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Int Deep: A Deep Learning Initialized Iterative Method for Nonlinear Problems

Speaker

Jianguo Huang, School of Mathematical Sciences, and LSC-MOE, Shanghai Jiao Tong University

Time

2019.11.20 12:20-13:50

Venue

Room 306, No.5 Science Building

Abstract

In this talk, we are concerned with a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an expectation minimization problem formulated from a given nonlinear PDE is approximately resolved with mesh-free deep neural networks to parametrize the solution space. In the second phase, a solution ansatz of the finite element method to solve the given PDE is obtained from the approximate solution in the first phase, and the ansatz can serve as a good initial guess such that Newton’s method for solving the nonlinear PDE is able to converge to the ground truth solution with high-accuracy quickly. Systematic theoretical analysis is provided to justify the Int-Deep framework for several classes of problems. Numerical results show that the Int-Deep outperforms existing purely deep learning-based methods or traditional iterative methods (e.g., Newton’s method and the Picard iteration method). This is a joint work with Hao-Qin Wang (SJTU) and Yaizhao Yang (Purdue University)