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

arXiv: https://arxiv.org/abs/2507.01795 · HTML: https://mlsanity.com/paper/2507.01795/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 2 Jul 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [dynoNet: a neural network architecture for learning dynamical systems](https://mlsanity.com/paper/2006.02250/) (2020)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [NeuralPDE: Modelling Dynamical Systems from Data](https://mlsanity.com/paper/2111.07671/) (2021)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [A TVD neural network closure and application to turbulent combustion](https://mlsanity.com/paper/2408.03413/) (2024)

## Built on by (most cited first, 3 of 3)

- [Parametric Hyperbolic Conservation Laws: A Unified Framework for Conservation, Entropy Stability, and Hyperbolicity](https://mlsanity.com/paper/2601.21080/) (2026)
- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables](https://mlsanity.com/paper/2607.14855/) (2026)

## Abstract (arXiv)

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.