KANO模型突破传统神经算子瓶颈，可精准建模变系数微分方程。
KANO: Kolmogorov-Arnold Neural Operator
- 融合频域与空域基的双域架构，具备符号可解释性
- 在变系数微分方程上表现稳定，而FNO在此类问题中失效
- 量子哈密顿量学习中精度达小数点后四位，优于理想数据下的FNO
我们提出科尔莫戈罗夫-阿诺尔德神经算子（KANO），一种在频域与空间域联合参数化的神经算子，具有内在符号可解释性。理论上证明，KANO克服了傅里叶神经算子（FNO）的纯频域瓶颈：对任意物理输入，KANO在位置相关动力学（变系数偏微分方程）下仍保持表达能力，而FNO仅适用于频谱稀疏算子，并严格要求输入傅里叶尾部快速衰减。我们在位置相关微分算子上验证了该结论，KANO表现出稳健泛化能力，而FNO则失败。在量子哈密顿量学习基准测试中，KANO以闭式符号形式重建真实哈密顿量，系数精度达小数点后四位，从投影测量数据获得约6×10⁻⁶的状态保真度误差，相较使用理想全波函数数据训练的FNO（约1.5×10⁻²）提升数个数量级。
原文摘要 · Abstract (English)
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.
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