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On the approximation of rough functions with deep neural networks

by T. De Ryck and S. Mishra and D. Ray

(Report number 2020-07)

Abstract
Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its high-order accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural networks show excellent performance for approximating solutions of nonlinear conservation laws and at data compression.

Keywords: ENO, Deep Nets, Subcell.

@Techreport{DMR20_880,
  author = {T. De Ryck and S. Mishra and D. Ray},
  title = {On the approximation of rough functions with deep neural networks},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-07.pdf },
  year = {2020}
}

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