KANO: Kolmogorov-Arnold Neural Operator

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 a generic position-dependent dynamics for any physical input, whereas FNO stays practical only to spectrally sparse operators and strictly imposes 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 $\approx6\times10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $\approx1.5\times10^{-2}$, by orders of magnitude.