主講人：郭汝馳 加州大學(xué)爾灣分校訪(fǎng)問(wèn)助理教授

時(shí)間：2023年4月26日10：00

地點(diǎn)：騰訊會(huì)議 345 803 597

舉辦單位：數(shù)理學(xué)院

主講人介紹：郭汝馳博士，于2019年在弗吉尼亞理工大學(xué)取得博士學(xué)位，后于俄亥俄州立大學(xué)擔(dān)任Zassenhaus Assistant Professor，現(xiàn)于加州大學(xué)爾灣分校擔(dān)任Visiting Assistant Professor。主要研究領(lǐng)域?yàn)榭茖W(xué)計(jì)算，特別是針對(duì)偏微分方程的數(shù)值方法，包括界面問(wèn)題的非匹配網(wǎng)格算法，以及界面反問(wèn)題的重構(gòu)算法，包括浸入有限元算法、虛擬元算法，以及反問(wèn)題的優(yōu)化算法、直接法和深度學(xué)習(xí)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.等計(jì)算數(shù)學(xué)領(lǐng)域雜志上發(fā)表多篇文章。

內(nèi)容介紹：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 equa-tion-based feature map to yield 2D harmonic extensions in different frequency input channels. Then, by introducing learnable non-local kernel, the approxima-tion of direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severe-ly ill-posed nonlinear inverse problem. The new method achieves superior accura-cy 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 math-ematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.