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A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks

arXiv 2406.02917 ↗First posted 5 Jun 2024cs.LG physics.comp-phcited by 104

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Abstract (arXiv)

Kolmogorov-Arnold Networks (KANs) were recently introduced as an alternative representation model to MLP. Herein, we employ KANs to construct physics-informed machine learning models (PIKANs) and deep operator models (DeepOKANs) for solving differential equations for forward and inverse problems. In particular, we compare them with physics-informed neural networks (PINNs) and deep operator networks (DeepONets), which are based on the standard MLP representation. We find that although the original KANs based on the B-splines parameterization lack accuracy and efficiency, modified versions based on low-order orthogonal polynomials have comparable performance to PINNs and DeepONet although they still lack robustness as they may diverge for different random seeds or higher order orthogonal polynomials. We visualize their corresponding loss landscapes and analyze their learning dynamics using information bottleneck theory. Our study follows the FAIR principles so that other researchers can use our benchmarks to further advance this emerging topic.

Key facts. Cited by 104 papers in the map (data through 14 Oct 2026). Builds on 40 of its 77 references that are in the map. First posted on arXiv 5 Jun 2024.

Builds on

40 of 77 references.

References · 40

  1. 2017Attention Is All You Needcited by 60,361
  2. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  3. 2022Generative Adversarial Networkscited by 5,627
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  6. 2018On the Spectral Bias of Neural Networkscited by 1,341
  7. 2017Visualizing the Loss Landscape of Neural Netscited by 1,130
  8. 2015Deep Learning and the Information Bottleneck Principlecited by 802
  9. 2019Hamiltonian Neural Networkscited by 712
  10. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  11. 2017Opening the Black Box of Deep Neural Networks via Informationcited by 503
  12. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
28 more
  1. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  2. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  3. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  4. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  5. 2024Kolmogorov-Arnold Networks are Radial Basis Function Networkscited by 128
  6. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  7. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 118
  8. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  9. 2024Kolmogorov-Arnold Networks (KANs) for Time Series Analysiscited by 98
  10. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 84
  11. 2023Residual-based attention in physics-informed neural networkscited by 72
  12. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 68
  13. 2020The Information Bottleneck Problem and Its Applications in Machine Learningcited by 66
  14. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
  15. 2021Learning Functional Priors and Posteriors from Data and Physicscited by 37
  16. 2022ExSpliNet: An interpretable and expressive spline-based neural networkcited by 33
  17. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  18. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  19. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  20. 2023Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regressioncited by 25
  21. 2022Information Flow in Deep Neural Networkscited by 22
  22. 2024Smooth Kolmogorov Arnold networks enabling structural knowledge representationcited by 22
  23. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  24. 2024Learning in PINNs: Phase transition, diffusion equilibrium, and generalizationcited by 13
  25. 2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
  26. 2023A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimescited by 8
  27. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  28. 2023Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learningcited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 35; 2025: 44; 2026: 25.

104 papers.

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  1. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 118
  2. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  3. 2024A Survey on Kolmogorov-Arnold Networkcited by 68
  4. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 67
  5. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  6. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 38
  7. 2024On the expressiveness and spectral bias of KANscited by 37
  8. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 35
  9. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  10. 2025TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecastingcited by 30
  11. 2024Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KANcited by 25
  12. 2025A Practitioner's Guide to Kolmogorov-Arnold Networkscited by 18
38 more
  1. 2024SineKAN: Kolmogorov-Arnold Networks Using Sinusoidal Activation Functionscited by 15
  2. 2024Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networkscited by 15
  3. 2024PointNet with KAN versus PointNet with MLP for 3D Classification and Segmentation of Point Setscited by 15
  4. 2024Deep Learning Alternatives of the Kolmogorov Superposition Theoremcited by 12
  5. 2024SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networkscited by 12
  6. 2024Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architecturescited by 12
  7. 2024Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networkscited by 11
  8. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  9. 2025Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Studycited by 11
  10. 2024COEFF-KANs: A Paradigm to Address the Electrolyte Field with KANscited by 10
  11. 2025Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacologycited by 10
  12. 2024EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomialscited by 9
  13. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  14. 2025Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNscited by 9
  15. 2024CF-KAN: Kolmogorov-Arnold Network-based Collaborative Filtering to Mitigate Catastrophic Forgetting in Recommender Systemscited by 8
  16. 2026Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelinescited by 8
  17. 2024DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networkscited by 7
  18. 2024Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equationscited by 7
  19. 2025Complex Physics-Informed Neural Networkcited by 7
  20. 2025High precision PINNs in unbounded domains: application to singularity formulation in PDEscited by 7
  21. 2025KANO: Kolmogorov-Arnold Neural Operatorcited by 7
  22. 2025ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithmscited by 7
  23. 2025Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddingscited by 6
  24. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  25. 2024Kolmogorov-Arnold Networks in Low-Data Regimes: A Comparative Study with Multilayer Perceptronscited by 5
  26. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  27. 2025Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsamplingcited by 5
  28. 2025BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamicscited by 5
  29. 2025A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networkscited by 5
  30. 2026FEKAN: Feature-Enriched Kolmogorov-Arnold Networkscited by 5
  31. 2024KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics examplecited by 4
  32. 2025Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinementcited by 4
  33. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  34. 2026Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learningcited by 4
  35. 2024Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiologycited by 3
  36. 2025Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefieldscited by 3
  37. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  38. 2025Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learningcited by 3

50 most cited of 104 All 104 (JSON)