From Initial Data to Boundary Layers: Neural Networks for Nonlinear Hyperbolic Conservation Laws

TL;DR

This paper develops a neural network framework for approximating entropy solutions to nonlinear hyperbolic conservation laws, demonstrating promising results in one-dimensional tests and potential for industrial applications.

Contribution

It introduces a systematic neural network-based approach for solving hyperbolic conservation laws, combining efficiency and accuracy in a novel way.

Findings

Effective neural network methodology for hyperbolic PDEs

Fast convergence and reliable predictions achieved

Potential applicability to complex industrial problems

Abstract

We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and reliable learning algorithms, combining fast convergence during training with accurate predictions. The methodology that relies on solving a certain relaxed related problem is assessed through a series of one-dimensional scalar test cases. These numerical experiments demonstrate the potential of the methodology developed in this paper and its applicability to more complex industrial scenarios.