This research investigates the numerical approximation of the two-dimensional convection-dominated singularly perturbed problem on square, circular, and elliptic domains. Singularly perturbed boundary value problems present a significant challenge due to the presence of sharp boundary layers in their solutions. Additionally, the considered domain exhibits characteristic points, giving rise to a degenerate boundary layer problem. The stiffness of the problem is attributed to the sharp singular layers, which can result in substantial computational errors if not appropriately addressed. Traditional numerical methods typically require extensive mesh refinements near the boundary to achieve accurate solutions, which can be computationally expensive. To address the challenges posed by singularly perturbed problems, we employ physics-informed neural networks (PINNs). However, PINNs may struggle with rapidly varying singularly perturbed solutions over a small domain region, leading to inadequate resolution and potentially inaccurate or unstable results. To overcome this limitation, we introduce a semi-analytic method that augments PINNs with singular layers or corrector functions. Through our numerical experiments, we demonstrate significant improvements in both accuracy and stability, thus demonstrating the effectiveness of our proposed approach.

翻译：本研究探讨了在正方形、圆形及椭圆形域上的二维对流主导奇异摄动问题的数值逼近。奇异摄动边值问题因其解中存在尖锐边界层而构成重大挑战。此外，所考虑的区域呈现出特征点，导致产生退化边界层问题。该问题的刚性源于尖锐奇异层，若处理不当可能导致显著的计算误差。传统数值方法通常需要在边界附近进行精细网格剖分以获得精确解，但这会带来高昂的计算成本。为应对奇异摄动问题的挑战，我们采用了物理信息神经网络（PINNs）。然而，PINNs在处理小区域上快速变化的奇异摄动解时可能出现困难，导致分辨率不足及潜在的不准确或不稳定结果。为克服这一局限，我们引入了一种半解析方法，将奇异层或修正函数增强至PINNs中。通过数值实验，我们证明了该方法在精度和稳定性方面的显著提升，从而验证了所提出方法的有效性。