A Priori Analysis to Numerical PDEs by Neural Network Functions

主讲人：洪庆国  美国宾州州立大学

时间：2021年11月17日10：00

地点：腾讯会议 845 355 532

举办单位：数理学院

主讲人介绍：洪庆国，博士，先后在奥地利科学院Radon研究所(RICAM)，德国Duisburg-Essen University，美国The Pennsylvania  State University 从事博士后研究。目前研究兴趣包括迭代法，间断有限元方法及应用。在SIAM J. Numer. Anal., Numer.  Math., Comput. Methods Appl. Mech. Engrg.和中国科学-数学等国内外期刊发表系列论文。

内容介绍：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.