Singular Layer Physics-Informed Neural Network Method for Convection-Dominated Boundary Layer Problems in Two Dimensions

TL;DR

This work addresses the challenge of solving 2D convection-dominated, singularly perturbed boundary value problems with sharp boundary layers on square, circular, and elliptical domains. It develops a semi-analytic SL-PINN by deriving boundary-layer corrector functions from classical perturbation theory and embedding them into a two-layer PINN with hard boundary constraints, mitigating stiffness in $L_{\varepsilon}u^{\varepsilon} = -\varepsilon \Delta u^{\varepsilon} - u^{\varepsilon}_y = f$. The approach yields explicit correctors for channel and circular geometries and extends to elliptical domains, supported by convergence results and extensive numerical demonstrations across time-independent, time-dependent, and nonlinear variants. The SL-PINN consistently produces stable, highly accurate approximations for very small $\varepsilon$ (e.g., $10^{-8}$), outperforming conventional PINNs and offering a robust mesh-free alternative for multi-domain convection-diffusion problems. This framework has potential applications to Burgers-type equations and more complex geometries where boundary layers govern the solution structure.

Abstract

This research explores neural network-based numerical approximation of two-dimensional convection-dominated singularly perturbed problems on square, circular, and elliptic domains. Singularly perturbed boundary value problems pose significant challenges due to sharp boundary layers in their solutions. Additionally, the characteristic points of these domains give rise to degenerate boundary layer problems. The stiffness of these problems, caused by sharp singular layers, can lead to substantial computational errors if not properly addressed. Conventional neural network-based approaches often fail to capture these sharp transitions accurately, highlighting a critical flaw in machine learning methods. To address these issues, we conduct a thorough boundary layer analysis to enhance our understanding of sharp transitions within the boundary layers, guiding the application of numerical methods. Specifically, we employ physics-informed neural networks (PINNs) to better handle these boundary layer problems. However, PINNs may struggle with rapidly varying singularly perturbed solutions in small domain regions, leading to inaccurate or unstable results. To overcome this limitation, we introduce a semi-analytic method that augments PINNs with singular layers or corrector functions. Our numerical experiments demonstrate significant improvements in both accuracy and stability, showcasing the effectiveness of our proposed approach.

Figures (13)

• Figure 2.1: Numerical prediction of \ref{['e:square']} with $\varepsilon = 10^{-6}$ and $f=\sin(2 \pi x)$. For our simulations, we select a uniform grid of $50$ discretized points in each of the $x$ and $y$ directions.
• Figure 2.2: The one-dimensional profile of predicted solutions along the line $x = 0.25$.
• Figure 3.1: Numerical prediction of \ref{['e:circle_eq']} with $\varepsilon = 10^{-6}$ and $f=(1-x^2)^2$. For our simulations, we select a uniform grid of $50$ discretized points in each of the $\eta$ and $\tau$ directions.
• Figure 3.2: The one-dimensional profile of predicted solutions along the line $\tau = \pi/2$.
• Figure 3.3: Numerical prediction of \ref{['e:circle_eq']} with $\varepsilon = 10^{-6}$ and $f=1$ (non-compatible case). For our simulations, we select a uniform grid of $50$ discretized points in each of the $\eta$ and $\tau$ directions.

Theorems & Definitions (14)

• Theorem 2.1
• Theorem 3.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.1
• Lemma 6.1
• Lemma 6.2
• proof
• Lemma 6.3
• proof