求解奇异摄动微分方程的广义同族物理信息神经网络
General-Kindred Physics-Informed Neural Network to the Solutions of Singularly Perturbed Differential Equations
AI总结：
针对传统PINN无法逼近奇异摄动微分方程边界层的问题，提出GKPINN方法，利用渐近分析构建辅助网络，使$L_2$误差降低两到四个数量级并大幅加快收敛，在极端参数下仍具出色泛化能力。
摘要：
物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）领域的一个有前景的研究方向。处理奇异摄动问题在PINN领域仍是一个艰巨的挑战。奇异摄动问题的解通常表现出尖锐的边界层和陡峭的梯度，传统PINN无法实现对边界层的逼近。在本文中，我们提出了广义同族物理信息神经网络（GKPINN）来求解奇异摄动微分方程（SPDEs）。该方法利用渐近分析从方程中获取边界层的先验知识，并建立了一个新型网络来辅助PINN逼近边界层。通过求解一维、二维和时变SPDE方程的实例，将其与传统PINN进行了比较。研究结果突显了我们新方法GKPINN的卓越性能，与既定的PINN方法相比，它在将$L_2$误差降低两到四个数量级方面取得了显著提升。这种显著改善伴随着收敛速度的大幅加快，且未牺牲应用所需的高精度。此外，GKPINN在摄动参数为${1\ imes10}^{-38}$的极端情况下依然表现良好，展现了其出色的泛化能力。
英文摘要：
Physics-Informed Neural Networks (PINNs) have become a promising research direction in the field of solving Partial Differential Equations (PDEs). Dealing with singular perturbation problems continues to be a difficult challenge in the field of PINN. The solution of singular perturbation problems often exhibits sharp boundary layers and steep gradients, and traditional PINN cannot achieve approximation of boundary layers. In this manuscript, we propose the General-Kindred Physics-Informed Neural Network (GKPINN) for solving Singular Perturbation Differential Equations (SPDEs). This approach utilizes asymptotic analysis to acquire prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approximating the boundary layer. It is compared with traditional PINN by solving examples of one-dimensional, two-dimensional, and time-varying SPDE equations. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the $L_2$ error by two to four orders of magnitude compared to the established PINN methodology. This significant improvement is accompanied by a substantial acceleration in convergence rates, without compromising the high precision that is critical for our applications. Furthermore, GKPINN still performs well in extreme cases with perturbation parameters of ${1\times10}^{-38}$, demonstrating its excellent generalization ability.
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