Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers
This addresses the challenge of improving solution accuracy for PDEs in scientific and engineering fields, representing an incremental advance by enhancing existing iterative solver methods.
The paper tackles the problem of reducing numerical errors in iterative PDE solvers by comparing machine learning approaches, finding that integrating the solver into the training loop outperforms previous methods, yielding stable rollouts over hundreds of steps and improved accuracy across various PDEs.
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.