Research

Deep Learning data-driven models for physical systems

I am interested into building deep learning models which mimic the physics of dynamical systems described by partial differential equations. In this area, we have investigated in [Dal Santo et al.] how to build efficient deep learning models which reliably embed the underlying physics and are able to identify hidden parameters which determine it. In [Mitra et al.], we have provided an experimental understanding of how the training process for building physical emulators is influenced by the initialisation of the weights and the training algorithm for scientific datasets, setting the ground for further understanding of this area.

Keywords: Deep learning, artificial neural networks, computational physics, physical emulators.

Hyper-reduction methods for computational fluid dynamics in complex geometries

The accurate numerical simulation of flows in complex three-dimensional geometries can be extremely expensive from a computational viewpoint, since it requires fine computational meshes that are able to capture the complex flow patterns which feature several real world problems, from arterial blood flows to aerodynamics.

In this context, I have worked on new, general and computationally efficient reduced basis strategies to tackle parametrized fluid flows modelled by the Navier-Stokes (NS) equations and defined on domains with variable shape. The procedure relies on two phases: (i) at first, we define the mesh motion by means of a solid extension and construct a RB approximation solely for this problem; (ii) we generate a RB model for the unsteady NS problem. This strategy has been applied to arterial blood flows on three-dimensional carotid bifurcations, being able to recover complex flow patterns which are featured by the blood dynamics. Additional details can be found in [Dal Santo, Manzoni].

A big obstacle when building reduced order models for CFD applications is given by the saddle-point nature of the differential equations system, which prevents from using a vanilla RB approximation for the coupled velocity/pressure system. We proposed a new, purely algebraic, Petrov-Galerkin RB method for the parametrized Stokes equations which does not require to enrich the velocity space or any knowledge of an analytical map between a reference domain and the physical, parameter-dependent domain. Additional details and numerical results can be found in [Dal Santo et al.].

Keywords: Reduced order modelling, reduced basis, hyper-reduction methods, computational physics, finite elements, computational fluid dynamics.

Preconditioning methods for finite elements simulations

This is the main project of my PhD thesis, where I investigated novel preconditioning strategies for parametrized systems which arise from the finite element discretization of parametrized PDEs. The proposed preconditioners combine multiplicatively a Reduced Basis (RB) coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. This technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioners directly exploit the parameter dependence, since they are tailored to the class of problems at hand, and significantly speed up the solution of the parametrized (non)linear systems arising in the context of CFD simulations of arterial blood flows.

Keywords: Preconditioning methods, Krylov methods, computational physics, finite elements, computational fluid dynamics.