GNRK: Graph Neural Runge-Kutta method for solving partial differential equations

TL;DR

The paper introduces GNRK, a graph neural network-based Runge-Kutta method for solving PDEs that is adaptable, efficient, and capable of handling various PDE types and resolutions, outperforming existing neural PDE solvers.

Contribution

This work presents a novel GNRK approach combining graph neural networks with classical Runge-Kutta methods, enabling versatile and resolution-invariant PDE solving.

Findings

GNRK outperforms existing neural PDE solvers in accuracy and model size.

GNRK demonstrates robustness to changes in spatial and temporal resolutions.

The method extends easily to coupled differential equations.

Abstract

Neural networks have proven to be efficient surrogate models for tackling partial differential equations (PDEs). However, their applicability is often confined to specific PDEs under certain constraints, in contrast to classical PDE solvers that rely on numerical differentiation. Striking a balance between efficiency and versatility, this study introduces a novel approach called Graph Neural Runge-Kutta (GNRK), which integrates graph neural network modules with a recurrent structure inspired by the classical solvers. The GNRK operates on graph structures, ensuring its resilience to changes in spatial and temporal resolutions during domain discretization. Moreover, it demonstrates the capability to address general PDEs, irrespective of initial conditions or PDE coefficients. To assess its performance, we benchmark the GNRK against existing neural network based PDE solvers using the 2-dimensional Burgers' equation, revealing the GNRK's superiority in terms of model size and accuracy. Additionally, this graph-based methodology offers a straightforward extension for solving coupled differential equations, typically necessitating more intricate models.