Universal approximation theorem for neural networks with inputs from a topological vector space

TL;DR

This paper extends the universal approximation theorem to neural networks with inputs from topological vector spaces, enabling them to approximate a wider class of functions beyond traditional finite-dimensional inputs.

Contribution

It introduces a universal approximation theorem for TVS-FNNs, broadening the theoretical foundation for neural networks handling diverse input types.

Findings

TVS-FNNs can approximate any continuous function on topological vector spaces.

The theorem generalizes classical results to infinite-dimensional and structured input spaces.

Potential applications include processing sequences, matrices, and functions.

Abstract

We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more. We prove a universal approximation theorem for TVS-FNNs, which demonstrates their capacity to approximate any continuous function defined on this expanded input space.