Nonlinear Parameter Space and Model Order Reductions enhanced by scientific machine learning

Two of the main critical issues of model order reduction are the curse of dimensionality and the slow Kolmogorov n-width decay. With this thesis we want to provide novel methodologies that can ameliorate them: for the former we develop nonlinear parametric space reduction methods and for the latter nonlinear model order reduction methods. The most successful strategies applied for both approaches are local linear approximants and nonlinear solution manifold learning methods from scientific machine learning. Despite many successful applications envision purely data-driven surrogate modelling, we focus our efforts towards the advancement of more interpretable frameworks that combine the numerical understanding of the models underneath with machine learning paradigms. The first part is devoted to nonlinear parameters space reduction. We introduce hierarchical active subspaces for local parameter space reduction: new metrics to perform the clustering and classify the regions of the parameters space based on the local Grassmannian manifold's intrinsic dimension are validated through academic benchmarks. Shape optimization studies for automotive applications are the target of new multi-fidelity regression methods: nonlinear autoregressive Gaussian processes are combined with the response surfaces designed on the active subspace. Finally, the generation with free form deformation of new computational geometries that satisfy multilinear geometrical constraints, such as the volume, is sped up with constrained generative models. New tailored metrics to validate the posterior distributions are defined. The second part is focused on nonlinear model order reduction. Friedrichs' systems are presented as a new class of structure-preserving reduced order models (ROMs) with the possibility to obtain optimally stable error estimates. At the same time, Friedrichs' systems are employed as testing ground for new repartitioning algorithms for domain decomposable ROMs based on the discontinuous Galerkin method. Vanishing viscosity solutions are inferred with graph neural networks as the limit of the predictions of high viscosity domain decomposable ROMs. The last two chapters are dedicated to the embedding inside residual-based reduced numerical schemes of nonlinear approximants of the solution manifold. Such nonlinear parametrization maps are represented by neural networks obtained with teacher-student training or a combination of neural networks and singular value decomposition's modes. Finally, local nonlinear manifold approximations and adaptive hyper-reduction methods are used to reduce the transient dynamics of numerical models affected by a slow Kolmogorov n-width decay. Every result is supported by numerical experiments performed on parametrized computational fluid dynamics benchmarks.