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Mathematics > Numerical Analysis

arXiv:2211.12393 (math)
[Submitted on 22 Nov 2022 (v1), last revised 19 Dec 2023 (this version, v2)]

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

Authors:Aidan Chaumet, Jan Giesselmann
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Abstract: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.
Comments: New version now contains extensions of weak PINNs to weak boundary conditions and applications for systems as well. The title was changed to better reflect the new content
Subjects: Numerical Analysis (math.NA)
MSC classes: 35L65, 65M99
Cite as: arXiv:2211.12393 [math.NA]
  (or arXiv:2211.12393v2 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2211.12393

Submission history

From: Aidan Chaumet [view email]
[v1] Tue, 22 Nov 2022 16:39:34 UTC (892 KB)
[v2] Tue, 19 Dec 2023 09:56:20 UTC (1,402 KB)
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