Learning data driven discretizations for partial differential equations

TL;DR

This paper introduces a data-driven neural network approach to discretize PDEs, enabling accurate solutions at coarser resolutions than traditional methods by learning optimized derivative approximations from actual solutions.

Contribution

It presents a novel neural network-based method for learning discretizations of PDEs directly from data, improving accuracy at lower resolutions compared to standard finite difference methods.

Findings

Achieves 4-8x coarser resolutions with high accuracy

Neural networks optimize spatial derivatives end-to-end

Allows efficient integration of nonlinear PDEs in 1D

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.