Solving Singularly Perturbed Differential Equations (SPDEs) presents challenges due to the rapid change of their solutions at the boundary layer. In this manuscript, We propose Asymptotic Physics-Informed Neural Networks (ASPINN), a generalization of Physics-Informed Neural Networks (PINN) and General-Kindred Physics-Informed Neural Networks (GKPINN) approaches. This is a decomposition method based on the idea of asymptotic analysis. Compared to PINN, the ASPINN method has a strong fitting ability for solving SPDEs due to the placement of exponential layers at the boundary layer. Unlike GKPINN, ASPINN lessens the number of fully connected layers, thereby reducing the training cost more effectively. Moreover, ASPINN theoretically approximates the solution at the boundary layer more accurately, which accuracy is also improved compared to GKPINN. We demonstrate the effect of ASPINN by solving diverse classes of SPDEs, which clearly shows that the ASPINN method is promising in boundary layer problems. Furthermore, we introduce Chebyshev Kolmogorov-Arnold Networks (Chebyshev-KAN) instead of MLP, achieving better performance in various experiments.

本研究针对奇异摄动微分方程（SPDEs）在边界层快速变化导致的求解难题，提出了渐近物理信息神经网络（ASPINN）。该方法通过在边界层设置指数层，提高了对SPDEs的拟合能力，并有效减少了全连接层的数量，降低了训练成本。实验结果表明，ASPINN在边界层问题上表现出色，相较于以往方法在准确性和性能上都有明显提升。

ASPINN：解决奇异摄动微分方程的渐近策略