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

A Theoretical Analysis of Deep Neural Networks and Parametric PDEs