Int-Deep: A Deep Learning Initialized Iterative Method for Nonlinear Problems

TL;DR

Int-Deep combines deep learning and iterative methods to efficiently solve nonlinear PDEs, providing a novel framework that improves convergence and accuracy over existing methods.

Contribution

The paper introduces a new deep learning initialized iterative framework for nonlinear PDEs, integrating neural networks with finite element methods for enhanced solution accuracy.

Findings

Outperforms existing deep learning-based methods.

Achieves high-accuracy solutions with faster convergence.

Provides theoretical justification for the framework.

Abstract

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).