mlsanityData to 7 Oct 2026

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

arXiv 2109.02035 ↗First posted 5 Sep 2021math.NA cs.LG cs.NAcited by 17

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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.

Key facts. Cited by 17 papers in the map (data through 7 Oct 2026). Builds on 18 of its 40 references that are in the map. First posted on arXiv 5 Sep 2021.

Builds on

18 of 40 references.

References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015Automatic differentiation in machine learning: a surveycited by 846
  3. 2015Cyclical Learning Rates for Training Neural Networkscited by 627
  4. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  5. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  6. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  7. 2020Physics-informed learning of governing equations from scarce datacited by 167
  8. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  9. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  10. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  11. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  12. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
6 more
  1. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  2. 2019Deep Neural Network Approximation Theorycited by 92
  3. 2021On the approximation of functions by tanh neural networkscited by 81
  4. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  5. 2020Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imagingcited by 40
  6. 2019VarNet: Variational Neural Networks for the Solution of Partial Differential Equationscited by 35

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 2; 2025: 6; 2026: 6.

17 papers.

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  1. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  2. 2023Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networkscited by 17
  3. 2023Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanicscited by 4
  4. 2025Prediction error certification for PINNs: Theory, computation, and application to Stokes flowcited by 4
  5. 2024Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEscited by 3
  6. 2026Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithmcited by 3
  7. 2024Computable Lipschitz Bounds for Deep Neural Networkscited by 2
  8. 2025Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networkscited by 2
  9. 2025Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebracited by 1
  10. 2025DPG loss functions for learning parameter-to-solution maps by neural networkscited by 1
  11. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 1
  12. 2026Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolutioncited by 1
5 more
  1. 2025The Finite Element Neural Network Method: One Dimensional Studycited by 0
  2. 2026Certified and accurate computation of function space norms of deep neural networkscited by 0
  3. 2026PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEscited by 0
  4. 2026Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equationscited by 0
  5. 2026Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Lawscited by 0