Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering

TL;DR

This paper demonstrates that deep ReLU neural networks can efficiently approximate solutions to high-frequency acoustic scattering problems with error bounds that are robust to increasing wavenumber, without prior shape information.

Contribution

It introduces a constructive method showing DNNs with fixed width and spectrally increasing depth can approximate scattering solutions with wavenumber-robust error bounds, unlike traditional methods.

Findings

DNN depth increases poly-logarithmically with wavenumber

DNN width remains fixed regardless of wavenumber

Error bounds are explicit and wavenumber-robust

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 $\textit{spectrally}$ with the target accuracy $\epsilon>0$. We show that for fixed $\epsilon>0$, the depth of these NNs should increase $\textit{poly-logarithmically}$ 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 $\textit{ansatz}$ spaces, our DNN approximations do not require any prior analytic information about the scatterer's shape.