Chang-Ock Lee, Professor , KAIST, South Koera

Chensong Zhang, Professor

Nonoverlapping Domain Decomposition Methods for Extreme Learning Machines

15:00-16:00 July 15(Tuesday), Zoom Meeting : 814 6993 3858, Passcode: 851014

Abstract: Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. By preseting hidden layer parameters and solving for last layer coefficients via a least squares method, it can typically solve partial differential equations faster and more accurately than Physics Informed Neural Networks. However, it remains computationally expensive when high accuracy requires large least squares problems to be solved. In this talk, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. We introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. Furthermore, we present a coarse space for ELMs, which enables further acceleration of their training. Key to the performance of the proposed method is a Neumann-Neumann acceleration that utilizes the coarse space.

Bio: Chang-Ock Lee received the Ph.D. degree in mathematics from the University of Wisconsin, Madison, WI, USA, in 1995. From 1995 to 2000, he was an Assistant Professor with the Department of Mathematics, Inha University, Incheon, Korea. In 2000, he joined the Division of Applied Mathematics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, where he is currently a Professor of mathematics with the Department of Mathematical Sciences. His research interests include efficient numerical solvers in parallel environment, image processing based on PDEs, and Scientific machine learning. Prof. Lee served as a President for the Korean Society for Industrial and Applied Mathematics (KSIAM) from 2021 to 2022.