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A Practitioner's Guide to Kolmogorov-Arnold Networks

This paper provides a systematic review of Kolmogorov-Arnold Networks (KANs) by clarifying their theoretical foundations and relationships to existing methods, analyzing their design components, summarizing recent performance advances, and offering a practical guide for researchers and practitioners.

Original authors: Amir Noorizadegan, Sifan Wang, Leevan Ling, Juan P. Dominguez-Morales

Published 2026-04-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amir Noorizadegan, Sifan Wang, Leevan Ling, Juan P. Dominguez-Morales

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 understand the world. For decades, the standard way to do this has been using MLPs (Multilayer Perceptrons). Think of an MLP as a factory assembly line where raw materials (data) go in, get mixed together by a conveyor belt (linear math), and then pass through a single, rigid machine at the end that decides if the product is "good" or "bad" (a fixed activation function like ReLU). It works well, but it's a bit like trying to sculpt a masterpiece using only a hammer and a chisel that never change shape.

Enter KANs (Kolmogorov–Arnold Networks). This paper is a "Practitioner's Guide" to a new kind of robot brain that flips the assembly line upside down.

Here is the breakdown of the paper in simple terms, using everyday analogies.

1. The Big Idea: Changing the Assembly Line

In a traditional MLP, the "decision-making" part (the activation function) is fixed and glued to the nodes (the workers).

MLP: Mix ingredients $\rightarrow$ Apply a fixed rule (e.g., "If it's hot, stop").
KAN: Apply a flexible, learnable rule to each ingredient first $\rightarrow$ Then mix them.

The Analogy:
Imagine you are baking a cake.

MLP is like taking flour, sugar, and eggs, dumping them in a bowl, mixing them, and then running the whole mixture through a single, unchangeable filter that decides if it's a cake.
KAN is like having a special, adjustable machine for each ingredient. You can tune the machine to perfectly sift the flour, whisk the eggs, and melt the sugar before you mix them. Because you can tune the machines for the specific ingredients, you get a much better cake with fewer ingredients.

2. The Secret Sauce: The "Basis Functions"

The paper spends a lot of time talking about "Basis Functions." Think of these as the tools in your toolbox.

In the old days, everyone used a B-spline (a smooth, flexible ruler) as their default tool. It works well for most things.
But this paper says: "Why use a ruler if you are trying to draw a circle? Use a compass!"

The authors cataloged many different "tools" (bases) you can swap into your KAN:

Chebyshev Polynomials: Great for smooth, wavy patterns (like sound waves).
Fourier: Perfect for anything that repeats (like tides or seasons).
Wavelets: Good for spotting sudden changes or "spikes" (like an earthquake).
Sinc: Excellent for sharp edges and sudden jumps.

The Lesson: There is no "one size fits all." If you are solving a physics problem about heat (smooth), use one tool. If you are solving a problem about shockwaves (sharp), use another. The paper gives you a guide on which tool to pick for which job.

3. The "Choose-Your-Own-Adventure" Guide

One of the most practical parts of the paper is the "Choose-Your-KAN" section. It's like a flowchart for a mechanic:

Is your data sparse (hard to get)? Use a KAN; they learn faster with less data.
Do you need to understand why the robot made a decision? Use a KAN; because the "rules" are visible and adjustable, you can actually read them.
Is the problem about smooth curves? Use Chebyshev.
Is the problem about jagged edges? Use Sinc or Rational functions.

4. Why Not Just Use the Old Way (MLP)?

The paper admits that MLPs are still kings in some areas. If you have a massive dataset and just need a quick answer (like recognizing a cat in a photo), a standard MLP is fast and efficient.

However, KANs shine in Scientific Machine Learning:

Physics Simulations: When you need to solve complex equations (like how air flows over a wing), KANs often find the answer faster and more accurately.
Interpretability: In science, you don't just want the answer; you want the formula. KANs can sometimes reveal the underlying mathematical equation hidden in the data, whereas MLPs are often "black boxes."

5. The Catch: It's Not Magic

The paper is very honest about the downsides.

Training is harder: Because the "tools" are adjustable, the robot has to learn how to tune them and how to mix them. This takes more computing power and time.
It's complex: You have to be careful about which "tool" you pick. If you pick the wrong one, the robot might get confused.

Summary: The "Chef's Table" vs. The "Fast Food Joint"

MLPs are like a Fast Food Joint: They are fast, consistent, and great for feeding a crowd (big data). But the menu is fixed, and you can't ask for a custom sauce.
KANs are like a Michelin-Star Chef's Table: They take longer to prepare, require a skilled chef (you, the user) to choose the right ingredients (basis functions), and are harder to manage. But if you do it right, they can create a dish (solution) that is more delicious (accurate), more nutritious (interpretable), and perfectly tailored to your specific taste (the problem at hand).

The Bottom Line:
This paper is a manual for the future of AI in science. It tells us that instead of forcing every problem into the same rigid box (MLP), we should build flexible, modular systems (KANs) where we can choose the specific mathematical "shape" that fits the problem we are trying to solve. It's about moving from "one size fits all" to "custom-tailored solutions."

1. Problem Statement

Multilayer Perceptrons (MLPs) have long been the standard for function approximation and scientific machine learning (e.g., Physics-Informed Neural Networks or PINNs). However, MLPs face several limitations:

Fixed Activations: They rely on fixed, pointwise activation functions (e.g., ReLU, Tanh) applied after linear mixing, limiting adaptability.
Interpretability: They act as "black boxes," making it difficult to extract symbolic equations or understand causal relationships.
Spectral Bias: They tend to learn low-frequency components faster than high-frequency ones, hindering convergence for oscillatory or sharp-gradient solutions.
Parameter Efficiency: Achieving high accuracy often requires massive parameter counts.

While Kolmogorov–Arnold Networks (KANs) were recently introduced as an alternative inspired by the Kolmogorov Superposition Theorem (KST), the field has rapidly expanded with numerous variants, basis functions, and applications. This creates a fragmented landscape where practitioners lack a systematic understanding of:

The theoretical relationship between KANs, KST, and classical methods.
The impact of different basis functions on performance.
Best practices for accuracy, efficiency, and convergence.

2. Methodology and Framework

The paper provides a comprehensive review and systematic analysis of the KAN literature, structured around three core themes:

A. Theoretical Bridging

The authors clarify the mathematical relationships between KANs and other paradigms:

KANs vs. KST: Modern KANs are not exact realizations of the Kolmogorov Superposition Theorem. The KST requires universal, non-smooth inner functions and a specific two-layer structure. KANs replace these with learnable, smooth basis functions (e.g., B-splines) and allow for arbitrary depth, making them data-driven approximations inspired by KST rather than constructive proofs of it.
KANs vs. Kernel Methods: Shallow, 1D KANs are mathematically equivalent to classical kernel regression (basis expansion). However, deep KANs diverge by using compositional layers to generate multivariate interactions, avoiding the "curse of dimensionality" inherent in tensor-product kernel methods.
KANs vs. MLPs: KANs and MLPs are expressively equivalent (any MLP can be represented as a KAN and vice versa). However, KANs relocate nonlinearity from nodes to edges (learnable univariate functions on edges vs. fixed activations on nodes). This structural shift ("activate-then-mix" vs. "mix-then-activate") leads to superior parameter efficiency and interpretability.

B. Basis Function Analysis

The paper identifies the choice of basis function as the central design axis of KANs. It categorizes and evaluates major families:

B-splines: The default choice; offers local control, smoothness, and numerical stability.
Chebyshev Polynomials: Ideal for smooth, global, or oscillatory functions; requires normalization (e.g., tanh) to prevent instability in deep stacks.
ReLU-based: Computationally efficient (fast GPU execution) but less smooth; variants like HRKAN improve smoothness for PDEs.
Gaussian (RBF): Smooth and infinitely differentiable; effective for local features and operator learning.
Fourier: Best for periodic structures; includes Random Fourier Features (RFF) for adaptive spectral learning.
Wavelets & Sinc: Specialized for multi-scale features, singularities, and sharp gradients.
Jacobi/General Polynomials: Offer flexible orthogonality and rational approximations (e.g., fKAN, rKAN).

C. Optimization and Enhancement Strategies

The review synthesizes techniques to improve KAN performance:

Accuracy: Physics-informed loss design, adaptive sampling (residual-based), domain decomposition (FBKAN), and function decomposition (multi-fidelity).
Efficiency: GPU-optimized kernels (CUDA/JAX), matrix optimizations, and replacing heavy bases with faster alternatives (e.g., ReLU-power).
Regularization: Sparsity penalties ( $L_1$ ), entropy balancing, Lipschitz constraints, and Bayesian formulations for uncertainty quantification.
Convergence: Analysis via Neural Tangent Kernels (NTK) shows KANs often have better-conditioned spectra than MLPs, leading to faster convergence, though this depends heavily on basis selection and grid resolution.

3. Key Contributions

Systematic Taxonomy: A unified framework connecting KANs to KST, kernel methods, and MLPs, resolving misconceptions about their theoretical foundations.
Basis Function Guide: A comprehensive comparison of basis functions (splines, polynomials, RBFs, etc.), detailing their mathematical properties, stability requirements, and ideal use cases.
Practical "Choose-Your-KAN" Guide: A 7-step algorithmic recipe for practitioners to select the optimal KAN configuration based on problem structure (smoothness, periodicity, discontinuities) and constraints.
Resource Consolidation: A curated list of open-source repositories, benchmarks, and implementations (Table 2), serving as a central hub for the community.
Research Roadmap: Identification of current gaps (e.g., lack of unified benchmarks, finite-width optimization theory) and a "Multi-Pillar Framework" for future research.

4. Results and Findings

Performance Trends: KANs generally match or outperform MLPs and PINNs in accuracy and convergence speed, particularly for problems with smooth, periodic, or structured solutions. However, for simple tasks with abundant data, optimized MLPs may remain competitive due to lower training overhead.
Basis Sensitivity: There is no "one-size-fits-all" basis.
- B-splines are robust defaults.
- Chebyshev excels in smooth PDEs but requires stabilization.
- Sinc/Wavelets are superior for discontinuities.
- Fourier is best for periodicity.
Convergence Dynamics: KANs exhibit reduced spectral bias compared to MLPs, allowing them to learn high-frequency components more effectively. However, this can lead to ill-conditioned loss landscapes if the basis resolution is too fine, necessitating careful initialization and optimizer scheduling (e.g., Adam $\to$ L-BFGS).
Parameter Efficiency: KANs achieve comparable accuracy to MLPs with significantly fewer parameters due to the expressivity of learnable edge functions.

5. Significance

This paper serves as a definitive practitioner's guide and theoretical bridge for the emerging field of Kolmogorov–Arnold Networks.

For Researchers: It clarifies the theoretical underpinnings, distinguishing KANs from exact KST constructions and providing a roadmap for future theoretical work (e.g., finite-width analysis, RKHS formulations).
For Practitioners: It moves beyond hype by offering a structured decision-making process. It emphasizes that KAN performance is not intrinsic to the architecture name but is a result of informed design choices (basis selection, grid adaptation, regularization).
For Scientific ML: It accelerates the adoption of KANs in physics-informed learning, operator learning, and symbolic regression by providing a unified toolkit of methods to handle complex, multi-scale, and discontinuous physical phenomena.

In summary, the paper argues that KANs represent a paradigm shift toward structured, interpretable, and modular neural architectures, but their success relies on aligning the mathematical properties of the chosen basis functions with the specific characteristics of the target problem.