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

TL;DR

This paper introduces a new Kolmogorov-Arnold network (KAN) that effectively approximates irregular functions in high dimensions, outperforming traditional neural networks and spline-based methods in accuracy and convergence.

Contribution

The paper proposes a novel KAN architecture with theoretical error bounds and universal approximation theorems, demonstrating superior performance for irregular and smooth functions.

Findings

KAN outperforms multilayer perceptrons in accuracy and convergence speed.

KAN outperforms existing networks for irregular functions.

KAN achieves similar accuracy to spline-based KAN for smooth functions.

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.