# An Algorithm for Approximating Continuous Functions on Compact Subsets with a Neural Network with one Hidden Layer

## TL;DR

This paper presents an algorithm for training a single-hidden-layer neural network to approximate any continuous function on compact sets, addressing the practical challenge of parameter determination highlighted by Cybenko's theoretical results.

## Contribution

It provides a concrete algorithm for constructing such neural networks, bridging the gap between theoretical approximation capabilities and practical implementation.

## Key findings

- The algorithm can reconstruct continuous functions with arbitrary accuracy.
- It demonstrates the feasibility of training single-hidden-layer networks for function approximation.
- The approach offers a practical method for function approximation using neural networks.

## Abstract

George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function $f$ on the unit hypercube. The paper did not address how to find the weight/parameters of such a network, or if finding them would be computationally feasible. This paper outlines an algorithm for a neural network with exactly one hidden layer to reconstruct any continuous scalar or vector valued continuous function.