A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations

TL;DR

This paper introduces a multi-level deep learning framework inspired by multigrid methods for solving PDEs, employing adaptive sampling and iterative training to improve accuracy in high-frequency regions.

Contribution

It presents a novel multi-level training approach with adaptive sampling for deep PDE solvers, enhancing focus on complex regions and leveraging neural network generalization.

Findings

Effective in capturing high-frequency PDE components

Improves convergence and accuracy over traditional methods

Validated through rigorous proofs and numerical experiments

Abstract

In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method.