Kolmogorov-Arnold Networks (KANs) have shown strong potential for efficiently approximating complex nonlinear functions. However, the original KAN formulation relies on B-spline basis functions, which incur substantial computational overhead due to De Boor's algorithm. To address this limitation, recent work has explored alternative basis functions such as radial basis functions (RBFs) that can improve computational efficiency and flexibility. Yet, standard RBF-KANs often sacrifice accuracy relative to the original KAN design. In this work, we propose Free-RBF-KAN, a RBF-based KAN architecture that incorporates adaptive learning grids and trainable smoothness to close this performance gap. Our method employs freely learnable RBF shapes that dynamically align grid representations with activation patterns, enabling expressive and adaptive function approximation. Additionally, we treat smoothness as a kernel parameter optimized jointly with network weights, without increasing computational complexity. We provide a general universality proof for RBF-KANs, which encompasses our Free-RBF-KAN formulation. Through a broad set of experiments, including multiscale function approximation, physics-informed machine learning, and PDE solution operator learning, Free-RBF-KAN achieves accuracy comparable to the original B-spline-based KAN while delivering faster training and inference. These results highlight Free-RBF-KAN as a compelling balance between computational efficiency and adaptive resolution, particularly for high-dimensional structured modeling tasks.

翻译：Kolmogorov-Arnold网络（KANs）在高效逼近复杂非线性函数方面展现出强大潜力。然而，原始KAN公式依赖于B样条基函数，其De Boor算法会带来显著的计算开销。为克服这一局限，近期研究探索了如径向基函数（RBFs）等替代基函数，以提升计算效率与灵活性。但标准RBF-KAN往往在精度上逊于原始KAN设计。本文提出Free-RBF-KAN，一种基于RBF的KAN架构，通过引入自适应学习网格与可训练平滑性来弥合这一性能差距。我们的方法采用自由可学习的RBF形状，动态调整网格表示以匹配激活模式，从而实现表达能力强且自适应的函数逼近。此外，我们将平滑性作为与网络权重联合优化的核参数处理，且不增加计算复杂度。我们为RBF-KAN提供了普适的通用性证明，该证明涵盖我们的Free-RBF-KAN公式。通过一系列广泛的实验，包括多尺度函数逼近、物理信息机器学习以及偏微分方程解算子学习，Free-RBF-KAN达到了与原始基于B样条的KAN相当的精度，同时实现了更快的训练与推理速度。这些结果表明，Free-RBF-KAN在计算效率与自适应分辨率之间取得了理想的平衡，尤其适用于高维结构化建模任务。