Analysis of approaches to the universal approximation of a continuous function using Kolmogorov’s superposition

Abstract

Kolmogorov and Arnold proved that any real continuous bounded function of many variables can be represented as a superposition of functions of one variable and addition. In subsequent works by Neht-Nielsen, it was shown that such a specific type of superposition can be interpreted as a two-layer forward neural network. Such a superposition can also be used as a universal approximation of the function of many variables. In the work, one of the variants of numerical implementation of Kolmogorov’s superposition, proposed by Sprecher and modified by Koppen, was investigated. In addition, the functions of the first layer were considered as a generator of space-filling curves (Peano curves). Numerical experiments were conducted to study the accuracy of the approximation of the function of many variables for the numerical implementation of Kolmogorov’s superposition, proposed by Sprecher and modified by Koppen, for a simplified version of this superposition and for an approach using filling curves. A comparative analysis showed that the best results are obtained using space-filling curves.