Learned Corrections for Discretization in Partial Differential Equations

Direct numerical solutions to partial differential equations are often computationally impractical due to the need for very fine grid sizes. As a result, one must rely upon effective equations on coarser grids that can approximate the small-scale dynamics. Traditional approaches to derive these dynamics posed significant challenges, but recent advances in differentiable physics and machine learning have enabled data-driven approaches to automatically learn them. In this work, we use convolutional neural networks to learn corrections for errors introduced by finite difference approximations of spatial derivatives on coarser grids. The correction network is trained against high-resolution solutions of the underlying equation, allowing the low-resolution solver to accurately simulate dynamics on small spatiotemporal scales. This entire process is implemented within a fully differentiable physics framework, integrating the correction network into the time integration steps and the training process. We demonstrate improved accuracy in various advection-diffusion and fluid systems and investigate the generalizability of these trained correction networks.