用可学习的径向基函数提升KAN模型效率与精度
Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learning
- 引入可学习的RBF形状和光滑参数，动态匹配激活模式
- 在多尺度回归等任务中达到与B样条相当的精度，训练更快
- 首个为RBF-KAN提供通用逼近理论证明，适合高维建模
Kolmogorov-Arnold网络（KAN）为复杂非线性函数逼近提供了有前景的框架，但原始的B样条形式因De Boor算法导致显著计算开销。尽管近期基于RBF的变体提升了效率，却常牺牲原样条设计的逼近精度。为此，我们提出Free-RBF-KAN，通过引入可学习的网格和可训练的光滑参数，实现高表达力、高分辨率的函数逼近。该方法采用可学习的RBF形状，能动态对齐激活模式，并首次为RBF-KAN家族提供通用逼近证明。在多尺度回归、物理信息偏微分方程及算子学习中的实证评估表明，Free-RBF-KAN在保持与B样条相当精度的同时，显著加快了训练与推理速度。结果确立了其作为高维结构化建模高效自适应替代方案的地位。
原文摘要 · Abstract (English)
Kolmogorov-Arnold Networks (KANs) offer a promising framework for approximating complex nonlinear functions, yet the original B-spline formulation suffers from significant computational overhead due to De Boor algorithm. While recent RBF-based variants improve efficiency, they often sacrifice the approximation accuracy inherent in the original spline-based design. To bridge this gap, we propose Free-RBF-KAN, an architecture that integrates adaptive learning grids and trainable smoothness parameters to enable expressive, high-resolution function approximation. Our method utilizes learnable RBF shapes that dynamically align with activation patterns, and we provide the first formal universal approximation proof for the RBF-KAN family. Empirical evaluations across multiscale regression, physics-informed PDEs, and operator learning demonstrate that Free-RBF-KAN can achieve accuracy comparable to its B-spline counterparts while delivering significantly faster training and inference. These results establish Free-RBF-KAN as an efficient and adaptive alternative for high-dimensional structured modeling tasks.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。