报告题目：Statistical limit and Computational Methods for Deep Learning Based PDE Solver

报告时间：2022-03-01  10:00 - 12:00

报告人：陆一平  Stanford University

ZOOMID: 937 0043 1941

密码：20220301

Abstract：In this talk, we aim study the statistical limits of machine learning from physical observations. Specifically, we consider two recently developed methods, the Deep Ritz Method (DRM) and Physics Informed Neural Networks (PINNs), for solving Elliptic partial differential equations (PDEs) from random samples. We establish upper and lower bounds for both methods and prove that PINN and a modified version of DRM can achieve minimax optimal bounds over Sobolev spaces, which improve upon concurrent developed upper bounds for this problem via a fast rate generalization bound. In the analysis, we find out that generalization of the gradient term is hard where need more samples for the deep Ritz methods and the sparsity of neurons (number of weights) is not a good generalization complexity measure for the gradients. In the second part of this talk, I’ll talk about computational feasible algorithm for optimize PINNs and DRM to achieve this statistical optimal bound. We consider a model parameterized by a Reproducing Kernel Hilbert space and observes this although both methods can achieve information theoretical optimal bound under gradient descent, PINN actually convergence faster. We refer this phenomenon “Sobolev implicit acceleration”.