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).

本文提出了一种针对低维非线性偏微分方程（PDE）的深度学习初始化迭代方法（Int-Deep）。相应的框架包括两个阶段。在第一阶段，使用无网格的深度神经网络近似解决由给定的非线性PDE公式化的期望最小化问题，以对求解空间进行参数化。在第二阶段中，从第一阶段的近似解中获得用于求解给定PDE的有限元方法的求解ansatz，并且该ansatz可以作为一个很好的初始猜测，从而可以使用牛顿法或其他迭代方法来求解。非线性PDE可以快速高精度地收敛到地面真值解。提供了系统的理论分析，以证明针对多种类型问题的Int-Deep框架的合理性。数值结果表明，Int-Deep优于现有的纯粹基于深度学习的方法或传统的迭代方法（例如，Newton方法和Picard迭代方法）。