Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws

TL;DR

This paper introduces NESCFN, a neural network framework that learns hyperbolic conservation laws and entropy functions directly from data, ensuring stability and physical consistency without predefined discretizations.

Contribution

It presents a novel neural network architecture that embeds entropy-stable principles, enabling data-driven discovery of conservation laws and entropy functions without prior knowledge.

Findings

Achieves stable long-term simulations of hyperbolic systems.

Accurately captures shock propagation speeds.

Ensures conservation and entropy dissipation in learned models.

Abstract

We propose a neural entropy-stable conservative flux form neural network (NESCFN) for learning hyperbolic conservation laws and their associated entropy functions directly from solution trajectories, without requiring any predefined numerical discretization. While recent neural network architectures have successfully integrated classical numerical principles into learned models, most rely on prior knowledge of the governing equations or assume a fixed discretization. Our approach removes this dependency by embedding entropy-stable design principles into the learning process itself, enabling the discovery of physically consistent dynamics in a fully data-driven setting. By jointly learning both the numerical flux function and a corresponding entropy, the proposed method ensures conservation and entropy dissipation, critical for long-term stability and fidelity in the system of hyperbolic conservation laws. Numerical results demonstrate that the method achieves stability and conservation over extended time horizons and accurately captures shock propagation speeds, even without oracle access to future-time solution profiles in the training data.