Message Passing Neural PDE Solvers

TL;DR

This paper introduces a neural message passing-based PDE solver that generalizes across various properties, demonstrating stability, accuracy, and flexibility in fluid flow problems in 1D and 2D.

Contribution

It presents a novel neural message passing framework for PDE solving that encompasses classical methods and ensures stability through a domain adaptation approach.

Findings

Achieves fast, stable, and accurate solutions across diverse domain properties.

Contains classical numerical methods within its neural message passing framework.

Demonstrates effectiveness on fluid flow problems in 1D and 2D.

Abstract

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.