用渐近分析思想改进神经网络，更准解边界层难题
ASPINN: An asymptotic strategy for solving singularly perturbed differential equations
- 将渐近分析融入神经网络，在边界层加指数层增强拟合能力
- 相比传统方法，训练成本更低且边界层解更精确
- 适合需要高精度求解奇异摄动方程的科研与工程场景
求解奇异摄动微分方程（SPDEs）因边界层处解的快速变化而面临挑战。本文提出渐近物理信息神经网络（ASPINN），作为物理信息神经网络（PINN）和广义同类物理信息神经网络（GKPINN）的推广。该方法基于渐近分析思想，通过在边界层设置指数层，显著提升对SPDEs的拟合能力。相比PINN，ASPINN具有更强的边界层建模能力；不同于GKPINN，其全连接层更少，训练成本更低。此外，理论分析表明，ASPINN在边界层解的逼近精度更高，实验验证了其在多种类型SPDEs上的有效性。进一步地，用切比雪夫型科尔莫戈罗夫-阿诺德网络（Chebyshev-KAN）替代MLP，在多个实验中获得更优性能。
原文摘要 · Abstract (English)
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.
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