  Within the framework of parameter dependent PDEs, we develop a constructive
approach based on Deep Neural Networks for the efficient approximation of the
parameter-to-solution map. The research is motivated by the limitations and
drawbacks of state-of-the-art algorithms, such as the Reduced Basis method,
when addressing problems that show a slow decay in the Kolmogorov n-width. Our
work is based on the use of deep autoencoders, which we employ for encoding and
decoding a high fidelity approximation of the solution manifold. To provide
guidelines for the design of deep autoencoders, we consider a nonlinear version
of the Kolmogorov n-width over which we base the concept of a minimal latent
dimension. We show that the latter is intimately related to the topological
properties of the solution manifold, and we provide theoretical results with
particular emphasis on second order elliptic PDEs, characterizing the minimal
dimension and the approximation errors of the proposed approach. The theory
presented is further supported by numerical experiments, where we compare the
proposed approach with classical POD-Galerkin reduced order models. In
particular, we consider parametrized advection-diffusion PDEs, and we test the
methodology in the presence of strong transport fields, singular terms and
stochastic coefficients.
