# Variational Physics Informed Neural Networks: the role of quadratures and test functions

arXiv: https://arxiv.org/abs/2109.02035 · HTML: https://mlsanity.com/paper/2109.02035/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 5 Sep 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)

## Built on by (most cited first, 17 of 17)

- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [Prediction error certification for PINNs: Theory, computation, and application to Stokes flow](https://mlsanity.com/paper/2508.07994/) (2025)
- [Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs](https://mlsanity.com/paper/2412.05719/) (2024)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)
- [Computable Lipschitz Bounds for Deep Neural Networks](https://mlsanity.com/paper/2410.21053/) (2024)
- [DPG loss functions for learning parameter-to-solution maps by neural networks](https://mlsanity.com/paper/2506.18773/) (2025)
- [Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks](https://mlsanity.com/paper/2509.07579/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2505.11638/) (2025)
- [Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution](https://mlsanity.com/paper/2603.09693/) (2026)
- [PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs](https://mlsanity.com/paper/2607.20378/) (2026)
- [The Finite Element Neural Network Method: One Dimensional Study](https://mlsanity.com/paper/2501.12508/) (2025)
- [Certified and accurate computation of function space norms of deep neural networks](https://mlsanity.com/paper/2603.06431/) (2026)
- [Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2607.25608/) (2026)
- [Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2608.10389/) (2026)

## Abstract (arXiv)

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.