Implicit Bias in Understanding Deep Learning for Solving PDEs Beyond Ritz-Galerkin Method (CSIAM Transactions on Applied Mathematics, 2022)
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 aparticular 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-Gmethod converges to a piecewise linear function for the one dimensional problem orfunctions 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 amongall functions that can fit the equation at available data points. This bias is explainedby the recent study of frequency principle. In addition to the similarity between thetraditional numerical methods and DNNs in the approximation perspective, our workshows that the implicit bias in the learning process, which is different from traditionalnumerical methods, could help better understand the characteristics of DNNs.