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Improving Weak PINNs for Hyperbolic Conservation Laws: Dual Norm Computation, Boundary Conditions and Systems

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arxiv 2211.12393 v2 pith:MWOXPHG7 submitted 2022-11-22 math.NA cs.NA

Aidan Chaumet , Jan Giesselmann This is my paper · ORCID
classification math.NAcs.NA
keywords conservationlawsweakhyperbolicpinnsboundarydualexperiments
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We consider the approximation of entropy solutions of nonlinear hyperbolic conservation laws using neural networks. We provide explicit computations that highlight why classical PINNs will not work for discontinuous solutions to nonlinear hyperbolic conservation laws and show that weak (dual) norms of the PDE residual should be used in the loss functional. This approach has been termed "weak PINNs" recently. We suggest some modifications to weak PINNs that make their training easier, which leads to smaller errors with less training, as shown by numerical experiments. Additionally, we extend wPINNs to scalar conservation laws with weak boundary data and to systems of hyperbolic conservation laws. We perform numerical experiments in order to assess the accuracy and efficiency of the extended method.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

    math.NA 2026-08 conditional novelty 6.0 of 10

    WEPINN enforces weak formulation and entropy condition with trigonometric test functions and fast Fourier transform integration, resolving shocks and rarefactions in conservation laws more accurately than Diff-PINN, V...

  2. Physics-Informed Neural Networks for Speech Production

    cs.SD 2025-11 conditional novelty 6.0 of 10

    A physics-informed neural network with differentiable glottal closure, learnable period, and hard glottis-tract coupling solves forward and inverse two-mass vocal fold plus vocal-tract problems.

  3. Discontinuity-aware KAN-based physics-informed neural networks

    physics.comp-ph 2025-07 conditional novelty 5.0 of 10

    A discontinuity-aware PINN with adaptive Fourier features, DyT-plus-spline KAN activations, and learned local viscosity captures shocks with errors between 0.9% and 5% on benchmark PDEs and airfoil flows.

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