  We consider the point evaluation of the solution to interface problems with
geometric uncertainties, where the uncertainty in the obstacle is described by
a high-dimensional parameter $\boldsymbol{y}\in[-1,1]^d$, $d\in\mathbb{N}$. We
focus in particular on an elliptic interface problem and a Helmholtz
transmission problem. Point values of the solution in the physical domain
depend in general non-smoothly on the high-dimensional parameter, posing a
challenge when one is interested in building surrogates. Indeed, high-order
methods show poor convergence rates, while methods which are able to track
discontinuities usually suffer from the so-called curse of dimensionality. For
this reason, in this work we propose to build surrogates for point evaluation
using deep neural networks. We provide a theoretical justification for why we
expect neural networks to provide good surrogates. Furthermore, we present
extensive numerical experiments showing their good performance in practice. We
observe in particular that neural networks do not suffer from the curse of
dimensionality, and we study the dependence of the error on the number of point
evaluations (that is, the number of discontinuities in the parameter space), as
well as on several modeling parameters, such as the contrast between the two
materials and, for the Helmholtz transmission problem, the wavenumber.

Numerical Analysis;Machine Learning;Numerical Analysis;Machine Learning