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

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.

Figures (10)

• Figure 1: Schematic illustration of the neural network architecture for the one-dimensional case. The network takes three inputs, $x$, $\phi_L/\varepsilon$, and $\phi_R/\varepsilon$, which are fed into three separate fully connected blocks with a variable number of hidden layers. The outputs of the hidden layers are subsequently combined and mapped to a single scalar output $u$.
• Figure 2: Neural network architecture for the two-dimensional regular domain. Six inputs, $x$, $y$, and the scaled level-set functions $\phi_L/\varepsilon$, $\phi_R/\varepsilon$, $\phi_B/\varepsilon$, and $\phi_T/\varepsilon$, are processed by seven fully connected blocks to realize the solution decomposition, and the final output $u$ is obtained through algebraic combinations of the block outputs.
• Figure 3: Neural network architecture for the two-dimensional irregular domain. The inputs consist of $x$, $y$, and the level-set function $\phi/\varepsilon$, and two fully connected blocks are combined additively to produce the solution.
• Figure 4: One-dimensional convection–diffusion–reaction problem with $\varepsilon=10^{-10}$ in Example 1. (a) Learned regular component $u_r$, which remains smooth over the entire domain. (b) Learned singular component $u_s$, accurately capturing the boundary layer near the left endpoint while remaining nearly constant elsewhere. (c) Neural-network approximation $u_{\mathcal{N}}$ with an zoom-in inset view near the boundary layer at $x=0$. (d) Absolute error $|u-u_{\mathcal{N}}|$, showing that the maximum error is localized near the boundary layer and remains of comparable magnitude throughout the domain.
• Figure 5: One-dimensional reaction–diffusion problem in Example 2 with $\varepsilon=10^{-10}$. (a) Neural-network approximation $u_{\mathcal{N}}$. An inset provides a zoomed-in view near $x=1$ to highlight the boundary-layer structure. (b) Absolute error $|u-u_{\mathcal{N}}|$.