KANO：柯尔莫戈罗夫-阿诺尔德神经算子
KANO: Kolmogorov-Arnold Neural Operator
AI总结：
KANO通过双域参数化克服FNO的纯谱瓶颈，在位置依赖动力学和量子哈密顿量学习中展现更优性能。
摘要：
我们介绍了柯尔莫戈罗夫-阿诺尔德神经算子（KANO），这是一种双域神经算子，由谱域和空间域基底共同参数化，具有内在的符号可解释性。我们理论证明KANO克服了傅里叶神经算子（FNO）的纯谱瓶颈：KANO在任何物理输入下都能保持对通用位置依赖动力学（变系数PDEs）的表达能力，而FNO仅适用于谱稀疏算子，并严格要求输入傅里叶尾部快速衰减。我们在位置依赖的微分算子上验证了我们的主张，KANO稳健地泛化，而FNO却失败。在量子哈密顿量学习基准测试中，KANO以闭式符号表示重构真实哈密顿量，系数精确到小数点后四位，并从投影测量数据中获得约6×10⁻⁶的状态不忠实度，显著优于由理想完整波函数数据训练的FNO，其不忠实度约为1.5×10⁻²，差距呈数量级。
英文摘要：
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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