Speaker：Juncai He (KAUST)

Time：3 PM - 4 PM, Tuesday, June 4^th, 2024

Tencent Meeting：546-789-198    

Bio: Dr. Juncai He is currently a research scientist at King Abdullah University of Science and Technology (KAUST). Before that, he received a B.S. degree in Pure and Applied Mathematics from Sichuan University in 2014 and a Ph.D. degree in Computational Mathematics under the supervision of Prof. Jinchao Xu and Prof. Jun Hu at Peking University. From 2019 to 2020, he worked as a Postdoctoral Scholar supervised by Prof. Jinchao Xu at Pennsylvania State University. From 2020 to 2022, he was an R.H. Bing instructor fellow working with Prof. Richard Tsai and Prof. Rachel Ward at UT Austin. His research focuses on mathematical analysis, algorithm development, and their applications in machine learning and scientific computing, spanning both data and physical sciences.

Abstract: In this talk, I will present recent results on applying multigrid structures to both general and specific operator learning problems in numerical PDEs. First, we will discuss some basic background on operator learning, including the problem setup, a uniform framework, and a general universal approximation result. Then, we will illustrate MgNet as a unified framework for convolutional neural networks and multigrid methods. Motivated by the general definition of neural operators and the MgNet structure, we propose MgNO, which utilizes multigrid structures to parameterize these linear operators between neurons, offering a new and concise architecture in operator learning. This approach provides both mathematical rigor and practical expressivity, with many interesting numerical properties and observations. Finally, I will offer some remarks on our most recent work in using multigrid structures to approximate and learn a specific operator in numerical PDEs. These results are based on joint work with Prof. Jinchao Xu and Dr. Xinliang Liu.