$L^p$ approximation results for infinite dimensional Neural Networks

Abstract

Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of $L^p(\mu )$, where $1\le p<\infty$ and $\mu$ is a Radon probability measure on a suitable infinite dimensional topological space $\mathfrak{X}$. Namely, any function $f:\mathfrak{X}\to \mathbb{R}$ in $L^p(\mu )$ can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where $f=f(x)\in E$ and $E$ is a Banach space) is also considered and analogous results are obtained.