FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

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 φ_m(x; d). (d) Hybrid FI-KAN: retains B-splines and adds a fractal correction (f_b^α = b + h). When d = 0, (d) reduces to (b).

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This repository provides a reference implementation of Fractal Interpolation KAN (FI-KAN), a neural architecture that incorporates learnable fractal interpolation function (FIF) bases from iterated function system (IFS) theory into the Kolmogorov-Arnold Network (KAN) framework.

FI-KAN introduces basis functions with a learnable fractal dimension that adapts to the regularity of the target function during training, providing a continuous, differentiable knob from smooth (dim_B = 1) to fractal (dim_B > 1) edge geometry.

Two variants are provided:

• Pure FI-KAN (Barnsley framework): replaces B-splines entirely with FIF bases.
• Hybrid FI-KAN (Navascues framework): retains the B-spline path and adds a learnable fractal correction, implementing the alpha-fractal decomposition f_b^alpha = b + h.

The architecture is particularly suited for function approximation targets with non-trivial Holder regularity, fractal self-similarity, or structured roughness inherited from PDE operators (corner singularities, rough coefficients, stochastic forcing).

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Paper

FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks Gnankan Landry Regis N'guessan

The preprint is available on arXiv.

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Key idea

Standard KAN uses B-spline basis functions that are near-optimal for smooth targets but operate at a single resolution with no intrinsic multi-scale decomposition. For targets with box-counting dimension dim_B > 1 (turbulence fields, fracture surfaces, natural terrain, rough PDE solutions), smooth bases are fundamentally mismatched.

FI-KAN resolves this by:

1. Embedding fractal interpolation function (FIF) bases (Barnsley, 1986) into KAN edges,
2. Treating the IFS contraction parameters {d_i} as differentiable, trainable quantities,
3. Providing a fractal dimension regularizer (geometry-aware Occam's razor) that penalizes unnecessary fractal complexity,
4. Learning both the interpolation ordinates (what values to hit) and the inter-grid-point geometry (how rough the interpolation should be).

When d = 0, FIF bases reduce to piecewise linear hat functions (Pure) or the architecture reduces exactly to standard KAN (Hybrid). The network develops fractal structure only where the data demands it.

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Architecture variants

• Pure FI-KAN (Barnsley framework): all-fractal bases, strongest on rough targets. When d = 0: order-1 spline KAN.
• Hybrid FI-KAN (Navascues framework): spline backbone + fractal correction. When d = 0, w_frac = 0: reduces to standard KAN. Recommended as the default variant with K = 2 (fractal depth).

Baselines include efficient-KAN (Blealtan, MIT License) and parameter-matched MLP.

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Key results

Target	KAN MSE	Hybrid FI-KAN MSE	Improvement
Takagi-Landsberg sawtooth (dim_B = 1.5)	1.14e-2	1.81e-3	6.3x
Rough-coeff diffusion (H=0.3)	1.19e-3	1.50e-5	79x
L-shaped corner singularity	2.89e-3	8.35e-4	3.5x
Holder alpha=1.5	1.58e-3	4.8e-5	33x
Noise robustness (sawtooth, clean SNR)	1.14e-2	1.87e-3	6.1x

On smooth targets (polynomial, exp sin), Hybrid FI-KAN remains competitive or superior (up to 220x improvement on polynomial). Pure FI-KAN underperforms on smooth targets by design, confirming the regularity-matching hypothesis.

The strongest results are on non-smooth PDE solutions with structured roughness: 65-79x improvement over KAN on rough-coefficient diffusion equations computed via scikit-fem, where the PDE operator transforms unstructured fBm coefficient roughness into structured, deterministic solution roughness that FI-KAN captures effectively.

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Install

pip install -r requirements.txt
pip install -e .

For PDE benchmarks (scikit-fem, fbm):

pip install scikit-fem fbm scipy

Quickstart

Fit the Takagi-Landsberg function (dim_B = 1.5, w = 2^{-1/2}):

python quickstart_fit_sawtooth.py

What is inside

• bases.py: fractal interpolation basis function evaluation via truncated Read-Bajraktarevic iteration (Algorithm 1), box-counting dimension computation
• layers.py: PureFIKANLinear (Barnsley framework), HybridFIKANLinear (Navascues framework) with learnable contraction parameters, fractal dimension regularizer, and fractal energy ratio diagnostic
• models.py: PureFIKAN, HybridFIKAN (multi-layer network wrappers)
• baselines.py: efficient-KAN (Blealtan, MIT), MLP with parameter matching
• targets.py: Weierstrass, Takagi-Landsberg (w = 2^{-1/2}, dim_B = 1.5), Holder family, Ackley 2D, multiscale, chirp, and smooth benchmarks
• training.py: multi-seed training loop with fractal dimension tracking
• benchmarks.py: PDE data generators (L-shaped domain FEM, rough-coefficient diffusion FEM, stochastic heat equation, fractal terrain, fBm paths, rough volatility)

Usage example

import torch
from models import HybridFIKAN
from training import train_model

device = "cuda" if torch.cuda.is_available() else "cpu"

# Create Hybrid FI-KAN: [input_dim, hidden, output_dim]
model = HybridFIKAN(
    [1, 16, 1],
    grid_size=8,
    spline_order=3,
    fractal_depth=2,    # K=2 recommended (Section 5.9)
    d_init_std=0.01,
).to(device)

# Train on your data
result = train_model(
    model, x_train, y_train, x_test, y_test,
    epochs=500, lr=1e-3,
    reg_frac=0.001,              # fractal dimension regularization
    model_type='fikan_hybrid',
)

# Inspect learned fractal dimensions
for i, d in enumerate(model.fractal_dimensions()):
    print(f"Layer {i}: dimB = {d.mean().item():.4f}")

# Inspect fractal energy ratio (rho ~ 0: spline-dominated)
for i, rho in enumerate(model.fractal_energy_ratios()):
    print(f"Layer {i}: rho = {rho:.4f}")

Citation

Will be updated soon

License

MIT