mlsanityData to 7 Oct 2026

KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics

arXiv 2407.04192 ↗First posted 5 Jul 2024cs.LGcited by 58

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Abstract (arXiv)

Kolmogorov-Arnold networks (KANs) as an alternative to multi-layer perceptrons (MLPs) are a recent development demonstrating strong potential for data-driven modeling. This work applies KANs as the backbone of a neural ordinary differential equation (ODE) framework, generalizing their use to the time-dependent and temporal grid-sensitive cases often seen in dynamical systems and scientific machine learning applications. The proposed KAN-ODEs retain the flexible dynamical system modeling framework of Neural ODEs while leveraging the many benefits of KANs compared to MLPs, including higher accuracy and faster neural scaling, stronger interpretability and generalizability, and lower parameter counts. First, we quantitatively demonstrated these improvements in a comprehensive study of the classical Lotka-Volterra predator-prey model. We then showcased the KAN-ODE framework's ability to learn symbolic source terms and complete solution profiles in higher-complexity and data-lean scenarios including wave propagation and shock formation, the complex Schr"odinger equation, and the Allen-Cahn phase separation equation. The successful training of KAN-ODEs, and their improved performance compared to traditional Neural ODEs, implies significant potential in leveraging this novel network architecture in myriad scientific machine learning applications for discovering hidden physics and predicting dynamic evolution.

Key facts. Cited by 58 papers in the map (data through 7 Oct 2026). Builds on 20 of its 47 references that are in the map. First posted on arXiv 5 Jul 2024.

Builds on

20 of 47 references.

References · 20

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  4. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  5. 2017Stable Architectures for Deep Neural Networkscited by 340
  6. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  7. 2017Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equationscited by 203
  8. 2017Sparse identification of nonlinear dynamics for model predictive control in the low-data limitcited by 139
  9. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 134
  10. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  11. 2024Kolmogorov-Arnold Networks are Radial Basis Function Networkscited by 125
  12. 2019Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noisecited by 98
8 more
  1. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  2. 2024Kolmogorov-Arnold Networks (KANs) for Time Series Analysiscited by 97
  3. 2021Stiff Neural Ordinary Differential Equationscited by 77
  4. 2024Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretabilitycited by 69
  5. 2020Bayesian Neural Ordinary Differential Equationscited by 44
  6. 2020Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Networkcited by 27
  7. 2022Precision Machine Learningcited by 14
  8. 2021Autonomous Kinetic Modeling of Biomass Pyrolysis using Chemical Reaction Neural Networkscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 14; 2025: 29; 2026: 15.

58 papers.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  3. 2024Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KANcited by 25
  4. 2024Reduced Effectiveness of Kolmogorov-Arnold Networks on Functions with Noisecited by 17
  5. 2025A Practitioner's Guide to Kolmogorov-Arnold Networkscited by 16
  6. 2024PointNet with KAN versus PointNet with MLP for 3D Classification and Segmentation of Point Setscited by 15
  7. 2025LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behaviorcited by 12
  8. 2024Deep Learning Alternatives of the Kolmogorov Superposition Theoremcited by 11
  9. 2024KAN versus MLP on Irregular or Noisy Functionscited by 7
  10. 2024Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equationscited by 7
  11. 2025MatrixKAN: Parallelized Kolmogorov-Arnold Networkcited by 7
  12. 2024Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Designcited by 6
38 more
  1. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  2. 2025KANO: Kolmogorov-Arnold Neural Operatorcited by 6
  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. 2025Multi-Exit Kolmogorov-Arnold Networks: enhancing accuracy and parsimonycited by 4
  7. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  8. 2024KACQ-DCNN: Uncertainty-Aware Interpretable Kolmogorov-Arnold Classical-Quantum Dual-Channel Neural Network for Heart Disease Detectioncited by 3
  9. 2025AF-KAN: Activation Function-Based Kolmogorov-Arnold Networks for Efficient Representation Learningcited by 3
  10. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  11. 2025SPIN-ODE: Stiff Physics-Informed Neural ODE for Chemical Reaction Rate Estimationcited by 3
  12. 2025FlashKAT: Understanding and Addressing Performance Bottlenecks in the Kolmogorov-Arnold Transformercited by 3
  13. 2025Improving Memory Efficiency for Training KANs via Meta Learningcited by 3
  14. 2025State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identificationcited by 3
  15. 2026Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learningcited by 3
  16. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  17. 2024Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximantscited by 2
  18. 2025Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertaintycited by 2
  19. 2025iTFKAN: Interpretable Time Series Forecasting with Kolmogorov-Arnold Networkcited by 2
  20. 2025System Identification Using Kolmogorov-Arnold Networks: A Case Study on Buck Converterscited by 2
  21. 2025On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimatorscited by 2
  22. 2025MetaCluster: Enabling Deep Compression of Kolmogorov-Arnold Networkcited by 2
  23. 2026How (Not) to Hybridize Neural and Mechanistic Models for Epidemiological Forecastingcited by 2
  24. 2024Uncertainty Quantification with Bayesian Higher Order ReLU KANscited by 1
  25. 2024Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularitiescited by 1
  26. 2024KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forwardcited by 1
  27. 2025Fredholm Neural Networks for inverse problems in elliptic PDEscited by 1
  28. 2025Discovering Generalizable Governing Equations for Graph Dynamical Systems with Interpretable Neural Networkscited by 1
  29. 2025Kolmogorov-Arnold Chemical Reaction Neural Networks for learning pressure-dependent kinetic rate lawscited by 1
  30. 2025Optimized Architectures for Kolmogorov-Arnold Networkscited by 1
  31. 2025MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamicscited by 1
  32. 2026A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networkscited by 1
  33. 2026Empirical Stability Analysis of Kolmogorov-Arnold Networks in Hard-Constrained Recurrent Physics-Informed Discoverycited by 1
  34. 2026Multilevel Training for Kolmogorov Arnold Networkscited by 1
  35. 2026PH-KAN: Port-Hamiltonian Kolmogorov-Arnold Networkcited by 1
  36. 2026SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functionscited by 1
  37. 2025Algorithms and data structures for automatic precision estimation of neural networkscited by 0
  38. 2025Learning continuous state of charge dependent thermal decomposition kinetics for Li-ion cathodes using Kolmogorov-Arnold Chemical Reaction Neural Networks (KA-CRNNs)cited by 0

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