用傅里叶谱重参数化改进可解释网络，提升高维高频建模能力。
Kolmogorov-Arnold Fourier Networks
- 将B样条转为自适应谱表示，参数量从O(G)降为O(1)
- 引入可训练随机傅里叶特征，精准捕捉高频成分
- 混合GELU-傅里叶激活，训练中逐步增强频率表达
尽管基于Kolmogorov-Arnold的可解释网络（KANs）具有强理论表达能力，但在高维任务中仍存在参数爆炸和难以捕捉高频特征的问题。为此，我们提出柯尔莫戈罗夫-阿诺德傅里叶网络（KAF），通过谱重参数化从根本上重构KAN范式。主要贡献包括：（1）将局部网格化的B样条表示转换为全局自适应谱表示，改变网络归纳偏置，使参数复杂度从$O(G)$降至$O(1)$，同时保持表达能力；（2）引入通过谱对齐初始化的可训练随机傅里叶特征（RFF），突破固定核的平滑性限制，准确捕捉高频成分；（3）实现自适应混合GELU-傅里叶激活机制，在训练过程中逐步增强频率表征。全面实验表明，KAF在计算机视觉（CV）、自然语言处理（NLP）、音频及偏微分方程（PDE）求解任务中均表现卓越，达到最新水平且效率更高。代码已开源：https://github.com/kolmogorovArnoldFourierNetwork/KAF。
原文摘要 · Abstract (English)
Although Kolmogorov-Arnold-based interpretable networks (KANs) possess strong theoretical expressiveness, they suffer from severe parameter explosion and limited ability to capture high-frequency features in high-dimensional tasks. To address these issues, we propose the Kolmogorov-Arnold Fourier Network (KAF), which fundamentally redefines the KAN paradigm through spectral reparameterization. Our key contributions include: (1) proposing a fundamental basis transformation from the local, grid-based B-spline representation to a global, adaptive spectral representation. This shift changes the network's inductive bias, reducing parameter complexity from $O(G)$ to $O(1)$ while preserving expressiveness; (2) introducing trainable Random Fourier Features (RFF) initialized via a spectral alignment strategy, which allows the model to break the smoothness limitation of fixed kernels and accurately capture high-frequency components; and (3) implementing an adaptive hybrid GELU-Fourier activation mechanism that progressively enhances frequency representation during training. Comprehensive experiments demonstrate the superiority of KAF across computer vision (CV), natural language processing (NLP), audio, and partial differential equation (PDE) solving tasks, achieving state-of-the-art performance with improved efficiency. The code is available at https://github.com/kolmogorovArnoldFourierNetwork/KAF.
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