Mathematics

This work aims to develop and investigate a computational framework to study parametrized Partial Differential Equations (PDEs) which model nonlinear systems undergoing bifurcations.Bifurcation analysis, i.e. following the coexisting bifurcating branches due to the non-uniqueness of the solution, as well as determining the bifurcation points themselves, are complex computational tasks. The combination of  Reduced Basis (RB) model reduction and Artificial Neural Network (ANN) can potentially reduce the computational burden by several orders of magnitude and shed light on new strategies.Following the POD-NN approach, we analyzed two CFD applications where both physical and geometrical parameters were considered. In particular, we studied the Navier-Stokes equations for a viscous, steady, and incompressible flow: (i) in a planar straight channel with a narrow inlet of varying width, and (ii) in a triangular parametrized cavity. Moreover, we reconstructed the branching behavior exploring a new empirical strategy to employ the ANN reduced coefficients for a non-intrusive detection of the critical points.This is a joint work with F. Ballarin, G. Rozza and J. S. Hesthaven.