Int-Deep: A deep learning initialized iterative method for nonlinear problems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:419

Article

Huang, Jianguo^1,2  Wang, Haoqin^1,2  Yang, Haizhao^3,4 

[1] Shanghai Jiao Tong Univ, Sch Math Sci, Shanghai 200240, Peoples R China

[2] Shanghai Jiao Tong Univ, MOE LSC, Shanghai 200240, Peoples R China

[3] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA

[4] Natl Univ Singapore, Dept Math, Singapore, Singapore

关键词: Deep learning;    Nonlinear problems;    Partial differential equations;    Eigenvalue problems;    Iterative methods;    Fast and accurate;   

DOI  :  10.1016/j.jcp.2020.109675

来源: Elsevier

【 摘 要 】

This paper proposes 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 or other iterative methods for solving the nonlinear PDE are 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). (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free