Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws
This work addresses the challenge of data-driven modeling for hyperbolic conservation laws, enabling physically consistent dynamics without prior knowledge of equations or fixed discretization, though it is incremental in integrating entropy-stable principles into neural networks.
The authors tackled the problem of learning hyperbolic conservation laws and their entropy functions directly from solution trajectories without predefined discretization, achieving stable and accurate shock propagation speeds over extended time horizons.
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.