Improving Weak PINNs for Hyperbolic Conservation Laws: Dual Norm Computation, Boundary Conditions and Systems
Year: 2024
Type: article
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.
DOI: 10.5802/smai-jcm.116
Authors: Aidan Chaumet, Jan Giesselmann
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404567190