Year: 2022
Author: Jihong Wang, Zhi-Qin John Xu, Jiwei Zhang, Yaoyu Zhang
CSIAM Transactions on Applied Mathematics, Vol. 3 (2022), Iss. 2 : pp. 299–317
Abstract
This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a particular Poisson problem, where the information of the right-hand side of the equation $f$ is only available at $n$ sample points, that is, $f$ is known at finite sample points. Through both theoretical and numerical studies, we show that solution of the R-G method converges to a piecewise linear function for the one dimensional problem or functions of lower regularity for high dimensional problems. With the same setting, DNNs however learn a relative smooth solution regardless of the dimension, this is, DNNs implicitly bias towards functions with more low-frequency components among all functions that can fit the equation at available data points. This bias is explained by the recent study of frequency principle. In addition to the similarity between the traditional numerical methods and DNNs in the approximation perspective, our work shows that the implicit bias in the learning process, which is different from traditional numerical methods, could help better understand the characteristics of DNNs.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2020-0006
CSIAM Transactions on Applied Mathematics, Vol. 3 (2022), Iss. 2 : pp. 299–317
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Deep learning Ritz-Galerkin method partial differential equations F-Principle.