Weak and entropy physics-informed neural networks for conservation laws

1. Problem Statement

Nonlinear hyperbolic conservation laws govern critical phenomena in fluid dynamics, gas dynamics, and shallow water flows. A defining characteristic of these equations is the spontaneous formation of shock waves (discontinuities), even from smooth initial conditions.

• Limitations of Standard PINNs: Traditional Physics-Informed Neural Networks (PINNs) enforce governing equations in their strong differential form (minimizing pointwise residuals of $u_t + \nabla \cdot F(u) = 0$). This approach fails for conservation laws because:
  • Classical derivatives do not exist at shocks, causing the strong-form residual to diverge.
  • The optimization landscape becomes unstable, often leading to "oversmoothing" where the network fails to capture sharp gradients.
  • Standard PINNs do not inherently enforce the entropy condition, which is required to select the unique physically admissible solution among multiple weak solutions.
• Existing Alternatives: Previous attempts to fix this (e.g., adaptive sampling, artificial viscosity, or weak-form PINNs using dual norms) often suffer from high algorithmic complexity (requiring adversarial networks), dependence on fixed meshes, or lack of rigorous convergence guarantees.

2. Methodology: Weak and Entropy PINNs (WE-PINNs)

The authors propose WE-PINNs, a framework that replaces strong-form residuals with a space-time weak formulation derived directly from the divergence theorem.

Core Formulation

Instead of minimizing pointwise differential residuals, the method enforces conservation and entropy admissibility via boundary flux integrals over dynamically sampled space-time control volumes ($D_k$).

1. Weak Conservation Loss ($L_{cons}$):
   Based on the divergence theorem, conservation is enforced as:
   $$\int_{\partial D_k} (U n_t + F(U) \cdot n_x) \, dS = 0$$
   where $n = (n_x, n_t)$ is the outward normal to the boundary of the control volume $D_k$. This integral form remains well-defined even when $U$ contains discontinuities, implicitly satisfying the Rankine-Hugoniot jump conditions without explicit shock tracking.

2. Entropy Admissibility Loss ($L_{ent}$):
   To ensure the solution is the unique entropy solution, an integral entropy inequality is enforced:
   $$\int_{\partial D_k} (\eta(U) n_t + q(U) \cdot n_x) \, dS \leq 0$$
   where $\eta$ is a convex entropy function and $q$ is the entropy flux. The loss penalizes only violations of this inequality (one-sided penalty), allowing for physical entropy production at shocks.

3. Total Variation Diminishing (TVD) Regularization ($L_{TVD}$):
   To guarantee the convergence analysis holds, a regularization term bounds the total variation of the solution, preventing non-physical oscillations.

4. Implementation Details:

   • Mesh-Free: Control volumes are dynamically sampled with variable spatial and temporal scales (multi-scale sampling) to capture both global balance and local shock structures.
   • Quadrature: Boundary integrals are evaluated using deterministic Gauss-Legendre quadrature, avoiding the noise of Monte Carlo estimates.
   • Architecture: Uses a simple feedforward neural network with a temporal ansatz to exactly satisfy initial conditions, requiring no auxiliary networks or saddle-point optimization.

3. Key Contributions

1. Novel Weak Formulation: A direct, mesh-free space-time flux formulation that is mathematically well-defined for discontinuous solutions, avoiding the divergence issues of strong-form residuals.
2. Rigorous Convergence Analysis: The paper establishes a theoretical link between the network's loss function and the $L^1$ error. Using the Bouchut-Perthame framework, the authors prove that for scalar conservation laws, the $L^1$ error towards the entropy solution is bounded by the weak and entropy losses:
   $$\|U_\theta - u\|_{L^1} \leq K \cdot (L_{cons}^{1/4} + L_{ent}^{1/4})$$
   This provides the first explicit convergence rate for a mesh-free control-volume PINN.
3. Simplicity and Robustness: Unlike other weak-form approaches (e.g., wPINNs), WE-PINNs do not require dual networks or complex min-max optimization, making them easy to implement while maintaining robustness in shock-dominated regimes.

4. Numerical Results

The method was validated on three benchmark problems involving scalar equations and systems:

• Inviscid Burgers Equation:
  • Riemann Problems: Successfully resolved moving shocks and rarefaction waves without spurious oscillations.
  • Performance: WE-PINNs achieved significantly lower $L^1$ and $L^\infty$ errors compared to standard PINNs (e.g., $L^\infty$ error of $0.46$ vs $0.99$ for a shock problem).
• Compressible Euler Equations (Sod Shock Tube):
  • Accurately captured the complex interaction of shock waves, rarefaction fans, and contact discontinuities in a coupled system.
  • Standard PINNs failed to resolve the contact discontinuity and shock strength, whereas WE-PINNs preserved sharp gradients.
• Shallow Water Equations (Dam-Break):
  • Demonstrated the ability to handle hydraulic jumps (shocks) and maintain the positivity of water depth.
  • Successfully simulated both 1D and 2D dam-break scenarios, including diagonal and circular interfaces.

5. Significance and Conclusion

This work addresses a fundamental structural limitation of standard PINNs in hyperbolic problems. By shifting from pointwise differential enforcement to integral flux balance, WE-PINNs provide a theoretically grounded, mesh-free approach that naturally handles discontinuities.

• Theoretical Impact: It bridges the gap between deep learning and classical finite-volume methods, providing the first rigorous $L^1$ convergence proof for a PINN formulation of scalar conservation laws.
• Practical Impact: The method offers a robust, easy-to-implement alternative for simulating complex fluid dynamics problems involving shocks, eliminating the need for artificial viscosity or complex shock-capturing schemes.

The authors conclude that while the current convergence proof is limited to scalar laws, the framework is highly promising for systems, with future work focusing on extending the theoretical analysis to multi-component systems and optimizing quadrature strategies.