mlsanityData to 14 Oct 2026

KANO: Kolmogorov-Arnold Neural Operator

arXiv 2509.16825 ↗First posted 20 Sep 2025cs.LG cs.AI cs.CEcited by 7

Not on the map. Why?

Abstract (arXiv)

We introduce Kolmogorov--Arnold Neural Operator (KANO), a dual-domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over generic position-dependent dynamics (variable coefficient PDEs) for any physical input, whereas FNO stays practical only for spectrally sparse operators and strictly imposes a fast-decaying input Fourier tail. We verify our claims empirically on position-dependent differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground-truth Hamiltonians in closed-form symbolic representations accurate to the fourth decimal place in coefficients and attains ≈ 6×10-6 state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, ≈ 1.5×10-2, by orders of magnitude.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 32 of its 63 references that are in the map. First posted on arXiv 20 Sep 2025.

Builds on

32 of 63 references.

References · 32

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  5. 2016Error bounds for approximations with deep ReLU networkscited by 574
  6. 2020Underspecification Presents Challenges for Credibility in Modern Machine Learningcited by 496
  7. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 297
  8. 2020Inductive Biases for Deep Learning of Higher-Level Cognitioncited by 275
  9. 2021Factorized Fourier Neural Operatorscited by 249
  10. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 118
  11. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  12. 2024Operator Learning: Algorithms and Analysiscited by 85
20 more
  1. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 84
  2. 2021On the approximation of functions by tanh neural networkscited by 81
  3. 2018Neural Arithmetic Logic Unitscited by 70
  4. 2024Neural Operators with Localized Integral and Differential Kernelscited by 62
  5. 2024KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physicscited by 59
  6. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  7. 2024On the expressiveness and spectral bias of KANscited by 37
  8. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 35
  9. 2024Data Complexity Estimates for Operator Learningcited by 26
  10. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
  11. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 24
  12. 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 11
  13. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 11
  14. 2025Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Studycited by 11
  15. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 10
  16. 2022Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equationscited by 8
  17. 2025Temporal Neural Operator for Modeling Time-Dependent Physical Phenomenacited by 8
  18. 2025EFKAN: A KAN-Integrated Neural Operator For Efficient Magnetotelluric Forward Modelingcited by 4
  19. 2025X-KAN: Optimizing Local Kolmogorov-Arnold Networks via Evolutionary Rule-Based Machine Learningcited by 2
  20. 2025System Identification Using Kolmogorov-Arnold Networks: A Case Study on Buck Converterscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 7.

7 papers.

  1. 2026A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networkscited by 1
  2. 2026Inverse Neural Operator for ODE Parameter Optimizationcited by 1
  3. 2026Debiased neural operators for estimating functionalscited by 1
  4. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  5. 2026Euclidean Fourier Neural Operatorscited by 0
  6. 2026LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Predictioncited by 0
  7. 2026Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learningcited by 0