Gaussian Processes for Scientific Machine Learning: PDE Solvers and Operator Learning

Author: Darcy, Matthieu D.

Year: 2027

Degree: Dissertation (Ph.D.)

Advisor: Owhadi, Houman

Committee Members: Stuart, Andrew M.; Hoffmann, Franca; Hou, Thomas Y.; Braverman, Amy J.; Owhadi, Houman

Option: Computing and Mathematical Sciences

DOI: 10.7907/fm13-my05

Abstract

We present applications of probabilistic machine learning, with a strong emphasis on Gaussian processes, to several problems arising from differential equations. In particular, we consider three topics: designing solvers for forward and inverse problems, operator learning, and learning and predicting dynamical systems from data. In Chapter 2 we develop a kernel based method to solve partial differential equations with rough forcing and in Chapter 3 we develop a variational inference approach to forward and inverse problems in partial differential equations. In Chapter 4 we develop a Gaussian process approach to operator learning, in Chapter 5 we apply this approach to function renormalization group equations, and in Chapter 6 we develop a new framework for machine precision operator learning. In Chapter 7 we develop a Gaussian process approach to learning stochastic differential equations from one trajectory and in Chapter 8 we study the non-parametric kernel flows approach to learning chaotic systems. In each case, we develop a novel computational methodology, run numerical examples to demonstrate the efficacy of the method, and, where applicable, derive theoretical guarantees for the method.

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