用分形插值构建可学习的神经网络基函数，提升非光滑函数拟合效果。
FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks
- 引入分形插值函数作为可学习基，实现多尺度自适应逼近。
- 在霍尔德正则性测试中性能优于传统KAN 1.3至33倍，分形目标上误差降低6.3倍。
- 分形维数可解释性强，适合处理奇异点或粗糙系数的偏微分方程问题。
Kolmogorov-Arnold Networks (KAN) 使用固定网格上的B样条基函数，缺乏对非光滑函数的内在多尺度分解能力。本文提出分形插值KAN（FI-KAN），将迭代函数系统（IFS）理论中的可学习分形插值函数（FIF）基引入KAN。提出两种变体：纯FI-KAN（Barnsley, 1986）完全替换B样条；混合FI-KAN（Navascues, 2005）保留B样条路径并添加可学习分形修正项。IFS压缩参数使每条边具有可微分的分形维数，训练中自适应匹配目标正则性。在霍尔德正则性基准（α∈[0.2, 2.0]）上，混合FI-KAN在所有正则性水平均优于KAN（1.3x至33x）。在分形目标上，FI-KAN相比KAN最高实现6.3倍均方误差降低，5 dB信噪比下仍保持4.7倍优势。在非光滑偏微分方程解（scikit-fem）中，混合FI-KAN对粗糙系数扩散问题提升达79倍，对L形域角奇点改善3.5倍。纯FI-KAN在粗糙目标占优而平滑目标表现较差，验证了基函数几何需匹配目标正则性的原则。分形维数正则化提供可解释的复杂度控制，学习值能恢复真实分形维数。结果确立正则性匹配基设计为神经函数逼近的合理策略。
原文摘要 · Abstract (English)
Kolmogorov-Arnold Networks (KAN) employ B-spline bases on a fixed grid, providing no intrinsic multi-scale decomposition for non-smooth function approximation. We introduce Fractal Interpolation KAN (FI-KAN), which incorporates learnable fractal interpolation function (FIF) bases from iterated function system (IFS) theory into KAN. Two variants are presented: Pure FI-KAN (Barnsley, 1986) replaces B-splines entirely with FIF bases; Hybrid FI-KAN (Navascues, 2005) retains the B-spline path and adds a learnable fractal correction. The IFS contraction parameters give each edge a differentiable fractal dimension that adapts to target regularity during training. On a Holder regularity benchmark ($α\in [0.2, 2.0]$), Hybrid FI-KAN outperforms KAN at every regularity level (1.3x to 33x). On fractal targets, FI-KAN achieves up to 6.3x MSE reduction over KAN, maintaining 4.7x advantage at 5 dB SNR. On non-smooth PDE solutions (scikit-fem), Hybrid FI-KAN achieves up to 79x improvement on rough-coefficient diffusion and 3.5x on L-shaped domain corner singularities. Pure FI-KAN's complementary behavior, dominating on rough targets while underperforming on smooth ones, provides controlled evidence that basis geometry must match target regularity. A fractal dimension regularizer provides interpretable complexity control whose learned values recover the true fractal dimension of each target. These results establish regularity-matched basis design as a principled strategy for neural function approximation.
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