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報告題目：Transferable Neural Networks for Partial Differential Equations

主講人：鞠立力教授（美國南卡萊羅納大學）

時間：2024年5月23日（周四）11:00 a.m.

地點：北院卓遠樓305會議室

主辦單位：統(tǒng)計與數(shù)學學院

摘要：Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training.In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model, and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

主講人簡介：

鞠立力,，1995年畢業(yè)于武漢大學數(shù)學系獲數(shù)學學士學位,，1998年在中國科學院計算數(shù)學與科學工程計算研究所獲得計算數(shù)學碩士學位,，2002年在美國愛荷華州立大學獲得應(yīng)用數(shù)學博士學位。2002-2004年在美國明尼蘇達大學數(shù)學與應(yīng)用研究所從事博士后研究,。隨后進入美國南卡羅萊納大學工作,，歷任數(shù)學系助理教授（2004-2008）、副教授（2008-2012）,、教授（2013-至今）,。主要從事偏微分方程數(shù)值方法與分析、非局部模型與計算,、深度學習方法,、計算機視覺、高性能科學計算及其材料與地球科學中的應(yīng)用等方面的研究工作,。已發(fā)表科研論文150多篇,，Google學術(shù)引用約6200多次。自2006年起已主持了十多項由美國國家科學基金會和能源部資助的科研項目,。2012至2017年曾任SIAM J. Numer. Anal.的副主編,，目前擔任Math. Comp., J. Sci. Comput.，Numer. Meths. PDEs等國際計算與應(yīng)用數(shù)學領(lǐng)域?qū)W術(shù)期刊的副主編,。