The paper introduces Fractal Interpolation Kolmogorov-Arnold Networks (FI-KAN), a neural architecture that augments or replaces the B-spline bases in KAN with fractal interpolation function (FIF) bases derived from iterated function system (IFS) theory. Two variants are proposed: Pure FI-KAN, which replaces B-splines entirely with FIF bases, and Hybrid FI-KAN, which retains B-splines and adds a learnable fractal correction path. The key innovation is that the vertical contraction parameters of the IFS are treated as trainable, giving each edge activation a differentiable fractal dimension that adapts to target regularity during training. A fractal dimension regularizer provides interpretable complexity control by penalizing unnecessary fractal structure.

Extensive experiments demonstrate that FI-KAN significantly outperforms standard KAN on non-smooth targets. On a Hölder regularity benchmark (α∈[0.2,2.0]), Hybrid FI-KAN achieves 1.3× to 33× MSE reduction over KAN. On fractal targets, FI-KAN achieves up to 6.3× MSE reduction, maintaining a 4.7× advantage at 55 dB SNR. On non-smooth PDE solutions, Hybrid FI-KAN achieves up to 79× improvement on rough-coefficient diffusion and 3.5× on L-shaped domain corner singularities. The results validate the regularity-matching hypothesis: basis geometry must match target regularity for optimal approximation. The learned fractal dimension from the regularizer recovers the true fractal dimension of targets, providing an interpretable diagnostic. FI-KAN is presented as a principled extension for function classes with non-trivial geometric regularity, not a general-purpose replacement for KAN.