A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales

TL;DR

This paper enhances a multi-scale deep neural network algorithm to effectively solve complex nonlinear elliptic equations across multiple scales, demonstrating improved accuracy and efficiency in various dimensional settings.

Contribution

The paper introduces a smooth, localized activation function to improve the existing MscaleDNN algorithm for multi-scale nonlinear elliptic problems.

Findings

Effective in low-dimensional and high-dimensional spaces

Handles separable and non-separable scales

Demonstrates improved accuracy over previous methods

Abstract

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.