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Weak and entropy physics-informed neural networks for conservation laws

Ismail Oubarka, Imad Kissami, Mohamed Boubekeur, Fayssal Benkhaldoun, Aziz Madrane, Zakaria Saadi

Abstract

We propose Weak and Entropy PINNs (WE-PINNs) for the approximation of entropy solutions to nonlinear hyperbolic conservation laws. Standard physics-informed neural networks enforce governing equations in strong differential form, an approach that becomes structurally inconsistent in the presence of discontinuities due to the divergence of strong-form residuals near shocks. The proposed method replaces pointwise residual minimization with a space--time weak formulation derived from the divergence theorem. Conservation is enforced through boundary flux integrals over dynamically sampled space--time control volumes, yielding a mesh-free control-volume framework that remains well-defined for discontinuous solutions. Entropy admissibility is incorporated in integral form to ensure uniqueness and physical consistency of the weak solution. The resulting loss functional combines space--time flux balance and entropy inequalities, without resorting to dual-norm saddle-point formulations, auxiliary potential networks, or fixed discretization meshes. This makes the proposed method remarkably easy to implement, requiring only a simple standard neural network architecture. We establish a rigorous convergence analysis linking the network's loss function to the $L^1$ error towards the entropy solution, providing the first explicit $L^1$ convergence rate for a mesh-free control-volume PINN formulation via the Bouchut-Perthame framework for scalar conservation laws. Numerical experiments on the Burgers equation, the shallow water equations, and the compressible Euler equations demonstrate accurate shock resolution and robust performance in both smooth and shock-dominated regimes.

Weak and entropy physics-informed neural networks for conservation laws

Abstract

We propose Weak and Entropy PINNs (WE-PINNs) for the approximation of entropy solutions to nonlinear hyperbolic conservation laws. Standard physics-informed neural networks enforce governing equations in strong differential form, an approach that becomes structurally inconsistent in the presence of discontinuities due to the divergence of strong-form residuals near shocks. The proposed method replaces pointwise residual minimization with a space--time weak formulation derived from the divergence theorem. Conservation is enforced through boundary flux integrals over dynamically sampled space--time control volumes, yielding a mesh-free control-volume framework that remains well-defined for discontinuous solutions. Entropy admissibility is incorporated in integral form to ensure uniqueness and physical consistency of the weak solution. The resulting loss functional combines space--time flux balance and entropy inequalities, without resorting to dual-norm saddle-point formulations, auxiliary potential networks, or fixed discretization meshes. This makes the proposed method remarkably easy to implement, requiring only a simple standard neural network architecture. We establish a rigorous convergence analysis linking the network's loss function to the error towards the entropy solution, providing the first explicit convergence rate for a mesh-free control-volume PINN formulation via the Bouchut-Perthame framework for scalar conservation laws. Numerical experiments on the Burgers equation, the shallow water equations, and the compressible Euler equations demonstrate accurate shock resolution and robust performance in both smooth and shock-dominated regimes.

Paper Structure

This paper contains 25 sections, 3 theorems, 80 equations, 15 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical background and weak formulation
  3. Weak conservation in space--time
  4. Entropy condition and admissibility
  5. Weak and Entropy PINN formulation (WE-PINNs)
  6. Space--time control-volume formulation
  7. Weak conservation loss
  8. Entropy inequality loss
  9. Total Variation Diminishing (TVD) regularization
  10. Total loss function
  11. Convergence Analysis
  12. Weak Entropy Truncation Error
  13. Bouchut-Perthame Error Estimate
  14. Main Convergence Theorem
  15. Training procedure
  16. ...and 10 more sections

Key Result

Lemma 1

For any partition $\{D_i\}_{i=1}^{N_D}$ of $\Omega \times (0,T)$, the continuous weak truncation error is bounded by the volume-weighted network losses:

Figures (15)

  • Figure 1: Local space--time control volumes. Each collocation point defines the center of a control volume $D_k$ over which the conservation identity \ref{['eq:weak_conservation']} is enforced. The illustrations show different sampling densities of the space--time control volumes: 100 (top), 500 (middle), and 5000 (bottom).
  • Figure 2: WE-PINNs solution of Riemann problem for the inviscid Burgers equation. The numerical solution is shown at selected times together with the exact entropy solution.
  • Figure 3: space--time evolution of the numerical solution for the Riemann shock problem.
  • Figure 4: WE-PINNs solution of the rarefaction problem for the inviscid Burgers equation.
  • Figure 5: space--time evolution of the numerical solution for the rarefaction problem.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Lemma 1: Link Between Discrete Losses and Weak Truncation Error
  • proof
  • Theorem 1: Bouchut-Perthame bouchut1998
  • Theorem 2: Convergence of WE-PINNs
  • proof
  • Remark 1