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Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems

arXiv 2406.19662 ↗First posted 28 Jun 2024cs.LG physics.comp-phcited by 35

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Abstract (arXiv)

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). Builds on 36 of its 59 references that are in the map. First posted on arXiv 28 Jun 2024.

Builds on

36 of 59 references.

References · 36

  1. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  3. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  4. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  5. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  6. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  7. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  8. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  9. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
  10. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  11. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  12. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
24 more
  1. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  2. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 82
  3. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  4. 2024Convolutional Kolmogorov-Arnold Networkscited by 76
  5. 2023Residual-based attention in physics-informed neural networkscited by 72
  6. 2024TKAN: Temporal Kolmogorov-Arnold Networkscited by 70
  7. 2022Failure-informed adaptive sampling for PINNscited by 64
  8. 2024fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functionscited by 60
  9. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  10. 2024GKAN: Graph Kolmogorov-Arnold Networkscited by 52
  11. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  12. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  13. 2024Kolmogorov-Arnold Network for Satellite Image Classification in Remote Sensingcited by 46
  14. 2024Suitability of KANs for Computer Vision: A preliminary investigationcited by 33
  15. 2024ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLUcited by 32
  16. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  17. 2021DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networkscited by 28
  18. 2021Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)cited by 25
  19. 2021Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Controlcited by 24
  20. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  21. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  22. 2023A multifidelity approach to continual learning for physical systemscited by 13
  23. 2023Physics-informed machine learning of redox flow battery based on a two-dimensional unit cell modelcited by 5
  24. 2024Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problemscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 18; 2025: 10; 2026: 7.

35 papers.

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  1. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 116
  2. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  3. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  4. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 37
  5. 2024Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networkscited by 15
  6. 2024PointNet with KAN versus PointNet with MLP for 3D Classification and Segmentation of Point Setscited by 15
  7. 2024Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and fastercited by 13
  8. 2024SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networkscited by 12
  9. 2025LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behaviorcited by 12
  10. 2024Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networkscited by 11
  11. 2024Data-driven model discovery with Kolmogorov-Arnold networkscited by 9
  12. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
23 more
  1. 2024Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Designcited by 6
  2. 2024KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics examplecited by 4
  3. 2025KAC: Kolmogorov-Arnold Classifier for Continual Learningcited by 4
  4. 2025ChemKANs for Combustion Chemistry Modeling and Accelerationcited by 4
  5. 2025Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinementcited by 4
  6. 2024KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problemscited by 3
  7. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  8. 2025Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learningcited by 3
  9. 2026Making Gaussian Kolmogorov-Arnold Networks Reliable and Accuratecited by 3
  10. 2024Explainable fault and severity classification for rolling element bearings using Kolmogorov-Arnold networkscited by 2
  11. 2025X-KAN: Optimizing Local Kolmogorov-Arnold Networks via Evolutionary Rule-Based Machine Learningcited by 2
  12. 2026SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networkscited by 2
  13. 2026Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradientscited by 2
  14. 2024Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularitiescited by 1
  15. 2024KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forwardcited by 1
  16. 2025Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networkscited by 1
  17. 2025Domain decomposition architectures and Gauss-Newton training for physics-informed neural networkscited by 1
  18. 2026Multilevel Training for Kolmogorov Arnold Networkscited by 1
  19. 2026Partition-of-Unity Gaussian Kolmogorov-Arnold Networkscited by 1
  20. 2024On the Convergence of (Stochastic) Gradient Descent for Kolmogorov--Arnold Networkscited by 0
  21. 2025Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheetscited by 0
  22. 2026Optimized Piecewise Affine Abstractions of Neural Networks with Learnable Activation Functionscited by 0
  23. 2026Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decompositioncited by 0