Solving and Learning Nonlinear PDEs with Gaussian Processes

In this talk I present a simple, rigorous, and interpretable framework for solution of nonlinear PDEs based on the framework of Gaussian Processes. The proposed approach provides a natural generalization of kernel methods to nonlinear PDEs; has guaranteed convergence; and inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. I will outline our approach by focusing on an example nonlinear elliptic PDE followed by further numerical examples. I will also briefly comment on extending our approach to solving inverse problems. 

Bio
Dr. Hosseini obtained his Ph.D. Mathematics from Simon Fraser University. After he graduated, he worked as a Senior Postdoctoral Fellow and a von Karman Instructor at California Institute of Technology. Dr. Hosseini joined Applied Math at University of Washington this year as an Assistant Professor. His research interests lie at the intersection of applied mathematics, probability theory, and statistics focusing on the analysis and development of computational methods for extracting meaningful information from data. Dr. Hosseini is particularly interested in rigorous theoretical understanding of machine learning algorithms and incorporation of uncertainty quantification in data science tasks. See more on his website: https://bamdadhosseini.org .