  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.
