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Neural Networks for Singular Perturbations

by J. A. A. Opschoor and Ch. Schwab and C. Xenophontos

(Report number 2024-03)

Abstract
We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $(-1,1)$. We assume that the given source term and reaction coefficient are analytic in $[-1,1]$. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation parameter for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $\tanh$- and sigmoid-activated NNs. The latter activations can represent ``exponential boundary layer solution features'' explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. We prove that all DNN architectures allow robust exponential solution expression in so-called `energy' as well as in `balanced' Sobolev norms, for analytic input data.

Keywords: Singular Perturbations, Exponential Convergence, ReLU Neural Networks, Spiking Neural Networks, Tanh Neural Networks

@Techreport{OSX24_1085,
  author = {J. A. A. Opschoor and Ch. Schwab and C. Xenophontos},
  title = {Neural Networks for Singular Perturbations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-03.pdf },
  year = {2024}
}

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