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Int-Deep: A deep learning initialized iterative method for nonlinear problems
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DOI：10.1016/j.jcp.2020.109675.png)
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摘要
En 中文
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.
Keyword:
Deep learning
Nonlinear problems
Partial differential equations
Eigenvalue problems
Iterative methods
Fast and accurate
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期刊
IF:
3.8
论文数:
1.6W
被引数:
7.4W
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引用论文
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