用变分物理信息神经网络高效求解二维奇异摄动问题
Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems
- 以神经网络构造试函数，张量积帽函数作检验函数
- 在最大范数和L2范数下均达到高精度，有效捕捉边界层
- 适合需要高精度求解多尺度奇异摄动问题的研究者
本研究提出一种基于Petrov-Galerkin的变分物理信息神经网络（VPINN），用于高效求解含一个或两个小摄动参数的二维奇异摄动问题。方法利用神经网络构建试解空间，采用张量积帽函数作为检验函数以实现变分形式。为精确解析尖锐边界层，采用Petrov-Galerkin格式实现变分形式。狄利克雷边界条件直接施加，源项通过自动微分计算。对标准二维问题的数值实验表明，该方法在最大范数和L2范数下均表现出高精度，验证了Petrov-Galerkin VPINN在准确捕捉二维奇异摄动问题多尺度特征方面的高效性与鲁棒性。
原文摘要 · Abstract (English)
This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters. The approach employs neural networks to construct the trial solution space, while tensor-product hat functions are adopted as test functions to enforce the variational form. To accurately resolve of sharp boundary layers, the variational form is implemented using a Petrov-Galerkin formulation. Dirichlet boundary conditions are imposed directly, while the source terms are computed using automatic differentiation. Computational experiments on standard two-dimensional problems demonstrate that the proposed method achieves high accuracy in both the maximum and L_2 norms. These results confirm the efficiency and robustness of the Petrov-Galerkin VPINN approach in accurately capturing the multiscale features of two-dimensional SPPs.
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