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

TL;DR

This paper introduces a novel approach that integrates iterative PDE solvers into the training loop of machine learning models, significantly improving solution accuracy across various complex PDEs by enabling the model to interact with the solver during training.

Contribution

The paper demonstrates that integrating the PDE solver into the training process outperforms previous methods, leading to more accurate and stable solutions for complex PDEs.

Findings

Integrated solver training improves accuracy over traditional methods.

Differentiable physics networks outperform supervised variants.

Method is effective across diverse PDEs, including Navier-Stokes flows.

Abstract

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.