Gaussian Process Techniques for Differential Equations

Gaussian Proceeses for Machine Learning (GP4ML) are an established kernel-based Bayesian machine learning method that has also been considered in the context of differential equations. From a mathematical point of view, however, the application of GP4ML for differential equaitons is not sufficiently justified, in particular if it comes to infinite rank conditioning. The latter conditioning, however, is somewhat natural either has an idealized limit of existing methods or as a methodologically independent approach of its own. The goal of this project is to address these shortcomings by establishing a rigorous and general conditioning theory for GP4MLs and to apply this theory in the context of differential equations. One particular focus lies on approximations with finite rank conditionings. In cooperation with PN5-6 we will further apply our insights to specific applications and evaluate the approaches numerically.