Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers

By Kiwon Um, Yun Fei, Philipp Holl, Robert Brand, Nils Thürey. Neural Information Processing Systems, 2020

363 citations recorded by the source corpus.

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.

Subfields: Artificial Intelligence, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eadc2e781abf80e478e8301c4a56b86c90ff4694

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Citations in the corpus

Physics-Informed Neural Operator for Learning Partial Differential Equations (2021, background): “They are even able to tackle previously intractable problems such as higher-dimensional, multiscale, high-contrast, and chaotic PDE systems [33, 3, 8, 22, 12, 2].”
Factorized Fourier Neural Operators (2021, methodology): “Hybrid methods typically aim to speed up traditional numerical solvers by using lower resolution grids (Bar-Sinai et al., 2019; Um et al., 2020; Kochkov et al., 2021), or by replacing computationally expensive parts of the solver with learned alternatives Tompson et al. (2017); Obiols-Sales et al.…”
Machine learning–accelerated computational fluid dynamics (2021, neutral): “…maintains the same symmetries and scaling properties (e.g., rescaling coordinates x → λ x ) as the original equations, and (2) as the mesh spacing vanishes, the interpolation module retains polynomial accuracy so that our model performs well in the regime where traditional numerical methods excel.”
Neural Operator: Learning Maps Between Function Spaces (2021, methodology): “…Similarly, another line of work proposes to enhance existing numerical solvers with neural networks by building hybrid models (Pathak et al., 2020; Um et al., 2020a; Greenfeld et al., 2019) These approaches suffer from the same computational issue as classical methods: one needs to solve an…”