@Article{CSIAM-AM-3-2,
author = {Wang, Jihong and Zhi-Qin, Xu, John and Zhang, Jiwei and Zhang, Yaoyu},
title = {Implicit Bias in Understanding Deep Learning for Solving PDEs Beyond Ritz-Galerkin Method},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2022},
volume = {3},
number = {2},
pages = {299--317},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2020-0006},
url = {https://global-sci.com/article/82319/implicit-bias-in-understanding-deep-learning-for-solving-pdes-beyond-ritz-galerkin-method}
}