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Neural Networks for Singular Perturbations - Finite Regularity

by F. Rohner and Ch. Schwab and C. Xenophontos

(Report number 2026-17)

Abstract
We study finite-element and deep feedforward neural network (DNN for short) expressivity rate bounds for solution sets of a model linear, second order singularly perturbed, elliptic two-point boundary value problem, in Sobolev norms on a bounded interval \((-1,1)\), with explicit dependence on the singular perturbation parameter \(\varepsilon \in (0,1]\). Emphasis is on low Sobolev regularity of the data, i.e., source term \(f\) and reaction coefficient \(b\). A proof of \(\varepsilon\)-explicit solution regularity based on exponentially weighted energy-norm bounds is developed, and \(\varepsilon\)-robust, algebraic expression rate bounds in Sobolev norms for \(\mathbb{P}_1\) Finite-Elements on exponential and Shishkin type meshes is proved. Expression rates for shallow (fixed depth) ReLU-NNs are shown which are robust w.r. to \(\varepsilon\) and explicit in terms of the NN size. Robust NN expression rate bounds are further studied for deep feedforward DNNs with ReLU and tanh-activations. As in J. A. A. Opschoor and Ch. Schwab and C. Xenophontos: Neural Networks for Singular Perturbations, Numer. Math., 157/5 (2025), tanh- and sigmoid-activated sub-NNs allow to include exponential boundary layer functions exactly into the NN feature space, leading to reduced NN sizes. Recent bitstring encoding techniques for deep NNs with ReLU activations afford, still under low data regularity \(f,b \in H^1(I)\) twice the (robust) convergence rate of \(\mathbb{P}_1\) Finite-Elements achievable with "eXp" or Shishkin meshes.

Keywords: Singular Perturbations, Robust Convergence, Shishkin Meshes, Neural Networks, Superconvergence

@Techreport{RSX26_1172,
  author = {F. Rohner and Ch. Schwab and C. Xenophontos},
  title = {Neural Networks for Singular Perturbations -
Finite Regularity},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-17.pdf },
  year = {2026}
}

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