主講人：洪慶國(guó)  美國(guó)賓州州立大學(xué)

時(shí)間：2021年11月17日10：00

地點(diǎn)：騰訊會(huì)議 845 355 532

舉辦單位：數(shù)理學(xué)院

主講人介紹：洪慶國(guó)，博士，先后在奧地利科學(xué)院Radon研究所(RICAM)，德國(guó)Duisburg-Essen University，美國(guó)The Pennsylvania  State University 從事博士后研究。目前研究興趣包括迭代法，間斷有限元方法及應(yīng)用。在SIAM J. Numer. Anal., Numer.  Math., Comput. Methods Appl. Mech. Engrg.和中國(guó)科學(xué)-數(shù)學(xué)等國(guó)內(nèi)外期刊發(fā)表系列論文。

內(nèi)容介紹：Methods for solving PDEs using neural networks have recently become a very  important topic. We provide an a priori error analysis for such methods which is  based on the K1(D)-norm of the solution. We show that the resulting constrained  optimization problem can be efficiently solved using a greedy algorithm, which  replaces stochastic gradient descent. Following this, we show that the error  arising from discretizing the energy integrals is bounded both in the  deterministic case, i.e. when using numerical quadrature, and also in the  stochastic case, i.e. when sampling points to approximate the integrals. In the  later case, we use a Rademacher complexity analysis, and in the former we use  standard numerical quadrature bounds. This extends existing results to methods  which use a general dictionary of functions to learn solutions to PDEs and  importantly gives a consistent analysis which incorporates the optimization,  approximation, and generalization aspects of the problem. In addition, the  Rademacher complexity analysis is simplified and generalized, which enables  application to a wide range of problems.