KANO: Kolmogorov-Arnold Neural Operator
This addresses a bottleneck in neural operators for solving variable coefficient PDEs, offering improved accuracy and interpretability for physics and quantum computing applications, though it is incremental as it builds on FNO.
The paper tackles the limitation of Fourier Neural Operator (FNO) in handling position-dependent dynamics by introducing KANO, a dual-domain neural operator with spectral and spatial bases, which achieves accurate symbolic Hamiltonian reconstruction with state infidelity of ≈6×10⁻⁶, outperforming FNO by orders of magnitude.
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 $\approx 6\times10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $\approx 1.5\times10^{-2}$, by orders of magnitude.