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Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws

by T. De Ryck and S. Mishra and R. Molinaro

(Report number 2022-35)

Abstract
Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (\emph{wPINNs}) for accurate approximation of entropy solutions of scalar conservation laws. \emph{wPINNs} are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by \emph{wPINNs} and illustrate their performance through numerical experiments to demonstrate that \emph{wPINNs} can approximate entropy solutions accurately.

Keywords: deep learning, physics informed neural networks, weak solutions of PDEs, conservation laws

@Techreport{DMM22_1023,
  author = {T. De Ryck and S. Mishra and R. Molinaro},
  title = {Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-35.pdf },
  year = {2022}
}

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