Int-Deep: A deep learning initialized iterative method for nonlinear problems
Int-Deep: A deep learning initialized iterative method for nonlinear problems
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DOI：
10.1016/j.jcp.2020.109675
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发表时间：
2020-10-15
影响因子：
4.1
通讯作者：
Yang, Haizhao
中科院分区：
文献类型：
--
作者：
Huang, Jianguo;Wang, Haoqin;Yang, Haizhao
提出了一种求解低维非线性偏微分方程的深度学习初始化迭代方法（Int-Deep）。相应的框架由两个阶段组成。在第一阶段，利用无网格深度神经网络对给定非线性偏微分方程的期望最小化问题进行近似求解，使解空间参数化。在第二阶段，由第一阶段的近似解得到求解给定偏微分方程的有限元法解的解析解，该解析解可以作为一个很好的初始猜测，使得牛顿法或其他求解非线性偏微分方程的迭代方法能够快速高精度地收敛到地面真值解。系统的理论分析证明了intdeep框架适用于几类问题。数值结果表明，Int-Deep优于现有的纯深度学习方法或传统的迭代方法（如牛顿法和皮卡德迭代法）。(C) 2020爱思唯尔公司版权所有。
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.