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

Lizuo Liu, Lu Zhang, Anne Gelb

TL;DR

SymCLaw introduces a data-driven framework for learning hyperbolic conservation laws that preserves conservation, entropy stability, and hyperbolicity by construction. It parameterizes entropy via an input-convex neural network and fluxes as gradients of entropy potentials, ensuring symmetric and positive-definite structures that guarantee real eigenvalues. An entropy-stable finite-volume scheme, built with a discrete entropy-conservative flux plus Rusanov-type dissipation, enables stable integration and seamless coupling with classical solvers. Numerical experiments across Burgers', shallow water, Euler, and 2D problems show robust long-time predictions, generalization to unseen data, and stability under noisy training, highlighting practical viability for principled data-driven hyperbolic modeling.

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.

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

TL;DR

SymCLaw introduces a data-driven framework for learning hyperbolic conservation laws that preserves conservation, entropy stability, and hyperbolicity by construction. It parameterizes entropy via an input-convex neural network and fluxes as gradients of entropy potentials, ensuring symmetric and positive-definite structures that guarantee real eigenvalues. An entropy-stable finite-volume scheme, built with a discrete entropy-conservative flux plus Rusanov-type dissipation, enables stable integration and seamless coupling with classical solvers. Numerical experiments across Burgers', shallow water, Euler, and 2D problems show robust long-time predictions, generalization to unseen data, and stability under noisy training, highlighting practical viability for principled data-driven hyperbolic modeling.

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.
Paper Structure (32 sections, 2 theorems, 64 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 32 sections, 2 theorems, 64 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic conservation laws
  4. Entropy solution
  5. Symmetrization
  6. Parametric hyperbolic conservation laws (SymCLaw)
  7. Finite volume methods for conservation laws
  8. Entropy-stable flux for SymCLaw
  9. Time integration
  10. Training procedure design
  11. Data generation
  12. Loss function
  13. Conservation and entropy metric
  14. Network architectures and training configuration
  15. Fully connected neural networks
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $\eta({\bf u})$ be a strictly convex function. Then $\eta({\bf u})$ serves as an entropy function for eq: conservation_law if and only if the matrix $\nabla_{\bf v}\mathbf{u}(\mathbf{v})$ is symmetric positive-definite and $\nabla_{\bf v}\mathbf{g}_\mathfrak{i}(\mathbf{v})$ is symmetric for each is similar to the symmetric matrix Consequently, the existence of a strictly convex entropy functi

Figures (11)

  • Figure 1: Illustration of full time span and training subintervals.
  • Figure 1: Comparison of the reference solution (black solid line) to the SymCLaw solution for 1D Burgers' equation at (left) $t = 1$ (middle) $t = 2$ (right) $t = 3$ for $\xi = 0,.25,.5,1$ in \ref{['eq: noise burgers']}.
  • Figure 2: The discrete conserved quantity remainder $\mathcal{C}(u)$ defined in \ref{['eq:conservemetric']} (left) and the discrete entropy remainder $\mathcal{J}(u)$ defined in \ref{['eq: discrete entropy']} (right) for the SymCLaw prediction of the 1D Burgers' equation over the time interval $t \in [0,3]$ with noise levels $\xi = 0, .25, .5, 1$ in \ref{['eq: noise burgers']}.
  • Figure 3: Comparison of the reference solution (black solid line) of height $h$ and momentum $hu$ for the SymCLaw of the shallow water equation with $\xi = 0, .25, .5, 1$ in \ref{['eq:noise_shallow']}: (left) $t = .5$ of $h$, (middle-left) $t= 1.5$ of $h$, (middle-right) $t = .5$ of $hu$, (right) $t = 1.5$ of $hu$.
  • Figure 4: Discrete conserved quantity remainder \ref{['eq:conservemetric']} of height $\mathcal{C}(h)$ (left), the momentum $\mathcal{C}(hu)$ (middle), and the discrete entropy remainder $\mathcal{J}([h,hu]^\top)$ (right) for the SymCLaw of the shallow water equations with noise levels $\xi = 0,.25,.5, 1$ in \ref{['eq:noise_shallow']}.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 3.1
  • Remark 4.1