Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

FEKAN: Feature-Enriched Kolmogorov-Arnold Networks

Sidharth S. Menon, Ameya D. Jagtap

TL;DR

This work introduces Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters.

Abstract

Kolmogorov-Arnold Networks (KANs) have recently emerged as a compelling alternative to multilayer perceptrons, offering enhanced interpretability via functional decomposition. However, existing KAN architectures, including spline-, wavelet-, radial-basis variants, etc., suffer from high computational cost and slow convergence, limiting scalability and practical applicability. Here, we introduce Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters. By incorporating these additional features, FEKAN accelerates convergence, increases representation capacity, and substantially mitigates the computational overhead characteristic of state-of-the-art KAN architectures. We investigate FEKAN across a comprehensive set of benchmarks, including function-approximation tasks, physics-informed formulations for diverse partial differential equations (PDEs), and neural operator settings that map between input and output function spaces. For function approximation, we systematically compare FEKAN against a broad family of KAN variants, FastKAN, WavKAN, ReLUKAN, HRKAN, ChebyshevKAN, RBFKAN, and the original SplineKAN. Across all tasks, FEKAN demonstrates substantially faster convergence and consistently higher approximation accuracy than the underlying baseline architectures. We also establish the theoretical foundations for FEKAN, showing its superior representation capacity compared to KAN, which contributes to improved accuracy and efficiency.

FEKAN: Feature-Enriched Kolmogorov-Arnold Networks

TL;DR

This work introduces Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters.

Abstract

Kolmogorov-Arnold Networks (KANs) have recently emerged as a compelling alternative to multilayer perceptrons, offering enhanced interpretability via functional decomposition. However, existing KAN architectures, including spline-, wavelet-, radial-basis variants, etc., suffer from high computational cost and slow convergence, limiting scalability and practical applicability. Here, we introduce Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters. By incorporating these additional features, FEKAN accelerates convergence, increases representation capacity, and substantially mitigates the computational overhead characteristic of state-of-the-art KAN architectures. We investigate FEKAN across a comprehensive set of benchmarks, including function-approximation tasks, physics-informed formulations for diverse partial differential equations (PDEs), and neural operator settings that map between input and output function spaces. For function approximation, we systematically compare FEKAN against a broad family of KAN variants, FastKAN, WavKAN, ReLUKAN, HRKAN, ChebyshevKAN, RBFKAN, and the original SplineKAN. Across all tasks, FEKAN demonstrates substantially faster convergence and consistently higher approximation accuracy than the underlying baseline architectures. We also establish the theoretical foundations for FEKAN, showing its superior representation capacity compared to KAN, which contributes to improved accuracy and efficiency.
Paper Structure (44 sections, 5 theorems, 69 equations, 42 figures, 11 tables)

This paper contains 44 sections, 5 theorems, 69 equations, 42 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Methodology -- FEKAN
  3. Feature Expansion in Spatio-temporal Learning
  4. NTK Evaluation of the FEKAN architecture
  5. Results
  6. Function Approximation
  7. High-Frequency and Discontinuous Test Function
  8. Spline Basis liu2024kan
  9. Fourier Basis:
  10. Radial Basis Function abueidda2025deepokan:
  11. Performance comparison for other basis functions:
  12. Dynamical Systems
  13. Physics-Informed FEKAN (PI-FEKAN)
  14. Test Case 1: Helmholtz Equation
  15. Spline Basis liu2024kan
  16. ...and 29 more sections

Key Result

Theorem 1

Let $\mathbf{x} = (x_1, \dots, x_n) \in [0,1]^n$ be the original inputs, and let be a continuous function of both the original inputs and $m$ additional continuous features where each $u_j: [0,1]^n \to \mathbb{R}$ is continuous (for example, $u_j(\mathbf{x}) = \sin(x_i), \cos(x_i), x_i^p$, etc.). Then there exist continuous functions and continuous functions such that More details on proof of

Figures (42)

  • Figure 1: Schematic representation of KAN vs. FEKAN architectures. In FEKAN, $\gamma(x)$ is the input feature map, $\gamma:x \xrightarrow{} \tilde{x}$.
  • Figure 2: Schematic representation of feature and enriched feature space.
  • Figure 3: Evolution of the NTK eigenvalue spectra during training for KAN and FEKAN in the function approximation task across different basis functions.
  • Figure 5: FEKAN applied to diverse problems, including function approximation, partial differential equations, and operator learning tasks.
  • Figure 6: Spline Basis: (Left) Scaling law of KAN and FEKAN with increasing grid size $G$ using the spline basis and (Center) Computational cost as a function of grid size $G$ for the spline basis. (Right) Convergence of KAN and FEKAN on a high-frequency test function using the spline basis.
  • ...and 37 more figures

Theorems & Definitions (8)

  • Theorem 1: Feature-Enriched Kolmogorov Superposition Theorem
  • Theorem 2: Complementary Gains in Representation Capacity
  • Theorem 3: Rademacher Complexity Under Feature Augmentation
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof