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.

翻译：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和两个小摄动参数的二维奇异摄动问题（SPPs）。该方法采用神经网络构建试解空间，同时使用张量积帽函数作为测试函数来强制执行变分形式。为精确解析尖锐边界层，变分形式采用Petrov-Galerkin格式实现。Dirichlet边界条件直接施加，而源项则通过自动微分计算。针对标准二维问题的计算实验表明，所提方法在最大范数和L_2范数下均能达到高精度。这些结果证实了Petrov-Galerkin VPINN方法在精确捕获二维SPPs多尺度特征方面的效率和鲁棒性。