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From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning

arXiv 2410.13228 ↗First posted 17 Oct 2024cs.LG cs.AI physics.comp-phcited by 73

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Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have emerged as a key tool in Scientific Machine Learning since their introduction in 2017, enabling the efficient solution of ordinary and partial differential equations using sparse measurements. Over the past few years, significant advancements have been made in the training and optimization of PINNs, covering aspects such as network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions. A notable recent development is the Physics-Informed Kolmogorov-Arnold Networks (PIKANS), which leverage a representation model originally proposed by Kolmogorov in 1957, offering a promising alternative to traditional PINNs. In this review, we provide a comprehensive overview of the latest advancements in PINNs, focusing on improvements in network design, feature expansion, optimization techniques, uncertainty quantification, and theoretical insights. We also survey key applications across a range of fields, including biomedicine, fluid and solid mechanics, geophysics, dynamical systems, heat transfer, chemical engineering, and beyond. Finally, we review computational frameworks and software tools developed by both academia and industry to support PINN research and applications.

Key facts. Cited by 73 papers in the map (data through 14 Oct 2026). Builds on 185 of its 389 references that are in the map. First posted on arXiv 17 Oct 2024.

Builds on

185 of 389 references.

References · 185

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2015Variational Inference with Normalizing Flowscited by 2,170
  3. 2018On the Spectral Bias of Neural Networkscited by 1,341
  4. 2015Automatic differentiation in machine learning: a surveycited by 855
  5. 2020A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challengescited by 819
  6. 2015Deep Learning and the Information Bottleneck Principlecited by 802
  7. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  8. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  9. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  10. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  11. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  12. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
38 more
  1. 2017Opening the Black Box of Deep Neural Networks via Informationcited by 503
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  3. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  4. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  5. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  6. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  7. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  8. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  9. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  10. 2019Towards Understanding the Spectral Bias of Deep Learningcited by 174
  11. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
  12. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  13. 2020Physics-informed learning of governing equations from scarce datacited by 167
  14. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 161
  15. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  16. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
  17. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  18. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  19. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 140
  20. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
  21. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  22. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
  23. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  24. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  25. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  26. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 118
  27. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  28. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  29. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  30. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  31. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  32. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
  33. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 87
  34. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  35. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 79
  36. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
  37. 2018Deep Learning of Vortex Induced Vibrationscited 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

50 most cited of 185 All 185 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 42; 2026: 30.

73 papers.

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  1. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 31
  2. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  3. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 11
  4. 2026Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networkscited by 11
  5. 2025Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacologycited by 10
  6. 2025Learning Turbulent Flows with Generative Models: Super-resolution, Forecasting, and Sparse Flow Reconstructioncited by 8
  7. 2026Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelinescited by 8
  8. 2025High precision PINNs in unbounded domains: application to singularity formulation in PDEscited by 7
  9. 2025ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithmscited by 7
  10. 2025Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddingscited by 6
  11. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  12. 2025A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networkscited by 5
38 more
  1. 2025Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinementcited by 4
  2. 2025Multi-Exit Kolmogorov-Arnold Networks: enhancing accuracy and parsimonycited by 4
  3. 2025RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equationscited by 4
  4. 2025Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefieldscited by 3
  5. 2025Physics-Informed Neural Networks for Vessel Trajectory Prediction: Learning Time-Discretized Kinematic Dynamics via Finite Differencescited by 3
  6. 2025LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blockscited by 3
  7. 2025Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problemscited by 3
  8. 2025Bayesian Physics Informed Neural Networks for Reliable Transformer Prognosticscited by 3
  9. 2025AutoHood3D: A Multi-Modal Benchmark for Automotive Hood Design and Fluid-Structure Interactioncited by 3
  10. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  11. 2026SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networkscited by 3
  12. 2025Physics-informed deep learning for infectious disease forecastingcited by 2
  13. 2025FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor designcited by 2
  14. 2025Trace Regularity PINNs: Enforcing H^frac12(partial Omega) for Boundary Datacited by 2
  15. 2025Training Deep Physics-Informed Kolmogorov-Arnold Networkscited by 2
  16. 2026PILD: Physics-Informed Learning via Diffusioncited by 2
  17. 2026Cell-Type Prototype-Informed Neural Network for Gene Expression Estimation from Pathology Imagescited by 2
  18. 2026A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentialscited by 2
  19. 2025Hybrid machine learning based scale bridging framework for permeability prediction of fibrous structurescited by 1
  20. 2025Unraveling particle dark matter with Physics-Informed Neural Networkscited by 1
  21. 2025Computational, Data-Driven, and Physics-Informed Machine Learning Approaches for Microstructure Modeling in Metal Additive Manufacturingcited by 1
  22. 2025Data-Driven Adaptive PID Control Based on Physics-Informed Neural Networkscited by 1
  23. 2025Gradient Enhanced Self-Training Physics-Informed Neural Network (gST-PINN) for Solving Nonlinear Partial Differential Equationscited by 1
  24. 2025Enforcing boundary conditions for physics-informed neural operatorscited by 1
  25. 2026A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networkscited by 1
  26. 2026Multilevel Training for Kolmogorov Arnold Networkscited by 1
  27. 2026Physics-informed offline reinforcement learning eliminates catastrophic fuel waste in maritime routingcited by 1
  28. 2026Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Datacited by 1
  29. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  30. 2026Decision-Aware Evaluation of Physics-Informed Surrogatescited by 1
  31. 2026An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networkscited by 1
  32. 2026Can AI Follow In Einstein's Footsteps?cited by 1
  33. 2024A Variational Computational-based Framework for Unsteady Incompressible Flowscited by 0
  34. 2025The Finite Element Neural Network Method: One Dimensional Studycited by 0
  35. 2025Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networkscited by 0
  36. 2025Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learningcited by 0
  37. 2025Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squarescited by 0
  38. 2025From Initial Data to Boundary Layers: Neural Networks for Nonlinear Hyperbolic Conservation Lawscited by 0

50 most cited of 73 All 73 (JSON)