How Much Can One Learn a Partial Differential Equation from a Single Solution Data?

Abstract

In this talk, the speaker and his research group will study a few basic questions for partial differential equation (PDE) learning from observed solution data. Using various types of evolutionary PDEs as examples, they will show 1) how large the data space spanned by all snapshots along a solution trajectory is, 2) if one can construct an arbitrary solution by superposition of snapshots of a single solution, and 3) identifiability of a differential operator from a single solution data on local patches. They propose a consistent and sparse local regression method for general PDE identification. Their method requires minimal amount of local measurements in space and time from a single solution trajectory.

About the Speaker

Prof. ZHAO Hongkai received his PhD in Mathematics from the University of California, Los Angeles in 1996. In 1996-1999, he was on the faculty at Stanford University. He then joined the University of California, Irvine (UCI) as an Assistant Professor and was promoted to Associate Professor in 2003 and to Full Professor in 2007. He was the Chair of the Department of Mathematics at UCI in 2010-2013 and 2016-2019. He moved to Duke University in 2020 and is currently the Ruth F. DeVarney Distinguished Professor of Mathematics.

Prof. Zhao’s research interest is in computational and applied mathematics that includes modeling, analysis and developing numerical methods for problems arising from science and engineering. He serves on the editorial boards of SIAM Journal on Imaging Sciences, Multiscale Modeling and Simulation, Journal of Computational Mathematics, and Geometry, Imaging and Computing.

Prof. Zhao was elected a Fellow of the Society for Industrial and Applied Mathematics in 2022. He is also the recipient of the 2007 Feng Kang Prize in Scientific Computing and the 2002 Alfred P. Sloan Fellowship.

For Attendees' Attention

This talk will be held online via Zoom. To attend, please join the Zoom meeting at https://hkust.zoom.us/j/93231034696 (Meeting ID: 932 3103 4696 / Passcode: iasip2022).

About the Program

For more information, please refer to the program website at https://iasprogram.hkust.edu.hk/inverseproblems/.