報告題目：Transformer meets boundary value inverse problems: structure-conforming operator learning
報 告 人：郭汝馳 助理教授 University of California, Irvine
報告時間：2023年 04月28日（星期 五）9:00-10:00
騰訊會議: 984-552-816
校内聯系人：王翔 wxjldx@jlu.edu.cn
報告摘要：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 case study for a fundamental and 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? 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 frequency 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.
報告人簡介：郭汝馳，于2019年在弗吉尼亞理工大學取得博士學位，後于俄亥俄州立大學擔任Zassenhaus Assistant Professor，現于加州大學爾灣分校擔任Visiting Assistant Professor。主要研究領域為科學計算，特别是針對偏微分方程的數值方法，包括界面問題的非匹配網格算法，以及界面反問題的重構算法，包括浸入有限元算法、虛拟元算法，以及反問題的優化算法、直接法和深度學習算法等。在 SIAM J. Numer. Anal., M3AS, SIAM J. Sci. Comput., J. Comput. Phys., IMA J. Numer. Anal., ESAIM:M2AN, J. Sci. Comput., Comput. Methods Appl. Mech. Eng.,等計算數學領域雜志，以及深度學習會議ICLR上發表多篇文章。
