🤖 AI Summary
This work proposes a physics-informed neural network method based on a Petrov–Galerkin variational framework for two-dimensional singularly perturbed problems involving one or two small perturbation parameters. It is the first to integrate this variational framework with neural networks to efficiently resolve sharp boundary layers and multiscale features. The approach constructs a neural network trial solution space, employs tensor-product hat functions as test functions, computes source terms via automatic differentiation, and enforces Dirichlet boundary conditions strongly. Numerical experiments on benchmark problems demonstrate high accuracy in both the maximum norm and the $L^2$ norm, confirming the method’s effectiveness, robustness, and precise resolution of boundary layers in multiscale modeling.
📝 Abstract
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.