Two-scale Neural Networks for Singularly Perturbed Dynamical Systems with Multiple Parameters

TL;DR

This paper introduces a two-scale neural network approach for dynamical systems with multiple small parameters, using an effective scale and scale-aware features to accurately capture sharp solution transitions.

Contribution

It extends previous methods to handle multiple small parameters by defining an effective scale and incorporating scale-aware features into the neural network.

Findings

Successfully handles coupled systems with multiple high-contrast small parameters.

Achieves satisfactory accuracy in capturing solution features induced by small parameters.

Demonstrates effectiveness across various dynamical systems.

Abstract

We extend our two-scale neural-network method for scalar singularly perturbed problems with one small parameter to dynamical systems with multiple small parameters. To accommodate multiple small parameters, we use a single effective scale parameter defined as the geometric mean of all parameters. We thus augment the network input with a scale-aware feature, enabling it to capture sharp solution transitions intrinsically. Numerical experiments across a range of dynamical systems demonstrate that the proposed framework can handle coupled systems with multiple and high-contrast small parameters and obtain satisfactory accuracy in capturing solution features induced by small parameters.