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This research explores neural network-based numerical approximation oftwo-dimensional convection-dominated singularly perturbed problems on square,circular, and elliptic domains. Singularly perturbed boundary value problemspose significant challenges due to sharp boundary layers in their solutions.Additionally, the characteristic points of these domains give rise todegenerate boundary layer problems. The stiffness of these problems, caused bysharp singular layers, can lead to substantial computational errors if notproperly addressed. Conventional neural network-based approaches often fail tocapture these sharp transitions accurately, highlighting a critical flaw inmachine learning methods. To address these issues, we conduct a thoroughboundary layer analysis to enhance our understanding of sharp transitionswithin the boundary layers, guiding the application of numerical methods.Specifically, we employ physics-informed neural networks (PINNs) to betterhandle these boundary layer problems. However, PINNs may struggle with rapidlyvarying singularly perturbed solutions in small domain regions, leading toinaccurate or unstable results. To overcome this limitation, we introduce asemi-analytic method that augments PINNs with singular layers or correctorfunctions. Our numerical experiments demonstrate significant improvements inboth accuracy and stability, showcasing the effectiveness of our proposedapproach.

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