-based solvers can be developed.
In future, we would like to addressthese challenges of generalizability and scalability by training autoencoders on random, applicationagnostic PDE solutions and enforcing PDE-based constraints in the iterative inferencing procedure.
Future work will also investigate the potential for hybrid solvers and extensions to transient PDEsand inverse problems.
Finally, we will also address extension of the current approach to unstructuredmeshes, which is a current limitation.
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