Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

The paper introduces a novel physics-informed neural network framework called Weak-Entropy PINN (WEPINN) for solving hyperbolic conservation laws with discontinuous solutions. The method is designed to overcome the limitations of existing neural network-based approaches, which either rely on strong prior assumptions (e.g., knowledge of discontinuity locations), introduce artificial smoothing that degrades accuracy, or suffer from training instability.

The core methodology of WEPINN is built on three key components:

1. Weak formulation: The governing equations are enforced in their weak (integral) form, which transfers all spatial and temporal derivatives onto smooth test functions (specifically, trigonometric polynomials). This inherently accommodates discontinuities without artificial smoothing or prior knowledge of their locations.

2. Entropy condition: The method enforces the entropy condition as integral inequalities to select the physically admissible solution among multiple weak solutions. This is crucial for correctly determining whether an initial discontinuity evolves into a shock or a rarefaction wave.

3. Discrete fast Fourier transform (DFFT): The DFFT is used for efficient and accurate evaluation of the integrals appearing in both the weak formulation and the entropy condition, significantly reducing computational cost compared to standard numerical integration.

A key feature of WEPINN is that it pre-selects a set of test functions and minimizes the residuals over all of them simultaneously via least-squares optimization, avoiding the adversarial training employed in WPINN, which leads to stable and efficient training.

The paper provides extensive numerical experiments covering:

• One-dimensional scalar conservation laws: Linear advection equation, Burgers' equation, and the LWR traffic model, under both Dirichlet and periodic boundary conditions, with various initial condition classes (Sigmoid, Riemann, Fourier, Trig, Bell, PWC).

• One-dimensional system of conservation laws: The compressible Euler equations (Sod shock tube problem).

• Two-dimensional scalar conservation laws: The 2D Burgers' equation with disk and trigonometric initial conditions.

The results demonstrate that WEPINN consistently outperforms baseline methods (Diff-PINN, VPINN, and WPINN) across all benchmark problems. For example, in the one-dimensional scalar cases under Dirichlet boundary conditions, WEPINN achieves substantially lower relative L2 errors and near-perfect shock detection rates (S-Rate) with the smallest shock position errors (S-Acc). In the Sod shock tube problem, WEPINN achieves the lowest L2 error on all three variables (density, velocity, pressure) and accurately reproduces all three waves (rarefaction, contact discontinuity, and shock). In the two-dimensional Burgers' equation, WEPINN achieves the lowest L2 error and accurately tracks the propagation of circular discontinuities and captures shock formation from smooth profiles.

The paper also includes ablation studies:

• Role of entropy loss: The entropy loss is essential for eliminating unphysical weak solutions. Without it, WEPINN converges to a propagating shock instead of the correct rarefaction in a Riemann problem for Burgers' equation.

• Test function selection: Trigonometric test functions outperform Chebyshev polynomials, and the optimal maximum degree is 16 or 32, depending on the scenario.

The paper introduces two new shock-aware evaluation metrics: the shock detection rate (S-Rate) and shock position accuracy (S-Acc), which complement traditional Lp error measures by directly assessing the presence and location of discontinuities predicted by a model.

The conclusion states that the proposed method consistently outperforms existing approaches across a variety of scalar conservation laws and systems of conservation laws in one and two dimensions and "notably excels in accurately resolving complex nonlinear wave phenomena, including formation and propagation of shocks, shock-shock merging, and shock-rarefaction interaction, under diverse initial and boundary conditions." Future work will explore extending this methodology to design neural operators for solving hyperbolic conservation laws and related PDEs involving discontinuities and moving interfaces.

Improvements to AI Systems:

1. Physics-Informed Neural Networks (PINNs) with Weak-Form Enforcement for Discontinuous Solutions

• Replace strong-form PDE residuals with integral weak-form constraints using trigonometric test functions, eliminating the need for artificial smoothing or prior knowledge of discontinuity locations.

• This enables PINNs to solve hyperbolic conservation laws (e.g., Burgers’ equation, Euler equations) with sharp shocks, rarefactions, and contact discontinuities without spurious oscillations or training divergence.

2. Entropy-Condition Regularization for Physically Admissible Solutions

• Add integral entropy inequalities as a loss term to automatically select the correct weak solution among multiple possibilities (e.g., shock vs. rarefaction).

• This prevents the AI from converging to unphysical solutions, a common failure mode in naive PINNs for nonlinear conservation laws.

3. Stable Least-Squares Training via Pre-Selected Test Functions

• Replace adversarial or minimax training (as in WPINN) with a fixed set of test functions and simultaneous least-squares residual minimization.

• This improves training stability, reduces computational overhead, and accelerates convergence, making the AI system more reliable for real-time or large-scale simulations.

4. Efficient Integral Evaluation via Discrete Fast Fourier Transform (DFFT)

• Use DFFT to compute weak-form and entropy integrals, reducing cost from O(N2) to O(N log N).

• This allows the AI to handle high-resolution meshes and higher-dimensional problems (e.g., 2D) with significantly lower memory and time requirements.

5. Shock-Aware Evaluation Metrics for Model Selection and Validation

• Integrate S-Rate (shock detection rate) and S-Acc (shock position accuracy) into the training loop or validation pipeline.

• This enables the AI to directly optimize for correct discontinuity location and presence, rather than relying solely on L2 error, which can mask poor shock resolution.

6. Generalization to Systems and Higher Dimensions

• Extend the framework to systems of conservation laws (e.g., compressible Euler) and multi-dimensional problems (e.g., 2D Burgers) by using tensor-product trigonometric test functions.

• This allows the AI to model complex wave interactions (shock-shock merging, shock-rarefaction interaction) in engineering and geophysical applications.

What the Improved AI System Can Do:

• Accurately solve hyperbolic PDEs with discontinuities in 1D and 2D, for both scalar and system equations, without prior knowledge of shock locations or artificial viscosity.

• Automatically select the physically correct solution (e.g., rarefaction vs. shock) via entropy constraints, avoiding non-physical predictions.

• Train faster and more stably than adversarial or strong-form PINNs, with lower computational cost due to DFFT-based integration.

• Provide interpretable shock metrics (S-Rate, S-Acc) alongside standard error norms, enabling precise quality control in scientific simulations.

• Serve as a foundation for neural operators that learn solution mappings for families of initial/boundary conditions, potentially enabling real-time prediction of discontinuities in fluid dynamics, traffic flow, and plasma physics.