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.

翻译：本研究探索了基于神经网络的二维对流主导奇异摄动问题在正方形、圆形和椭圆域上的数值逼近方法。奇异摄动边值问题因其解中存在尖锐边界层而带来重大挑战。此外，这些域的特征点会引发退化边界层问题。由尖锐奇异层引起的这些问题刚度，若处理不当可能导致显著的计算误差。传统的基于神经网络的方法通常无法准确捕捉这些急剧过渡，这凸显了机器学习方法的一个关键缺陷。为解决这些问题，我们进行了深入的边界层分析，以增强对边界层内急剧过渡的理解，从而指导数值方法的应用。具体而言，我们采用物理信息神经网络（PINNs）来更好地处理这些边界层问题。然而，PINNs在小区域域内快速变化的奇异摄动解方面可能遇到困难，导致结果不准确或不稳定。为克服这一局限，我们引入了一种半解析方法，通过奇异层或校正函数来增强PINNs。我们的数值实验表明，该方法在精度和稳定性方面均有显著提升，证明了所提方法的有效性。