Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

翻译：奇异摄动边值问题因其边界层剧烈变化而对数值逼近构成重大挑战。这些剧烈的边界层导致解具有刚硬性，若处理不当将引发较大计算误差。众所周知，经典数值方法及物理信息神经网络（PINNs）均需在边界附近采取特殊处理（例如采用密集网格细化或更密集的配点），才能获得精确的近似解，尤其在刚硬边界层内部。本文改进了PINNs方法，构建了适用于奇异摄动边值问题的新型半解析SL-PINNs方法。通过进行边界层分析，我们首先找到描述边界层内刚硬解奇异行为的校正函数。随后通过将显式校正函数嵌入PINNs结构，或将校正函数与PINNs近似联合训练，获得奇异摄动问题的SL-PINN近似解。数值实验证实，我们提出的新型SL-PINN方法能够为刚硬解提供稳定且精确的近似。