报告题目：Transformer Meets Boundary Value Inverse Problems

报 告 人：Long Chen 教授

所在单位：University of California at Irvine (UCI)

报告时间：2022年11月04日 星期五 上午11:00

报告地点：#腾讯会议：108 807 618

校内联系人：贾继伟 jiajiwei@jlu.edu.cn

报告摘要： 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).

报告人简介：陈龙目前是加州大学欧文分校（UCI）的数学教授。 1997年毕业于南京大学，2000年获北京大学硕士学位，2005年获宾夕法尼亚州立大学博士学位。博士生导师为许进超教授。 2005年至2007年在加州大学圣地亚哥分校和马里兰大学帕克分校从事博士后研究。 2007年起在UCI工作，2011年获得终身教职，2015年晋升为正教授。

陈教授的研究领域是偏微分方程的数值解，尤其是有限元方法的设计与分析。此外，陈教授还开发了iFEM有限元软件包，为有限元方法的教学和研究提供了极大的便利。陈教授在国际知名期刊发表学术论文60余篇，担任多个SCI期刊编委。

更多详情请访问陈教授网站：https://www.math.uci.edu/~chenlong/。