Neural Networks in Fr\'echet spaces

TL;DR

This paper introduces neural networks in infinite dimensional Fréchet spaces, proving their universal approximation capabilities for continuous functions and demonstrating their practical projection to finite dimensions for functional data prediction.

Contribution

It develops a new class of neural networks in infinite dimensional spaces with proven universal approximation properties, extending classical results to a broader functional setting.

Findings

Neural networks in Fréchet spaces can approximate continuous functions universally.

Networks can be projected onto finite-dimensional subspaces with arbitrary accuracy.

Applicable for prediction tasks involving functional data.

Abstract

We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\'echet space $\X$ into a Banach space $\Y$. The approximation results are generalising the well known universal approximation theorem for continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$, where approximation is done with (multilayer) neural networks [15, 25, 18, 29]. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Several examples are given of such activation functions. We show furthermore that our neural networks on infinite dimensional spaces can be projected down to finite dimensional subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data.