A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I

TL;DR

This paper introduces a deep learning framework inspired by discontinuous Galerkin methods to accurately solve nonlinear hyperbolic PDEs with shocks and discontinuities, overcoming limitations of existing physics-informed neural networks.

Contribution

It develops a novel physics-informed deep learning approach based on discontinuous Galerkin principles for hyperbolic PDEs, handling shocks without prior knowledge of discontinuity locations.

Findings

Demonstrates high accuracy in approximating solutions with shocks.

Shows robustness across various nonlinear hyperbolic PDEs.

Validates effectiveness through numerical experiments.

Abstract

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.