Parametric Hyperbolic Conservation Laws: A Unified Framework for Conservation, Entropy Stability, and Hyperbolicity

2026-01-28

AI Summary

This work addresses the challenge of simultaneously preserving conservation, entropy stability, and hyperbolicity in data-driven learning of hyperbolic conservation laws. To this end, the authors propose the SymCLaw framework, which parameterizes the flux function and jointly learns a convex entropy function along with its associated entropy potential. Without requiring prior knowledge of the governing equations, SymCLaw is the first data-driven approach to unify these three fundamental physical properties and automatically select physically admissible weak solutions. By integrating entropy-stable numerical fluxes with standard discretization-compatible techniques, the method demonstrates strong generalization to unseen initial conditions, robustness to noise, and high-accuracy long-time predictions across benchmark problems including Burgers’, shallow water, Euler, and KPP equations.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Security and Privacy: Data transparency and provenance

Abstract

We propose a parametric hyperbolic conservation law (SymCLaw) for learning hyperbolic systems directly from data while ensuring conservation, entropy stability, and hyperbolicity by design. Unlike existing approaches that typically enforce only conservation or rely on prior knowledge of the governing equations, our method parameterizes the flux functions in a form that guarantees real eigenvalues and complete eigenvectors of the flux Jacobian, thereby preserving hyperbolicity. At the same time, we embed entropy-stable design principles by jointly learning a convex entropy function and its associated flux potential, ensuring entropy dissipation and the selection of physically admissible weak solutions. A corresponding entropy-stable numerical flux scheme provides compatibility with standard discretizations, allowing seamless integration into classical solvers. Numerical experiments on benchmark problems, including Burgers, shallow water, Euler, and KPP equations, demonstrate that SymCLaw generalizes to unseen initial conditions, maintains stability under noisy training data, and achieves accurate long-time predictions, highlighting its potential as a principled foundation for data-driven modeling of hyperbolic conservation laws.

Problem

• hyperbolic conservation laws
• conservation
• entropy stability
• hyperbolicity
• data-driven modeling

Innovation

• hyperbolicity
• entropy stability
• conservation laws
• data-driven modeling
• flux parameterization