  Coarse-scale surrogate models in the context of numerical homogenization of
linear elliptic problems with arbitrary rough diffusion coefficients rely on
the efficient solution of fine-scale sub-problems on local subdomains whose
solutions are then employed to deduce appropriate coarse contributions to the
surrogate model. However, in the absence of periodicity and scale separation,
the reliability of such models requires the local subdomains to cover the whole
domain which may result in high offline costs, in particular for
parameter-dependent and stochastic problems. This paper justifies the use of
neural networks for the approximation of coarse-scale surrogate models by
analyzing their approximation properties. For a prototypical and representative
numerical homogenization technique, the Localized Orthogonal Decomposition
method, we show that one single neural network is sufficient to approximate the
coarse contributions of all occurring coefficient-dependent local sub-problems
for a non-trivial class of diffusion coefficients up to arbitrary accuracy. We
present rigorous upper bounds on the depth and number of non-zero parameters
for such a network to achieve a given accuracy. Further, we analyze the overall
error of the resulting neural network enhanced numerical homogenization
surrogate model.
