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

2025-07-02

AI Summary

This work addresses hyperbolic conservation laws by proposing a **neural entropy-stable conservative flux network**, enabling fully data-driven, joint learning of both the governing conservation law and its associated entropy function directly from solution trajectories—without assuming any predefined numerical scheme, prior equation form, or discretization. Methodologically, entropy stability is explicitly enforced via architectural design and loss regularization: the flux function is parameterized by a neural network, subject to strict conservation constraints and an entropy dissipation penalty derived from the entropy inequality. The key contribution is the first data-driven framework that simultaneously guarantees conservation, entropy dissipation, and long-term numerical stability. Experiments demonstrate that the model preserves mass/energy conservation over extended time horizons, accurately captures shock propagation speeds, and exhibits strong robustness to unseen temporal evolutions.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Consent frameworks and practices on the web
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• Learning hyperbolic conservation laws from data
• Ensuring entropy stability without predefined discretization
• Jointly learning flux functions and entropy dynamics

Innovation

• Neural entropy-stable conservative flux networks
• Data-driven discovery of hyperbolic conservation laws
• Joint learning of flux functions and entropy