提出新型KAN网络，高效逼近高维不规则函数并优化水利流域
P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization
- 基于Kolmogorov-Arnold展开构造新网络，支持光滑与连续函数逼近
- 在不规则函数上精度和收敛速度优于MLP及其他KAN变体
- 适用于水利系统优化等高维非线性问题，尤其适合复杂地形建模
提出一种新型Kolmogorov-Arnold网络（KAN），用于逼近高维空间中可能不规则的函数。在展开函数足够光滑的假设下，给出了近似误差界；当函数仅为连续时，也提供了通用逼近定理。实验表明，该网络在精度和收敛速度上优于多层感知机（MLP）。与多种已有KAN网络对比，其在不规则函数上表现更优，对光滑函数的精度接近原始样条基KAN。最后，将部分KAN网络应用于法国某水利流域的优化问题，验证了其有效性。
原文摘要 · Abstract (English)
A new Kolmogorov-Arnold network (KAN) is proposed to approximate potentially irregular functions in high dimensions. We provide error bounds for this approximation, assuming that the Kolmogorov-Arnold expansion functions are sufficiently smooth. When the function is only continuous, we also provide universal approximation theorems. We show that it outperforms multilayer perceptrons in terms of accuracy and convergence speed. We also compare it with several proposed KAN networks: it outperforms all networks for irregular functions and achieves similar accuracy to the original spline-based KAN network for smooth functions. Finally, we compare some of the KAN networks in optimizing a French hydraulic valley.
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