mlsanityData to 7 Oct 2026

hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition

arXiv 2003.05385 ↗First posted 11 Mar 2020cs.NE cs.LG cs.NAcited by 236

Not on the map. Why?

Abstract (arXiv)

We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the whole computational domain, while the test space contains the piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in accuracy and training cost for several numerical examples of function approximation and solving differential equations.

Key facts. Cited by 236 papers in the map (data through 7 Oct 2026). First posted on arXiv 11 Mar 2020.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 29; 2022: 35; 2023: 46; 2024: 41; 2025: 42; 2026: 43.

236 papers.

Sort:

By area:

  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  3. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 169
  4. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  5. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
  6. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  7. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  8. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 86
  9. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 84
  10. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  11. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  12. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
38 more
  1. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  2. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  3. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  4. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 64
  5. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 55
  6. 2021Meta-learning PINN loss functionscited by 51
  7. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  8. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  9. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 43
  10. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  11. 2021Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domainscited by 37
  12. 2021A novel meta-learning initialization method for physics-informed neural networkscited by 35
  13. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 34
  14. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  15. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  16. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  17. 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
  18. 2023Physics informed WNOcited by 26
  19. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  20. 2022Scalable algorithms for physics-informed neural and graph networkscited by 21
  21. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  22. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  23. 2023Improving physics-informed neural networks with meta-learned optimizationcited by 19
  24. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  25. 2023Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networkscited by 17
  26. 2022SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domaincited by 16
  27. 2021An extended physics informed neural network for preliminary analysis of parametric optimal control problemscited by 15
  28. 2022Neural parameter calibration for large-scale multi-agent modelscited by 14
  29. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  30. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  31. 2023Finite Element Operator Network for Solving Elliptic-type parametric PDEscited by 14
  32. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  33. 2025A multi-agentic framework for real-time, autonomous freeform metasurface designcited by 14
  34. 2025AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learningcited by 13
  35. 2021A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticitycited by 12
  36. 2022State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learningcited by 12
  37. 2022Calibrating constitutive models with full-field data via physics informed neural networkscited by 12
  38. 2023Interpretability of Machine Learning: Recent Advances and Future Prospectscited by 12

50 most cited of 236 All 236 (JSON)