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Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering

by F. Henriquez and Ch. Schwab

(Report number 2024-18)

Abstract
We obtain wavenumber-robust error bounds for the deep neural network (DNN) emulation of the solution to the time-harmonic, sound-soft acoustic scattering problem in the exterior of a smooth, convex obstacle in two physical dimensions. The error bounds are based on a boundary reduction of the scattering problem in the unbounded exterior region to its smooth, curved boundary \(\Gamma\) using the so-called combined field integral equation (CFIE), a well-posed, second-kind boundary integral equation (BIE) for the field's Neumann datum on \(\Gamma\). In this setting, the continuity and stability constants of this formulation are explicit in terms of the (non-dimensional) wavenumber \(\kappa\). Using wavenumber-explicit asymptotics of the problem's Neumann datum, we analyze the DNN approximation rate for this problem. We use fully connected NNs of the feed-forward type with Rectified Linear Unit (ReLU) activation. Through a constructive argument we prove the existence of DNNs with an \(\epsilon\)-error bound in the \(L^\infty(\Gamma)\)-norm having a small, fixed width and a depth that increases spectrally with the target accuracy \(\epsilon>0\). We show that for fixed \(\epsilon>0\), the depth of these NNs should increase poly-logarithmically with respect to the wavenumber \(\kappa\) whereas the width of the NN remains fixed. Unlike current computational approaches such as wavenumber-adapted versions of the Galerkin Boundary Element Method (BEM) with shape- and wavenumber-tailored solution ansatz spaces, our DNN approximations do not require any prior analytic information about the scatterer's shape.

Keywords: Boundary Integral Equations, Acoustic Scattering, High Frequency, Deep Neural Networks

@Techreport{HS24_1100,
  author = {F. Henriquez and Ch. Schwab},
  title = {Deep ReLU Neural Network Emulation 
in High-Frequency Acoustic Scattering},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-18.pdf },
  year = {2024}
}

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