Two-scale Neural Networks for Partial Differential Equations with Small Parameters

TL;DR

This paper introduces a two-scale neural network approach for efficiently solving PDEs with small parameters, directly embedding these parameters into the network architecture to improve accuracy without complex feature searches.

Contribution

The paper presents a novel two-scale neural network architecture that directly incorporates small parameters, simplifying the solution process for PDEs with small parameters compared to existing methods.

Findings

Achieves accurate solutions capturing large derivatives

Avoids complex Fourier feature searches

Demonstrates effectiveness across multiple numerical examples

Abstract

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.