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P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization

Xavier Warin

TL;DR

This paper addresses the challenge of high-dimensional function approximation, especially for irregular functions, by introducing P1-KAN, a layered Kolmogorov-Arnold network using 1D finite-element-like bases. It provides rigorous convergence bounds under Lipschitz outer functions and universal approximation theorems, including both adaptation-enabled and non-adaptation variants, and demonstrates empirical strength against MLPs and other KANs on synthetic functions. The method is further validated through a hydraulic valley optimization case, where P1-KAN with adapting lattices achieves faster convergence and robust performance, approaching or surpassing spline-based KAN in irregular regimes. Overall, P1-KAN offers a theoretically grounded, practically effective framework for high-dimensional, potentially irregular function approximation and stochastic optimization in engineering applications.

Abstract

A new Kolmogorov-Arnold network (KAN) is proposed to approximate potentially irregular functions in high dimensions. We provide error bounds for this approximation, assuming that the Kolmogorov-Arnold expansion functions are sufficiently smooth. When the function is only continuous, we also provide universal approximation theorems. We show that it outperforms multilayer perceptrons in terms of accuracy and convergence speed. We also compare it with several proposed KAN networks: it outperforms all networks for irregular functions and achieves similar accuracy to the original spline-based KAN network for smooth functions. Finally, we compare some of the KAN networks in optimizing a French hydraulic valley.

P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization

TL;DR

This paper addresses the challenge of high-dimensional function approximation, especially for irregular functions, by introducing P1-KAN, a layered Kolmogorov-Arnold network using 1D finite-element-like bases. It provides rigorous convergence bounds under Lipschitz outer functions and universal approximation theorems, including both adaptation-enabled and non-adaptation variants, and demonstrates empirical strength against MLPs and other KANs on synthetic functions. The method is further validated through a hydraulic valley optimization case, where P1-KAN with adapting lattices achieves faster convergence and robust performance, approaching or surpassing spline-based KAN in irregular regimes. Overall, P1-KAN offers a theoretically grounded, practically effective framework for high-dimensional, potentially irregular function approximation and stochastic optimization in engineering applications.

Abstract

A new Kolmogorov-Arnold network (KAN) is proposed to approximate potentially irregular functions in high dimensions. We provide error bounds for this approximation, assuming that the Kolmogorov-Arnold expansion functions are sufficiently smooth. When the function is only continuous, we also provide universal approximation theorems. We show that it outperforms multilayer perceptrons in terms of accuracy and convergence speed. We also compare it with several proposed KAN networks: it outperforms all networks for irregular functions and achieves similar accuracy to the original spline-based KAN network for smooth functions. Finally, we compare some of the KAN networks in optimizing a French hydraulic valley.
Paper Structure (19 sections, 4 theorems, 32 equations, 36 figures, 4 tables)

This paper contains 19 sections, 4 theorems, 32 equations, 36 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The P1-KAN Networks
  3. The P1-KAN Layers
  4. The P1-KAN Layer with a Regular Lattice
  5. The P1-KAN Layer with an Adapting Lattice
  6. Importance of Adaptation
  7. The Global P1-KAN Network
  8. Convergence theorems
  9. Supposing regularity of the outer functions
  10. Universal approximation theorems
  11. Numerical Results for Function Approximation
  12. Results for the A Function
  13. Results for the B Function
  14. Application to Hydraulic Valley Optimization
  15. The Flow Equation During a Week
  16. ...and 4 more sections

Key Result

Proposition 2.1

Supposing that the $(\psi_i)_{i=1, \ldots, 2n+1}$ and the $(\Phi_{i,j})_{i=1,\ldots,2n+1 , j=1, \ldots,n}$ in the Kolmogorov-Arnold expansion eq:kom are $K$-Lipschitz. For $h>0$ given, let us define for $i=1, \ldots 2n+1$, then there exists a constant $C$ such that

Figures (36)

  • Figure 1: Uniform $P_1$ basis functions on $[0,1]$ with $P=5$.
  • Figure 3: Function A in 2D.
  • Figure 4: Function B in 2D.
  • Figure 9: Structure of the valley
  • Figure 10: Convergence of the expected gain in Euros (NbL: number of layers, NbN: number of neurons)
  • ...and 31 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2