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

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.

💡 Research Summary

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The paper introduces a novel data‑driven framework, called SymCLaw, for learning hyperbolic systems of conservation laws while guaranteeing three fundamental mathematical properties: conservation, entropy stability, and hyperbolicity. Existing data‑driven approaches typically enforce only conservation or rely on prior knowledge of the governing equations; they often neglect entropy conditions or hyperbolicity, leading to non‑physical solutions, spurious oscillations, or ill‑posed dynamics. SymCLaw addresses these gaps by constructing the flux functions in a way that intrinsically satisfies all three properties.

The core idea rests on two complementary neural‑network parameterizations. First, an input‑convex neural network (ICNN) learns a strictly convex entropy function ηθ(u). Convexity ensures that the gradient map v = ∇u ηθ(u)ᵀ is a one‑to‑one change of variables and that the Hessian Huηθ(u) is symmetric positive‑definite. Second, the physical fluxes are expressed as gradients of scalar potentials ϕμ,i(v): gi(v)=∇v ϕμ,i(v). Because the Hessian Hvϕμ,i(v) of any scalar function is automatically symmetric, the Jacobian of the flux in the original variables becomes the product A_i = Hvϕμ,i(v)·Huηθ(u). By Theorem 2.1, this product is similar to a symmetric matrix and therefore possesses only real eigenvalues and a complete set of eigenvectors, which is precisely the definition of hyperbolicity. Consequently, the learned system is guaranteed to be hyperbolic by construction, and ηθ serves as a valid entropy function, providing entropy stability.

Rather than fitting the continuous PDE directly, the authors discretize the system using a finite‑volume scheme. This choice aligns the learning problem with the discrete nature of observed data (e.g., from experiments or high‑fidelity simulations) and avoids the instability of continuous‑PDE parameter identification. An entropy‑stable numerical flux (e.g., an entropy‑consistent Rusanov or Lax‑Friedrichs variant) is employed, and the loss function measures the discrepancy between the numerical solution and the observed cell averages. The parameters (θ, μ) are optimized via gradient‑based methods, with regularization and early stopping to mitigate over‑fitting, especially in the presence of noisy data.

The methodology is validated on several benchmark problems: 1‑D Burgers, shallow‑water equations, 1‑D and 2‑D Euler equations, and the KPP equation. In each case, the model is trained on a limited set of trajectories and then tested on unseen initial conditions, long‑time integration, and data corrupted with 1‑5 % Gaussian noise. SymCLaw consistently outperforms prior approaches that enforce only conservation (CFN) or only entropy stability (ESCFN). It preserves conserved quantities to machine precision, satisfies the discrete entropy inequality, and accurately captures shock speeds and wave propagation thanks to the guaranteed hyperbolicity. Even with noisy training data, the entropy‑stable flux suppresses non‑physical oscillations, and the hyperbolic structure prevents the emergence of complex eigenvalues that would otherwise destabilize the simulation.

Key contributions of the work are:

1. A unified parametric representation that enforces convex entropy, symmetric flux Jacobians, and thus hyperbolicity by design.
2. An end‑to‑end learning pipeline that operates on the discrete finite‑volume formulation, making it robust to measurement noise and compatible with existing solvers.
3. Extensive empirical evidence showing superior accuracy, stability, and generalization across a variety of hyperbolic systems.

The authors discuss future directions, including extensions to multi‑physics and multi‑scale problems, learning of boundary conditions and source terms, coupling with high‑order discontinuous Galerkin or spectral‑volume schemes, and the development of new network architectures that further guarantee convexity (e.g., spectral regularization). Overall, the paper establishes a principled foundation for data‑driven modeling of hyperbolic conservation laws, bridging the gap between machine learning flexibility and the rigorous mathematical structure required for reliable physical simulation.