🤖 AI Summary
This work addresses the limitations of traditional Kolmogorov–Arnold Networks (KANs), which rely on fixed-grid B-spline bases and struggle to approximate nonsmooth or multiscale functions effectively. The authors introduce, for the first time, fractal geometry into neural function approximation by proposing Pure FI-KAN and Hybrid FI-KAN architectures. These models employ learnable fractal interpolation function (FIF) bases constructed via iterated function systems (IFS) and incorporate a differentiable fractal dimension as an adaptive parameter to dynamically align the Hölder regularity of the basis functions with that of the target function. Empirical results demonstrate substantial improvements over conventional KANs: on Hölder continuity benchmarks, performance gains range from 1.3× to 33×; mean squared error on fractal targets is reduced by 6.3×; and accuracy in approximating solutions to nonsmooth partial differential equations improves by up to 79×.
📝 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.