  We derive upper bounds on the complexity of ReLU neural networks
approximating the solution maps of parametric partial differential equations.
In particular, without any knowledge of its concrete shape, we use the inherent
low-dimensionality of the solution manifold to obtain approximation rates which
are significantly superior to those provided by classical neural network
approximation results. Concretely, we use the existence of a small reduced
basis to construct, for a large variety of parametric partial differential
equations, neural networks that yield approximations of the parametric solution
maps in such a way that the sizes of these networks essentially only depend on
the size of the reduced basis.
