用径向基函数构建分层网络，提升高维函数与随机场的逼近能力。
Hierarchical RBF-KAN and RBF-SKAN Architectures for Multidimensional Function Approximation and Random Field Learning
- 采用径向基函数作为激活函数，构建分层Kolmogorov-Arnold网络结构
- 理论证明可缓解高维函数学习的维度灾难问题，有效降低近似复杂度
- 适用于高维函数逼近与随机场建模，适合需要高效泛化能力的研究者
本文提出并分析了基于径向基函数的分层柯尔莫哥洛夫-阿诺德神经网络架构，用于逼近确定性函数和随机场模型。具体而言，我们设计了分层径向基函数柯尔莫哥洛夫-阿诺德网络（hierarchical RBF-KAN）用于多维确定性函数逼近，以及分层径向基函数随机柯尔莫哥洛夫-阿诺德网络（hierarchical RBF-SKAN）用于随机场学习。理论上，我们建立了两种架构的通用逼近性质，并为 hierarchical RBF-KAN 推导出定量逼近误差估计，表明该框架可通过降低近似问题的有效维度，部分缓解高维函数学习中的维度灾难。此外，我们证明 hierarchical RBF-SKAN 可在 Wasserstein-2 度量下逼近随机场模型。实验上，所提出的基于径向基函数的神经网络结构能有效学习多元函数与随机场模型。
原文摘要 · Abstract (English)
In this manuscript, we propose and analyze hierarchical Kolmogorov--Arnold neural network architectures employing radial basis functions as activation functions for approximating deterministic functions and random field models. Specifically, we develop a hierarchical radial-basis-function Kolmogorov--Arnold network (hierarchical RBF-KAN) for multidimensional deterministic function approximation and a hierarchical radial-basis-function stochastic Kolmogorov--Arnold network (hierarchical RBF-SKAN) for random field learning. From a theoretical perspective, we establish universal approximation results for both architectures. In particular, we derive quantitative approximation estimates for the hierarchical RBF-KAN, showing that the proposed framework has the potential to partially alleviate the curse of dimensionality in learning high-dimensional functions by reducing the effective dimensionality of the approximation problem. Furthermore, we show that the hierarchical RBF-SKAN can approximate random field models under the Wasserstein-2 metric. Empirically, we show that our proposed radial-basis-function-based neural network structure could effectively learn multivariate functions and random field models.
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