A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function

TL;DR

This paper demonstrates that a single neuron in a hidden layer with a specially constructed sigmoidal activation function can approximate any continuous univariate function on a finite interval, providing a constructive approach.

Contribution

It introduces a method to construct a smooth sigmoidal activation function enabling a single hidden neuron to approximate any continuous univariate function.

Findings

Constructed a smooth sigmoidal activation function for approximation

Developed an algorithm to compute the activation function values

Proved the approximation capability of a single neuron with this function

Abstract

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this paper, we consider constructive approximation on any finite interval of $\mathbb{R}$ by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function $\sigma$ providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of $\sigma$ at any reasonable point of the real axis.