FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Abstract

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.

Figures (24)

• Figure 1: MLPs vs. KANs vs. FI-KANs. (a) MLP: scalar weights on edges, fixed activations on nodes. (b) KAN: learnable B-spline functions on edges. (c) Pure FI-KAN: replaces B-splines with fractal interpolation bases $\varphi_m(x;\mathbf{d})$. (d) Hybrid FI-KAN: retains B-splines and adds a fractal correction ($f_b^\alpha = b + h$). When $\mathbf{d} = \mathbf{0}$, (d) reduces to (b).
• Figure 2: FI-KAN: learning basis geometry across the smooth-to-rough spectrum. Top row: FI-KAN combines smooth basis functions (B-spline/hat-type), fractal interpolation with learnable contraction parameters $\mathbf{d}$, and the KAN edge-function graph to produce regularity-matched edge geometry. Middle row: Smooth-to-fractal basis morphing as $d_i$ increases from $0$ to $0.9$. At $d_i = 0$ the basis is a smooth hat function ($\dim_B = 1$); as $d_i$ increases, the basis acquires progressively finer self-affine structure with $\dim_B > 1$. Bottom row: The two FI-KAN variants. Pure FI-KAN (Barnsley framework) uses all-fractal bases, strongest on rough targets. Hybrid FI-KAN (Navascués framework) retains a spline backbone and adds a fractal correction, providing robustness across both smooth and rough regimes.
• Figure 3: Edge function architecture for the three models. (a) KAN: base activation plus B-spline path. (b) Pure FI-KAN (Barnsley): replaces B-splines with fractal interpolation bases $\varphi_m(x; \mathbf{d})$. When $\mathbf{d} = \mathbf{0}$, the FIF bases reduce to piecewise linear hat functions. (c) Hybrid FI-KAN (Navascués): retains the B-spline path and adds a fractal correction, implementing the $\alpha$-fractal decomposition $f_b^\alpha = b + h$. When $\mathbf{d} = \mathbf{0}$ and fractal weights vanish, Hybrid reduces to standard KAN.
• Figure 4: Fractal basis function $\varphi_2(x; \mathbf{d})$ (the middle basis on a 5-interval grid) as the contraction parameters $d_i$ vary uniformly from 0 to 0.9. At $d_i = 0$, the basis is the standard piecewise linear hat function ($\dim_{\mathrm{B}} = 1$). As $d_i$ increases, the basis acquires increasingly fine-scale fractal structure ($\dim_{\mathrm{B}} > 1$) while maintaining the Kronecker property $\varphi_2(x_j) = \delta_{2j}$ at all grid points (black dots). Recursion depth $K = 8$.
• Figure 5: Pure FI-KAN (Barnsley framework) architecture detail. (a) Network graph: all edges carry fractal interpolation function (FIF) bases $\varphi_m(x;\mathbf{d})$ with learnable contraction parameters; nodes perform summation. (b) Edge activation: the input $x$ splits into a base SiLU path and a fractal path $\sum w_m^{\mathrm{frac}} \varphi_m(x;\mathbf{d})$, where $\mathbf{d}\in(-1,1)^N$ controls the fractal character of the basis. When $\mathbf{d}=\mathbf{0}$, the FIF bases reduce to piecewise linear hat functions (order-1 KAN). (c) Fractal basis $\varphi_2(x;\mathbf{d})$ at four contraction values $d_i \in \{0, 0.3, 0.6, 0.9\}$, computed via $K=10$ truncated Read--Bajraktarević iterations. The Kronecker property $\varphi_2(x_j) = \delta_{2j}$ is preserved at all $d$ values (dots). (d) Box-counting dimension $\dim_B(\mathbf{d})$ as a function of uniform $d_i$, with the transition surface $\sum|d_i| = 1$ marked. (e) Fractal dimension regularizer $\mathcal{R}(\mathbf{d}) = (\dim_B(\mathbf{d}) - 1)^2$: a geometry-aware Occam's razor that penalizes unnecessary fractal complexity.

Theorems & Definitions (33)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 3.1
• proof
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Theorem 4.1