Approximation of functions by neural networks

TL;DR

This paper demonstrates that neural networks can approximate any measurable function on a hypercube with arbitrary precision using a bounded number of neurons, independent of the function or dimension.

Contribution

It proves that neural networks can approximate any measurable function on the hypercube with a fixed neuron count depending only on the desired accuracy.

Findings

Neural networks can approximate any measurable function on the hypercube.

The number of neurons needed depends only on the approximation precision.

Approximation is independent of the function's complexity or dimension.

Abstract

We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function $f \colon W_n \to [-1,1]$ up to a prescribed precision $\varepsilon>0$ by a bounded number of neurons, depending only on $\varepsilon$ and not on the function $f$ or $n \in \mathbb N$.