Non-diffusive neural network method for hyperbolic conservation laws

TL;DR

This paper introduces a non-diffusive neural network method for accurately solving hyperbolic conservation laws by constructing weak solutions through smooth local solutions, effectively handling shock waves and their interactions.

Contribution

The paper presents a novel neural network algorithm that constructs weak solutions to hyperbolic conservation laws without diffusion, accommodating multiple shock waves and complex wave interactions.

Findings

Successfully captures entropic shock waves

Handles shock wave generation and interactions

Demonstrates high accuracy in numerical experiments

Abstract

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms.