  This paper focuses on proposing 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).
