# Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems

arXiv: https://arxiv.org/abs/2606.16510 · HTML: https://mlsanity.com/paper/2606.16510/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods](https://mlsanity.com/paper/2503.10021/) (2025)

## Abstract (arXiv)

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.