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

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.