KANO: Kolmogorov-Arnold Neural Operator

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 (variable coefficient PDEs) 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 ≈ 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.