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.

翻译：本文针对二维物理空间中光滑凸障碍物外部的时谐软声散射问题，获得了深度神经网络（DNN）仿真的波数鲁棒误差界。该误差界基于利用所谓的组合场积分方程（CFIE）将无界外部区域中的散射问题约化至其光滑弯曲边界$\Gamma$，CFIE是用于$\Gamma$上场诺伊曼数据的适定性第二类边界积分方程（BIE）。在此框架下，该公式的连续性和稳定性常数显式依赖于（无量纲）波数$\kappa$。利用问题诺伊曼数据的波数显式渐近性质，我们分析了该问题的DNN逼近速率。我们采用前馈型全连接神经网络，并使用修正线性单元（ReLU）激活函数。通过构造性论证，我们证明了存在具有$\epsilon$误差界的DNN（在$L^\infty(\Gamma)$范数下），其宽度较小且固定，深度随目标精度$\epsilon>0$呈谱增长。研究表明：对于固定$\epsilon>0$，这些神经网络的深度应随波数$\kappa$呈多对数增长，而宽度保持固定。与现有计算方法（如采用形状和波数定制解假设空间的波数自适应Galerkin边界元法（BEM））不同，本文的DNN逼近无需任何关于散射体形状的先验解析信息。