The minimal width of universal $p$-adic ReLU neural networks

TL;DR

This paper establishes the smallest width required for $p$-adic ReLU neural networks to universally approximate continuous functions over compact sets, extending neural network theory into the $p$-adic number domain.

Contribution

It determines the minimal width of $p$-adic neural networks with universal approximation capabilities, a novel extension of neural network theory to $p$-adic analysis.

Findings

Identifies the minimal width for $p$-adic neural networks to approximate functions.

Provides conditions under which $p$-adic ReLU networks are universal.

Extends neural network approximation theory into the $p$-adic setting.

Abstract

We determine the minimal width of $p$-adic neural networks with the universal approximation property for continuous $\mathbb Q_p$-valued functions on compact open subsets with respect to the $L_q$ norms and the $C_1$ norm, where the activation function is a natural $p$-adic analogue of the ReLU function.