FI-KAN：分形插值Kolmogorov-Arnold网络
FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks
AI总结：
FI-KAN通过引入可学习的分形插值函数基底改进了KAN，使其在不同正则性目标上表现优异，特别是在处理分形目标和非光滑PDE解时效果显著。
摘要：
Kolmogorov-Arnold Networks (KAN) 使用固定网格上的B-样条基底，无法提供内在的多尺度分解以近似非光滑函数。我们引入了分形插值KAN (FI-KAN)，将其从迭代函数系统 (IFS) 理论中学习到的分形插值函数 (FIF) 基底整合到KAN中。提出了两种变体：纯FI-KAN (Barnsley, 1986) 完全用FIF基底替代B-样条；混合FI-KAN (Navascues, 2005) 保留B-样条路径并添加可学习的分形修正。IFS收缩参数为每条边赋予可微的分形维度，该维度在训练过程中适应目标的正则性。在Hölder正则性基准（α∈[0.2, 2.0]）上，混合FI-KAN在每个正则性水平上均优于KAN（1.3倍到33倍）。在分形目标上，FI-KAN相比KAN实现了高达6.3倍的MSE减少，保持在5 dB SNR时4.7倍的优势。在非光滑PDE解（scikit-fem）上，混合FI-KAN在粗糙系数扩散和L形域角落奇异性上分别实现了79倍和3.5倍的改进。纯FI-KAN的互补行为，主导在粗糙目标上但表现欠佳于光滑目标，提供了受控证据，表明基底几何必须匹配目标正则性。一个分形维度正则化器提供可解释的复杂度控制，其学习值恢复每个目标的真实分形维度。这些结果确立了正则性匹配的基底设计作为神经函数近似的原理性策略。
英文摘要：
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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