🤖 machine learning

FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

This paper introduces Fractal Interpolation Kolmogorov-Arnold Networks (FI-KAN), a novel architecture that integrates learnable fractal interpolation function bases into KANs to achieve superior approximation accuracy and regularity adaptation for non-smooth and fractal targets compared to traditional B-spline-based KANs.

Original authors: Gnankan Landry Regis N'guessan

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gnankan Landry Regis N'guessan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to draw pictures.

For a long time, we've used two main types of "pens" for this robot:

The Smooth Pen (MLPs): This pen draws perfect, flowing curves. It's great for drawing a sunset or a smooth hill, but if you ask it to draw a jagged lightning bolt or a crumpled piece of paper, it struggles. It keeps trying to smooth out the rough edges, making the drawing look blurry and wrong.
The Flexible Pen (KANs): This is a newer, smarter pen. Instead of just drawing one smooth curve, it can learn to draw different shapes (like little bumps and waves) along its path. It's much better at capturing details than the Smooth Pen.

But here's the problem:
Even the Flexible Pen (KAN) is built on a specific kind of math that assumes things are mostly smooth. If you ask it to draw something truly "rough"—like a fractal coastline, a stock market crash, or the jagged edge of a broken rock—it still tries to force smoothness onto it. It's like trying to measure the length of a coastline with a ruler; you miss all the tiny nooks and crannies.

Enter FI-KAN: The "Shape-Shifting" Pen

This paper introduces a new tool called FI-KAN (Fractal Interpolation Kolmogorov–Arnold Network). Think of FI-KAN as a magic pen that can change its own texture.

Instead of just having a smooth tip or a bumpy tip, this pen has a "dial" that the robot can turn while it's learning.

Turn the dial to "Smooth": The pen acts like a normal, smooth pen (great for drawing a ball or a smooth curve).
Turn the dial to "Rough": The pen suddenly becomes jagged, self-repeating, and fractal (great for drawing a fern leaf, a lightning bolt, or a mountain range).

The genius of FI-KAN is that the robot learns exactly where to turn the dial. It doesn't need a human to tell it, "This part is smooth, that part is rough." The robot figures it out on its own.

The Two Versions of the Magic Pen

The paper presents two ways to use this magic pen:

1. The "Pure" Version (The Specialist)

How it works: This pen only uses the rough, fractal texture. It throws away the smooth part entirely.
Best for: Drawing things that are always rough, like a fractal pattern or a very jagged signal.
Weakness: If you ask it to draw a smooth circle, it gets confused. It tries to make the circle out of tiny jagged steps, which looks terrible. It's like trying to build a smooth wall out of broken bricks.

2. The "Hybrid" Version (The All-Rounder)

How it works: This is the superstar. It keeps the smooth pen and adds the rough fractal pen as a "correction layer."
The Analogy: Imagine you are painting a wall.
- First, you use a smooth roller to cover the big, flat areas (the B-spline part).
- Then, you take a rough sponge and dab it on top to add texture where the wall is cracked or bumpy (the fractal part).
Why it wins: If the wall is smooth, the sponge stays quiet, and the smooth roller does the work. If the wall is rough, the sponge kicks in and captures all the jagged details. This version is better at everything because it can switch between smooth and rough instantly.

Why Does This Matter? (Real-World Examples)

The paper tested this on some very difficult problems where "smooth" math usually fails:

The "Broken" Math Problems (PDEs): In physics, some equations describe things like fluid flowing around a sharp corner or heat spreading through a material with cracks. These solutions are naturally "rough" and jagged.
- Result: The Hybrid FI-KAN was up to 79 times more accurate than the old methods at solving these problems. It's like the difference between a blurry photo and a high-definition 4K image.
The "Noisy" Data: When data is messy (like stock prices or earthquake sensors), the old pens get confused by the noise. The FI-KAN pen is naturally good at separating the "signal" (the real pattern) from the "noise" because it understands how roughness works.
The "Fractal" Test: When asked to draw a true fractal (a shape that looks the same no matter how much you zoom in), the FI-KAN pen got it right, while the others just gave up and drew a blurry mess.

The "Occam's Razor" Dial

One of the coolest features is a built-in "complexity filter."
Imagine the robot is lazy and wants to do the least amount of work. The FI-KAN has a rule: "Don't make the pen rough unless you absolutely have to."

If the target is a smooth curve, the robot sets the dial to "Smooth" and saves energy.
If the target is a jagged mountain, the robot only then turns the dial to "Rough."

This prevents the robot from overthinking and making simple things complicated.

The Bottom Line

For decades, AI has been trying to solve complex, messy, real-world problems with tools designed for smooth, perfect math. It's like trying to fix a car engine with a hammer.

FI-KAN gives the AI a Swiss Army Knife. It can be a smooth screwdriver when it needs to be, or a jagged saw when the job requires it. By letting the AI learn how rough the problem is, it solves difficult scientific and engineering tasks much faster and more accurately than ever before.

In short: If the world is smooth, use the smooth pen. If the world is rough, use the fractal pen. With FI-KAN, the AI learns to pick the right pen for the job automatically.

1. Problem Statement

Standard neural function approximation architectures, including Multi-Layer Perceptrons (MLPs) and Kolmogorov–Arnold Networks (KANs), rely on smooth basis functions (e.g., ReLU, SiLU, or B-splines). While these are near-optimal for smooth targets (Sobolev or Besov spaces with high regularity), they suffer from a fundamental regularity mismatch when approximating non-smooth functions.

Many scientific and engineering targets—such as turbulence fields, financial time series, fracture surfaces, and solutions to Partial Differential Equations (PDEs) with corner singularities or rough coefficients—exhibit Hölder regularity ( $\alpha < 1$ ) or fractal self-similarity. Approximating such functions using smooth bases requires an exponential increase in basis elements ( $O(h^{-1/\alpha})$ ) to achieve a given accuracy, rendering standard architectures inefficient and prone to underfitting or over-smoothing.

The core problem addressed is the lack of a neural architecture capable of learning the geometric regularity of the target function, specifically adapting from smooth to fractal structures during training.

2. Methodology: FI-KAN Architecture

The authors introduce Fractal Interpolation KAN (FI-KAN), which integrates Fractal Interpolation Functions (FIFs) derived from Iterated Function Systems (IFS) theory into the KAN framework.

Core Mechanism

Instead of fixed B-spline bases, FI-KAN employs learnable fractal bases. The key innovation is treating the vertical contraction parameters ( $d_i$ ) of the IFS as trainable parameters.

Fractal Dimension Control: The box-counting dimension ( $dim_B$ $d i m_{B}$ ) of the basis functions is a differentiable function of the contraction parameters $d$ $d$ .
- When $d = 0$ , the basis reduces to standard piecewise linear "hat" functions (smooth, $dim_B = 1$ ).
- As $|d|$ increases, the basis acquires self-affine, fractal structure ( $dim_B > 1$ ).
Differentiability: The evaluation of FIF bases uses a truncated Read–Bajraktarević (RB) iteration, which is fully differentiable, allowing for end-to-end gradient-based optimization.

Two Variants

The paper proposes two architectures based on different mathematical frameworks:

Pure FI-KAN (Barnsley Framework):
- Structure: Completely replaces B-splines with FIF bases.
- Function: $\phi(x) = \sum w_m \phi_m(x; d)$ .
- Behavior: Highly specialized for rough/fractal targets. It struggles on smooth targets because it cannot efficiently reproduce high-degree polynomials without learning $d \approx 0$ , which limits its approximation power for smooth curvature.
Hybrid FI-KAN (Navascués Framework):
- Structure: Implements an $\alpha$ $α$ -fractal decomposition: $f^\alpha_b = b + h$ $f_{b}^{α} = b + h$ .
  - $b$ (Base): A standard B-spline path (classical approximant).
  - $h$ (Correction): A parallel FIF path acting as a fractal residual.
- Function: $\phi(x) = \text{Spline}(x) + \sum w_m \phi_m(x; d)$ .
- Behavior: Robust across the entire spectrum. It defaults to a standard KAN when $d=0$ and learns fractal corrections only where the data demands it.

Regularization

A Fractal Dimension Regularizer is introduced to penalize unnecessary fractal complexity:
$R_{frac}(d) = (\dim_B(d) - 1)^2$
This acts as a "geometry-aware Occam's razor," encouraging the network to learn smooth bases ( $dim_B \approx 1$ ) unless the data provides strong evidence for fractal structure.

3. Key Contributions

Learnable Basis Geometry: The first neural architecture where the geometric regularity (fractal dimension) of the basis functions is a continuous, learnable parameter derived from IFS theory.
Theoretical Foundation: Proves that FIF bases can exactly represent self-affine targets with $O(N)$ parameters, whereas B-splines require $O(\epsilon^{-1/\alpha})$ . It establishes that Pure FI-KAN suffers from infinite total variation on smooth targets, explaining its performance drop.
Hybrid Architecture: Demonstrates that a smooth-rough decomposition (Hybrid FI-KAN) outperforms both pure smooth and pure fractal approaches, providing a universal approximator for mixed regularity.
Interpretability: The learned contraction parameters provide a direct diagnostic for target regularity. The learned $dim_B$ correlates monotonically with the true Hölder exponent of the target.

4. Experimental Results

The authors validate FI-KAN across 1D/2D regression, noise robustness, and non-smooth PDE solutions.

Hölder Regularity Sweep: On targets $f_\alpha(x) = |x|^\alpha$ ( $\alpha \in [0.2, 2.0]$ ), Hybrid FI-KAN outperforms KAN at every level, with improvements ranging from 1.3× to 33× (MSE reduction).
Fractal Targets: On Weierstrass and Takagi-Landsberg functions, Hybrid FI-KAN achieves up to 6.3× MSE reduction over KAN. Pure FI-KAN also outperforms KAN on these specific targets but fails on smooth ones.
Non-Smooth PDE Solutions:
- Rough-Coefficient Diffusion: Hybrid FI-KAN achieves 65× to 79× improvement over KAN.
- L-shaped Domain (Corner Singularity): 3.5× improvement in approximating the $r^{2/3}$ singularity.
- Stochastic Heat Equation: Up to 25× improvement on noisy solutions.
Noise Robustness: FI-KAN acts as an implicit multi-scale denoiser, maintaining a 4.7× to 6.1× advantage over KAN even at low Signal-to-Noise Ratios (5 dB).
Parameter Efficiency: Even when matched for parameter count, Hybrid FI-KAN outperforms KAN, proving the gain is architectural, not just due to capacity.

5. Significance and Implications

Principled Regularity Matching: The paper establishes that basis geometry must match target regularity. The contrast between Pure and Hybrid variants provides controlled evidence that simply increasing the number of parameters is insufficient; the shape of the basis functions is critical.
Scientific Computing: FI-KAN offers a principled solution for problems where non-smoothness is intrinsic (e.g., fluid dynamics, material science, geophysics), addressing a gap where standard PINNs and KANs fail due to smoothness assumptions.
Interpretable Complexity: The ability to learn and regularize the fractal dimension offers a new dimension of interpretability in neural networks, allowing the model to "discover" the roughness of the underlying physical process.
Limitations: The method introduces computational overhead (approx. 2.5× per forward pass for $K=2$ ) due to sequential recursion. It is most effective for structured roughness (deterministic or PDE-inherited) rather than unstructured stochastic noise.

In conclusion, FI-KAN represents a significant shift from "smooth-first" neural approximation to regularity-adaptive approximation, leveraging fractal geometry to solve problems that were previously intractable for standard deep learning architectures.