A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations

Within the framework of parameter dependent Partial Differen-tial Equations (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 ver-sion 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 topo-logical 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 Principal Orthogonal De-composition (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.