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This paper aims at studying the difference between Ritz-Galerkin (R-G) methodand deep neural network (DNN) method in solving partial differential equations(PDEs) to better understand deep learning. To this end, we consider solving aparticular Poisson problem, where the information of the right-hand side of theequation f is only available at n sample points, that is, f is known at finitesample points. Through both theoretical and numerical studies, we show thatsolution of the R-G method converges to a piecewise linear function for the onedimensional (1D) problem or functions of lower regularity for high dimensionalproblems. With the same setting, DNNs however learn a relative smooth solutionregardless of the dimension, this is, DNNs implicitly bias towards functionswith more low-frequency components among all functions that can fit theequation at available data points. This bias is explained by the recent studyof frequency principle (Xu et al., (2019) [17] and Zhang et al., (2019) [11,19]). In addition to the similarity between the traditional numerical methodsand DNNs in the approximation perspective, our work shows that the implicitbias in the learning process, which is different from traditional numericalmethods, could help better understand the characteristics of DNNs.

Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs