Singular Layer Physics-Informed Neural Networks for Stiff Reaction-Diffusion Equations on Smooth Convex Domains

Abstract

In this thesis, we develop and apply a novel machine learning approach, the Singular Layer Physics- Informed Neural Networks (sl-PINN), for singularly perturbed elliptic boundary value problems. These problems, which arise in the analysis of reaction-reaction systems, are challenging to approximate nu- merically due to the boundary layers and sharp transitions. We address these challenges by adding boundary layer correctors into the physics-informed neural networks framework, improving its ability to approximate solutions in smooth domains with curved boundaries. Our method is shown to effec- tively capture the solutions, offering a significant improvement over traditional numerical methods. The effectiveness of sl-PINN is validated through numerical experiments, in which it accurately solves stiff reaction-diffusion equations with high computational efficiency.