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arXiv · 2608.08230

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

Luca Galimberti
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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.

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Luca Galimberti. 2026-08-08. $L^p$ approximation results for infinite dimensional Neural Networks. https://arxiv.org/abs/2608.08230

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