KANO: Kolmogorov-Arnold Neural Operator

2025-09-20

AI Summary

Fourier Neural Operators (FNOs) struggle to model non-spectrally-sparse, position-dependent dynamical systems due to their exclusive reliance on spectral representations. To address this limitation, we propose the Kolmogorov–Arnold Neural Operator (KANO), the first neural operator architecture that jointly parameterizes operators in both spectral and spatial domains. Grounded in the Kolmogorov–Arnold representation theorem, KANO incorporates interpretable symbolic structure into its design and is theoretically proven to possess universal approximation capability for arbitrary physical inputs—thereby overcoming FNO’s strict requirements on Fourier tail decay and spectral sparsity. In quantum Hamiltonian learning, KANO reconstructs closed-form symbolic Hamiltonians directly from projective measurement data alone, achieving coefficient accuracy on the order of 10⁻⁴ and state fidelity error as low as ≈6×10⁻⁶—substantially outperforming FNO, which requires full wavefunction supervision.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Knowledge Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 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 $approx 6 imes10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $approx 1.5 imes10^{-2}$, by orders of magnitude.

Problem

• Overcomes spectral bottlenecks in Fourier Neural Operators
• Learns position-dependent differential operators robustly
• Reconstructs quantum Hamiltonians with high symbolic accuracy

Innovation

• Dual-domain neural operator with spectral-spatial bases
• Overcomes Fourier Neural Operator's spectral limitations
• Learns symbolic Hamiltonian representations from measurement data