A Unified Weighted-Loss Physics-Informed Neural Network for Boundary Layer Problems in Singularly Perturbed PDEs

TL;DR

This paper introduces a unified weighted-loss physics-informed neural network framework for solving boundary layer problems in singularly perturbed PDEs, automatically identifying boundary layers without complex architectures.

Contribution

It presents a novel, generalizable neural network approach that only requires boundary layer thickness as prior knowledge, avoiding problem-specific modifications.

Findings

Robust performance for boundary layer thickness as small as 10^{-10}

Automatically identifies boundary layer locations during training

Maintains high solution accuracy across various systems and domains

Abstract

Singularly perturbed partial differential equations arise in many applications, including magnetohydrodynamic duct flows, chemical reaction transport systems, and Poisson Boltzmann electrostatics. These problems are characterized by sharp boundary layers and pronounced multiscale behavior, posing significant challenges for numerical methods. Existing approaches, particularly machine learning based methods, often rely on explicit asymptotic decompositions or specialized architectures, increasing implementation complexity and leading to optimization imbalance in stiff regimes. In this work, we propose a unified learning framework based on a weighted loss formulation within the standard physics informed neural network setting. The proposed method requires only prior knowledge of the boundary layer thickness, while the boundary layer locations are automatically identified during training. The resulting formulation avoids problem specific architectural modifications and remains applicable across different equation types. Numerical experiments on both scalar and coupled reaction diffusion and convection diffusion reaction systems, defined on regular and irregular domains, demonstrate robust performance for boundary layer thickness as small as $10^{-10}$ while maintaining high solution accuracy.