A two stages Deep Learning Architecture for Model Reduction of Parametric Time-Dependent Problems

Time-dependent systems, especially in the parametrized setting, describe a huge number of problems and are therefore a pervasive topic of extended scientific interest and industrial value. Indeed, parametric dynamical systems modeling and control play a fundamental role in many research fields, as in the case of fluid dynamics, chemical reactions, biological problems and more. In the majority of scenarios, the most suitable way to study such dynamics passes through numerical simulation. Especially for what concerns problems modelled by differential and partial differential equations, numerical approximation represent the standard to compute the system's response. However, a problem of dimensionality of the system's numerical discretization often appears significant, as performing multiple simulations in large-scale settings typically reveals demands of computational resources difficult to handle. This gives rise to the need of finding alternatives to classical numerical methods (Finite Element Method, Finite Volume Method, Finite Difference Method) in order to approximate the parametric response of a given system at a reduced computational cost. Reduced order models (ROMs) demonstrated to be a powerful tools in this regard and nowadays it is possible to find a large variety of applications in a number of different fields as heat transfer, fluid dynamics, shape optimization, uncertainty quantification. The main idea of ROMs is to approximate a high dimensional model, usually referred as full order model (FOM), with a low dimensional one still preserving the solution's key features. There mainly exist two different techniques to obtain a ROM: intrusive and non-intrusive approaches.