# A Variational Physics-Informed Neural Network Framework Using Petrov-Galerkin Method for Solving Singularly Perturbed Boundary Value Problems

arXiv: https://arxiv.org/abs/2509.12271 · HTML: https://mlsanity.com/paper/2509.12271/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 13 Sep 2025.

## 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)

## Built on by (most cited first, 1 of 1)

- [Unified generalization analysis for physics informed neural networks](https://mlsanity.com/paper/2605.13260/) (2026)

## Abstract (arXiv)

This work proposes a Variational Physics-Informed Neural Network (VPINN) framework that integrates the Petrov-Galerkin formulation with deep neural networks (DNNs) for solving one-dimensional singularly perturbed boundary value problems (BVPs) and parabolic partial differential equations (PDEs) involving one or two small parameters. The method adopts a nonlinear approximation in which the trial space is defined by neural network functions, while the test space is constructed from hat functions. The weak formulation is constructed using localized test functions, with interface penalty terms introduced to enhance numerical stability and accurately capture boundary layers. Dirichlet boundary conditions are imposed via hard constraints, and source terms are computed using automatic differentiation. Numerical experiments on benchmark problems demonstrate the effectiveness of the proposed method, showing significantly improved accuracy in both the L2 and maximum norms compared to the standard VPINN approach for one-dimensional singularly perturbed differential equations (SPDEs).