“Learning Solution Manifolds of Parametric Partial Differential Equations”

ABSTRACT:

Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this talk we will discuss the potential of neural operators – a class of methods for supervised learning in function spaces – in learning nonlinear solution manifolds of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of such methods in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional numerical solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.

BIOGRAPHY:

Paris Perdikaris is an Assistant Professor in the Department of Mechanical Engineering and Applied Mechanics at the University of Pennsylvania. He received his PhD in Applied Mathematics at Brown University in 2015, and, prior to joining Penn in 2018, he was a postdoctoral researcher at the department of Mechanical Engineering at the Massachusetts Institute of Technology. His current research interests include physics-informed machine learning, uncertainty quantification, and engineering design optimization. His work and service has received several distinctions including the DOE Early Career Award (2018), the AFOSR Young Investigator Award (2019), the Ford Motor Company Award for Faculty Advising (2020), the SIAG/CSE Early Career Prize (2021), and the Scialog Fellowship (2021).