Learning data-driven discretizations for partial differential equations

TLDR

Numerical solutions of PDEs are difficult because resolving wide spatiotemporal scales is computationally infeasible, so coarse‑grained approximations are used but are hard to derive. The authors aim to introduce data‑driven discretization that learns optimized PDE approximations from actual solutions. They employ neural networks to estimate spatial derivatives, optimizing them end‑to‑end to satisfy equations on a low‑resolution grid. The resulting methods achieve high accuracy, enabling integration of nonlinear one‑dimensional equations at 4–8× coarser resolutions than standard finite difference schemes.

Abstract

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small scale physics. Deriving such coarse grained equations is notoriously difficult, and often \emph{ad hoc}. Here we introduce \emph{data driven discretization}, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end-to-end to best satisfy the equations on a low resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in one spatial dimension at resolutions 4-8x coarser than is possible with standard finite difference methods.