Solving Nonlinear Conservation Laws of Partial Differential Equations Using Graph Neural Networks

Abstract

Nonlinear Conservation Laws of Partial Differential Equations (PDEs) are widely used in different domains. Solving these types of equations is a significant and challenging task. Graph Neural Networks (GNNs) have recently been established as fast and accurate alternatives for principled solvers when applied to standard equations with regular solutions. There have been few investigations on GNNs implemented for complex PDEs with nonlinear conservation laws. Herein, we explore GNNs to solve the following problem

ut + f(u, β)x = 0

where f(u, β) is the nonlinear flux function of the scalar conservation law, u is the main variable, and β is the physical parameter. The main challenge of nonlinear conservation laws is that solutions typically create shocks. That is, one or several jumps in the form (uL, uR) with uL ≠ uR moving in space and probably changing over time such that information about f(u) in the interval associated with this jump is not present in the observation data. We demonstrate that GNNs could achieve accurate estimates of PDEs solutions based on new initial conditions and physical parameters within a specific parameter range.