Title: Hierarchical RBF-KAN and RBF-SKAN Architectures for Multidimensional Function Approximation and Random Field Learning

Abstract:

This study introduces and evaluates hierarchical Kolmogorov--Arnold neural network (KAN) frameworks that utilize radial basis functions (RBFs) as activation mechanisms. These architectures are designed to model both deterministic functions and random field structures. We specifically formulate two distinct models: the hierarchical RBF-KAN, which targets the approximation of multidimensional deterministic functions, and the hierarchical RBF-SKAN, which is tailored for learning random field models.

From a theoretical standpoint, we prove universal approximation capabilities for both proposed architectures. Notably, we provide quantitative approximation bounds for the hierarchical RBF-KAN, demonstrating that the framework offers a pathway to mitigate the curse of dimensionality. This is achieved by effectively lowering the dimensionality required for approximating high-dimensional functions. Additionally, we establish that the hierarchical RBF-SKAN is capable of approximating random field models with convergence under the Wasserstein-2 metric. Our empirical results confirm that these RBF-based neural network structures are effective tools for learning both multivariate functions and random field models.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC