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報(bào)告題目：Transferable Neural Networks for Partial Differential Equations

主講人：鞠立力教授（美國(guó)南卡萊羅納大學(xué)）

時(shí)間：2024年5月23日（周四）11:00 a.m.

地點(diǎn)：北院卓遠(yuǎn)樓305會(huì)議室

主辦單位：統(tǒng)計(jì)與數(shù)學(xué)學(xué)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(jiǎn)介：

鞠立力,，1995年畢業(yè)于武漢大學(xué)數(shù)學(xué)系獲數(shù)學(xué)學(xué)士學(xué)位，1998年在中國(guó)科學(xué)院計(jì)算數(shù)學(xué)與科學(xué)工程計(jì)算研究所獲得計(jì)算數(shù)學(xué)碩士學(xué)位,，2002年在美國(guó)愛(ài)荷華州立大學(xué)獲得應(yīng)用數(shù)學(xué)博士學(xué)位,。2002-2004年在美國(guó)明尼蘇達(dá)大學(xué)數(shù)學(xué)與應(yīng)用研究所從事博士后研究。隨后進(jìn)入美國(guó)南卡羅萊納大學(xué)工作,，歷任數(shù)學(xué)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(jì)算、深度學(xué)習(xí)方法,、計(jì)算機(jī)視覺(jué),、高性能科學(xué)計(jì)算及其材料與地球科學(xué)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(xué)術(shù)引用約6200多次,。自2006年起已主持了十多項(xiàng)由美國(guó)國(guó)家科學(xué)基金會(huì)和能源部資助的科研項(xiàng)目,。2012至2017年曾任SIAM J. Numer. Anal.的副主編，目前擔(dān)任Math. Comp., J. Sci. Comput.,，Numer. Meths. PDEs等國(guó)際計(jì)算與應(yīng)用數(shù)學(xué)領(lǐng)域?qū)W術(shù)期刊的副主編,。