A multi-scale framework for neural network enhanced methods to the solution of partial differential equations

TL;DR

This paper introduces a multi-scale framework combining traditional numerical methods and neural networks to efficiently approximate solutions to partial differential equations by decomposing the problem into coarse and fine scales.

Contribution

It presents a novel multi-scale approach integrating neural networks with finite element methods for PDE solutions, considering interactions across scales.

Findings

Effective decomposition into coarse and fine solutions.

Neural networks improve fine-scale approximation.

Framework tested successfully on various PDE cases.

Abstract

In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of the target problem could be decomposed into two parts, i.e. the coarse scale solution and the fine scale solution. In the coarse scale, the conventional numerical methods (e.g. finite element methods) are applied and the coarse scale solution could be obtained. In the fine scale, the neural networks is introduced to formulate the solution. The custom loss functions are developed by taking into account the governing equations and boundary conditions of PDEs, the constraints and the interaction from coarse scale. The proposed methods are illustrated and examined by various of testing cases.