A neural network method for scalar conservation laws with convergence rates for shock-wave solutions

TL;DR

This paper introduces a neural network method for scalar conservation laws that achieves explicit convergence rates for shock-wave solutions, combining entropy stability with approximation theory.

Contribution

It develops a novel entropy-compatible neural network approach with proven convergence rates for solutions with shocks, including explicit error bounds and stability analysis.

Findings

Achieves explicit L^1 convergence rates for shock solutions.

Constructs neural networks with small loss using shock-adapted functions.

Numerical results support theoretical error bounds and suggest improved accuracy.

Abstract

We propose a new entropy-compatible neural network method for scalar hyperbolic conservation laws and establish, to our knowledge, the first explicit \(L^1\) convergence rates in this setting that apply to piecewise smooth entropy solutions, including those with discontinuities. The method is based on a computable approximation of the Kru\v{z}kov entropy residual that sits between the strong and weak forms of the entropy inequality. For piecewise smooth entropy solutions containing shocks, rarefactions, compound waves, regular shock interactions, and, in one space dimension, nondegenerate shock formation from smooth initial data, we construct explicit neural networks with provably small loss by combining shock-adapted continuous piecewise linear functions with known approximation properties of \(\tanh\) neural networks. Together with entropy-based stability estimates, this gives rigorous \(L^1\) error bounds for minimizers of the proposed loss. In particular, when the network size grows in proportion to the number of degrees of freedom of a space--time mesh of size \(h\), the analysis recovers the classical Kuznetsov rate \(O(h^{1/2})\) in shock-dominated cases. Numerical experiments in one and two space dimensions support the theory and suggest that the actual accuracy of the method can be better than the rate guaranteed by the analysis.