Reduced order models for parametric bifurcation problems in nonlinear PDEs

This work is concerned with the analysis and the development of efficient Reduced Order Models (ROMs) for the numerical investigation of complex bifurcating phenomena held by nonlinear parametrized Partial Differential Equations (PDEs) in many physical contexts, from Continuum Mechanics to Quantum Mechanics passing through Fluid Dynamics. Indeed, the reconstruction of the bifurcation diagrams, which highlight the singularities of the equations and the possible coexisting states, requires a huge computational effort, especially in the multi-parameter context. To overcome this issue, we developed a reduced order branch-wise algorithm for the efficient investigation of such complex behaviour, with a focus on the stability properties of the solutions. We applied our approach to the Von Kármán equations for buckling plates, the Gross-Pitaevskii equations in Bose-Einstein condensates, the Hyperelastic models for bending beams and the Navier-Stokes model for the flow in a channel. Several issues and questions arise when dealing with the approximation and the reduction of bifurcating phenomena, we addressed them by considering new models and emerging methodologies. In particular, we developed a reduced order approach to deflated continuation method, to efficiently discover new solution branches. We proposed and discussed different Optimal Control Problems (OCPs) to steer the bifurcating behaviour towards desired states. Finally, we exploited a Neural Network approach based on the Proper Orthogonal Decomposition (POD-NN), as an alternative to the Empirical Interpolation Method (EIM), to develop a reduced manifold based algorithm for the efficient detection of the bifurcation points.