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Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

arXiv 2201.05624 ↗First posted 14 Jan 2022cs.LG cs.AI cs.NAcited by 603

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Abstract (arXiv)

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

Key facts. Cited by 603 papers in the map (data through 7 Oct 2026). Builds on 89 of its 220 references that are in the map. First posted on arXiv 14 Jan 2022.

Builds on

89 of 220 references.

References · 89

  1. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2018Reconciling modern machine learning practice and the bias-variance trade-offcited by 919
  4. 2019N-BEATS: Neural basis expansion analysis for interpretable time series forecastingcited by 744
  5. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  6. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  7. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  8. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  9. 2018GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Accelerationcited by 603
  10. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  11. 2016Error bounds for approximations with deep ReLU networkscited by 567
  12. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
38 more
  1. 2017A Brief Survey of Deep Reinforcement Learningcited by 466
  2. 2021Interpretable Machine Learning: Fundamental Principles and 10 Grand Challengescited by 364
  3. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  4. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  5. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  6. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  7. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  8. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  9. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  10. 2018Horovod: fast and easy distributed deep learning in TensorFlowcited by 275
  11. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  12. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  13. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  14. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  15. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  16. 2021Towards Causal Representation Learningcited by 207
  17. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  18. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  19. 2019Neural Tangents: Fast and Easy Infinite Neural Networks in Pythoncited by 149
  20. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  21. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  22. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  23. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  24. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  25. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  26. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  27. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  28. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  29. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  30. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  31. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  32. 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
  33. 2019Deep Neural Network Approximation Theorycited by 92
  34. 2019A Survey of Optimization Methods from a Machine Learning Perspectivecited by 90
  35. 2021On the approximation of functions by tanh neural networkscited by 81
  36. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  37. 2016Understanding Deep Convolutional Networkscited by 76
  38. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 38; 2023: 117; 2024: 150; 2025: 175; 2026: 123.

603 papers.

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  1. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 137
  2. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  3. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  4. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  5. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 84
  6. 2024ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEscited by 78
  7. 2023A Mathematical Guide to Operator Learningcited by 74
  8. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  9. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  10. 2023Graph Neural Networks for temporal graphs: State of the art, open challenges, and opportunitiescited by 60
  11. 2024Neural Operators with Localized Integral and Differential Kernelscited by 59
  12. 2024KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physicscited by 58
38 more
  1. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  2. 2025Evaluating Newtonian Mechanics in Video Generative Models with Real Physical Systemscited by 44
  3. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 43
  4. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  5. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  6. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  7. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  8. 2024TimeDiT: General-purpose Diffusion Transformers for Time Series Foundation Modelcited by 33
  9. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  10. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  11. 2023Physics-Informed Computer Vision: A Review and Perspectivescited by 28
  12. 2023Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorialcited by 27
  13. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  14. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  15. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  16. 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
  17. 2023A Survey on Physics Informed Reinforcement Learning: Review and Open Problemscited by 22
  18. 2023Efficient physics-informed neural networks using hash encodingcited by 21
  19. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  20. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  21. 2024Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networkscited by 21
  22. 2025Comprehensive Review of Neural Differential Equations for Time Series Analysiscited by 20
  23. 2022Variationally Mimetic Operator Networkscited by 19
  24. 2023Improving physics-informed neural networks with meta-learned optimizationcited by 19
  25. 2022Neural Implicit Surface Evolutioncited by 18
  26. 2023A neural operator-based surrogate solver for free-form electromagnetic inverse designcited by 18
  27. 2025Vid2Sim: Generalizable, Video-based Reconstruction of Appearance, Geometry and Physics for Mesh-free Simulationcited by 18
  28. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  29. 2023Data-Driven Design for Metamaterials and Multiscale Systems: A Reviewcited by 17
  30. 2024Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantificationcited by 17
  31. 2025Terminal Velocity Matchingcited by 17
  32. 2025A Practitioner's Guide to Kolmogorov-Arnold Networkscited by 16
  33. 2023I-PHYRE: Interactive Physical Reasoningcited by 15
  34. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  35. 2023Multi-scale Time-stepping of Partial Differential Equations with Transformerscited by 14
  36. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  37. 2023NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Trainingcited by 13
  38. 2023Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibilitycited by 13

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