Learn Singularly Perturbed Solutions via Homotopy Dynamics

TL;DR

This paper introduces a homotopy dynamics method to improve neural network training for singularly perturbed PDEs, addressing near-singularities and enhancing convergence and accuracy.

Contribution

The paper proposes a novel homotopy dynamics approach that effectively manages parameters causing singularities in PDEs, with theoretical analysis and empirical validation.

Findings

Accelerates convergence in singularly perturbed PDEs

Improves solution accuracy for challenging PDE problems

Provides a robust framework for neural PDE solvers

Abstract

Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters in the PDEs that introduce near-singularities in the loss function. In this study, we overcome this challenge by introducing a novel method based on homotopy dynamics to effectively manipulate these parameters. From a theoretical perspective, we analyze the effects of these parameters on training difficulty in these singularly perturbed problems and establish the convergence of the proposed homotopy dynamics method. Experimentally, we demonstrate that our approach significantly accelerates convergence and improves the accuracy of these singularly perturbed problems. These findings present an efficient optimization strategy leveraging homotopy dynamics, offering a robust framework to extend the applicability of neural networks for solving singularly perturbed differential equations.