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Deep ReLU neural network expression for elliptic multiscale problems

by V.H. Hoang and Ch. Schwab

(Report number 2020-24)

Abstract
We analyze expression rates of deep ReLU neural network (DNN) approximations for several solution families of two-scale, linear, second order elliptic boundary value problems with either locally periodic or quasi-periodic setting. We prove that DNNs can approximate the multiscale solution families with error \(\delta >0\) in the norm of the Sobolev space \(H^1\) at an NN expression rate which is essentially independent of the scale parameter \(\varepsilon\).

Keywords:

@Techreport{HS20_897,
  author = {V.H. Hoang and Ch. Schwab},
  title = {Deep ReLU neural network expression for elliptic multiscale problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-24.pdf },
  year = {2020}
}

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