Deep ReLU neural network emulation in high-frequency acoustic scattering^★

¹ Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
² Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 May 2025
Accepted: 22 January 2026

Abstract

We obtain wavenumber-robust approximation 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 approximation error bounds are proved via a boundary reduction of the scattering problem in the unbounded exterior region to its smooth, curved boundary Γ 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 Γ. In this setting, the continuity and stability constants of this formulation are explicit in terms of the (non-dimensional) wavenumber k. Using known, wavenumber-explicit asymptotics of the Neumann datum of the BIE, we develop a wavenumber-explicit DNN approximation rate bound for the surface density of the BIE. Our DNN approximation is based on fixed-width, deep, fully connected feedforward NNs with strict Rectified Linear Unit (ReLU) activation: the approximation architecture is, in particular, geometry- and wavenumber-independent. We prove the existence of DNNs with an ∊-error bound in the L^∞(Γ)-norm having fixed width and depth that increases algebraically, with arbitrary small exponent in terms of the target accuracy ∊ > 0. We also show that for fixed target accuracy ∊ > 0, the depth of these NNs should increase poly-logarithmically with respect to the wavenumber k. Unlike current computational approaches such as wavenumber-adapted versions of the Galerkin Boundary Element Method (BEM) with shape- and wavenumber-tailored solution ansatz spaces, the presently considered DNN approximation architectures do not require prior analytic information about the scatterer's shape: wavenumber- and geometry-dependent solution features can be learned accurately with moderate-sized, feedforward ReLU NNs.