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.

翻译：我们提出了 Kolmogorov–Arnold 神经算子（KANO），这是一种双域神经算子，通过谱基和空间基联合参数化，具有内在的符号可解释性。我们从理论上证明，KANO 克服了傅里叶神经算子（FNO）的纯谱瓶颈：对于任何物理输入，KANO 在一般的位置相关动力学（变系数偏微分方程）上仍保持表达能力，而 FNO 仅适用于谱稀疏算子，并严格强制输入傅里叶尾快速衰减。我们在位置相关微分算子上通过实验验证了我们的主张，KANO 在这些算子上表现出稳健的泛化能力，而 FNO 则失败。在量子哈密顿量学习基准测试中，KANO 以闭式符号表示重构了真实哈密顿量，其系数精度达到小数点后第四位，并且从投影测量数据中获得了约 $6\times10^{-6}$ 的状态保真度损失，这显著优于使用理想全波函数数据训练的 FNO 所达到的约 $1.5\times10^{-2}$ 的损失，提升幅度达到数量级。