A novel paradigm for solving PDEs: multi scale neural computing

TL;DR

This paper introduces multi scale neural computing (MSNC), a hybrid approach combining neural networks and finite difference methods to efficiently solve PDEs with higher accuracy and lower computational costs.

Contribution

The paper presents a novel hybrid paradigm that leverages spectral bias and local approximation to improve PDE solving efficiency and accuracy over traditional methods.

Findings

Achieves 10-20 times higher accuracy than standard FDM

Reduces computational costs by 4-16 times

Demonstrates stable convergence and effective boundary condition satisfaction

Abstract

Numerical simulation is dominant in solving partial difference equations (PDEs), but balancing fine-grained grids with low computational costs is challenging. Recently, solving PDEs with neural networks (NNs) has gained interest, yet cost-effectiveness and high accuracy remains a challenge. This work introduces a novel paradigm for solving PDEs, called multi scale neural computing (MSNC), considering spectral bias of NNs and local approximation properties in the finite difference method (FDM). The MSNC decomposes the solution with a NN for efficient capture of global scale and the FDM for detailed description of local scale, aiming to balance costs and accuracy. Demonstrated advantages include higher accuracy (10 times for 1D PDEs, 20 times for 2D PDEs) and lower costs (4 times for 1D PDEs, 16 times for 2D PDEs) than the standard FDM. The MSNC also exhibits stable convergence and rigorous boundary condition satisfaction, showcasing the potential for hybrid of NN and numerical method.