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arXiv · 2405.12624

Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering

Fernando Henríquez·Christoph Schwab
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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 $Γ$ 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 $κ$. 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 $ε$-error bound in the $L^\infty(Γ)$-norm having a small, fixed width and a depth that increases $\textit{spectrally}$ with the target accuracy $ε>0$. We show that for fixed $ε>0$, the depth of these NNs should increase $\textit{poly-logarithmically}$ with respect to the wavenumber $κ$ 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.

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BibTeXRIS

Fernando Henríquez, Christoph Schwab. 2024-05-21. Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering. https://arxiv.org/abs/2405.12624

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