# KANO: Kolmogorov-Arnold Neural Operator

arXiv: https://arxiv.org/abs/2509.16825 · HTML: https://mlsanity.com/paper/2509.16825/

## 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 (most cited first, 30 of 32)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Underspecification Presents Challenges for Credibility in Modern Machine Learning](https://mlsanity.com/paper/2011.03395/) (2020)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Inductive Biases for Deep Learning of Higher-Level Cognition](https://mlsanity.com/paper/2011.15091/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Neural Arithmetic Logic Units](https://mlsanity.com/paper/1808.00508/) (2018)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [rKAN: Rational Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.14495/) (2024)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective](https://mlsanity.com/paper/2406.18794/) (2024)
- [Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study](https://mlsanity.com/paper/2509.03417/) (2025)
- [Enhancing Fourier Neural Operators with Local Spatial Features](https://mlsanity.com/paper/2503.17797/) (2025)
- [Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations](https://mlsanity.com/paper/2201.11967/) (2022)
- [Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena](https://mlsanity.com/paper/2504.20249/) (2025)
- [EFKAN: A KAN-Integrated Neural Operator For Efficient Magnetotelluric Forward Modeling](https://mlsanity.com/paper/2502.02195/) (2025)

## Built on by (most cited first, 7 of 7)

- [A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.18672/) (2026)
- [Inverse Neural Operator for ODE Parameter Optimization](https://mlsanity.com/paper/2603.11854/) (2026)
- [Debiased neural operators for estimating functionals](https://mlsanity.com/paper/2604.19296/) (2026)
- [Fourier Feature Pyramids for Physics-Informed Neural Networks](https://mlsanity.com/paper/2605.24278/) (2026)
- [Euclidean Fourier Neural Operators](https://mlsanity.com/paper/2608.28425/) (2026)
- [LLN: Learnable Lens Networks for Parameter-Efficient Long-Horizon Dynamical Prediction](https://mlsanity.com/paper/2609.34493/) (2026)
- [Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning](https://mlsanity.com/paper/2610.13028/) (2026)

## 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.