Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems
2026-06-15 • Machine Learning
Machine Learning
AI summaryⓘ
The authors created a new method using neural networks to solve tricky math problems with very sharp changes in two dimensions. They combined their neural network approach with a special technique called Petrov-Galerkin to better handle these sharp edges. Their method also uses a clever way to apply boundary conditions and compute other parts automatically. Tests showed their approach is accurate and reliable for these complex problems. Overall, their technique helps capture small details that are usually hard to solve in such problems.
Petrov-Galerkin methodVariational Physics-Informed Neural Networksingularly perturbed problemstensor-product hat functionsboundary layersDirichlet boundary conditionsautomatic differentiationtrial and test functionsmultiscale featurestwo-dimensional PDEs
Authors
Vijay Kumar, Gautam Singh
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.