Régression par processus Gaussiens pour des entrées non Euclidiennes : graphes et variables catégorielles

In the design phases of aircraft engine parts, numerical simulation is traditionally used to identify the right characteristics (geometry, materials, etc.) to meet desired specifications. Meta-models like Gaussian processes can achieve this objective at lower cost by learning from physics-based simulations, with the added benefit of uncertainty quantification. The goal of this thesis is to improve the predictivity of Gaussian process meta-models for non-Euclidean spaces. We propose the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel combining continuous Weisfeiler-Lehman iterations and an optimal transport for inputs that are graphs with continuous node attributes. This new kernel is used to learn scalar outputs with Gaussian processes when the inputs are finite element meshes. In the context of numerical simulations, it is also customary to work with output fields defined on meshes, which are challenging to handle due to changes between input geometries in terms of both size and adjacency structure. We develop a new methodology to address such limitations by relying on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. We illustrate the efficiency of the method to solve real problems in fluid dynamics and solid mechanics with uncertainty quantification. A last axis focuses on Gaussian process regression with categorical input variables. We carry out a reproducible comparative study of existing categorical kernels with novel evaluation metrics. We also introduce a new clustering-based strategy using target encodings of categorical variables in order to identify groups of levels when they are unknown.