  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. In order to
fully exploit the approximation capabilities of neural networks, we consider a
nonlinear version of the Kolmogorov n-width over which we base the concept of a
minimal latent dimension. We show that this minimal dimension is intimately
related to the topological properties of the solution manifold, and we provide
some theoretical results with particular emphasis on second order elliptic
PDEs. Finally, we report 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.
