Dynamics and geometry from high dimensional data

The role of dimension, order, and regularity in the learning of PDE inputs

Daniel Sanz-Alonso

Brown University

The aim of this talk is to explore some theoretical, computational, and methodological aspects of the Bayesian learning of PDE models by combining ideas from PDE theory, statistics, machine learning, and beyond. On the theoretical side I will show ---relying on recently developed regularity theory--- well-posedness of the Bayesian learning of fractional order elliptic PDEs. On the computational side I will argue that the key challenge facing sampling algorithms is large distance between prior and posterior, rather than large dimension of the unknown or the data. Finally, and time permitting, I will present a new methodology to construct priors based on random fields of spatially varying regularity. This is joint work with Nicolás García Trillos.