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Error analysis for deep neural network approximations of parametric hyperbolic conservation laws

by T. De Ryck and S. Mishra

(Report number 2022-34)

Abstract
We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments.

Keywords: deep learning, neural networks, conservation laws

@Techreport{DM22_1022,
  author = {T. De Ryck and S. Mishra},
  title = {Error analysis for deep neural network approximations of parametric hyperbolic conservation laws},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-34.pdf },
  year = {2022}
}

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