报告人：陈龙教授
报告题目：Transformer Meets Boundary Value Inverse Problems
报告时间：2022年11月2日9:00-10:00
地点：Zoom会议号:84822472533，密码：221102
报告人简介：
Long Chen is currently a Professor of Mathematics at University of California at Irvine (UCI). He graduated from Nanjing University in 1997, obtained a master's degree from Peking University in 2000, and finished Ph.D from Pennsylvania State University in 2005. His doctoral supervisor is Professor Jinchao Xu. From 2005 to 2007, he worked as a postdoctoral fellow at the University of California, San Diego and the University of Maryland, College Park. Since 2007, he has been working at UCI, and was tenured in 2011, and promoted to full professor in 2015.
Professor Chen's research field is the numerical solution of partial differential equations, especially the design and analysis of finite element methods. In addition, Professor Chen developed the iFEM finite element software package, which provided great convenience for the teaching and research of finite element methods. Professor Chen has published more than 60 academic papers in internationally renowned journals, and serves on the editorial board of several SCI journals. Since his work till now, Professor Chen has been constantly supported by the National Science Foundation.
For more detail, please visit Professor Chen's Website: https://www.math.uci.edu/~chenlong/.
报告内容简介：
A Transformer-based deep direct sampling method is proposed for solving a class of boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental but critical question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural network? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions in different frequencies as different input channels. Then, by introducing learnable non-local kernel, the approximation of direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severely ill-posed nonlinear inverse problem. The new method achieves superior accuracy over its predecessors and contemporary operator learners, as well as shows robustness with respect to noise.
This research shall strengthen the insights that the attention mechanism, despite being invented for natural language processing tasks, offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.
This is a joint work with Ruchi Guo (UCI) and Shuhao Cao (University of Missouri-Kansas City)