p-adic Cellular Neural Networks

TL;DR

This paper introduces p-adic cellular neural networks, a mathematical generalization of classical CNNs with hierarchical infinite structures modeled by integro-differential equations, and explores their properties and numerical solutions.

Contribution

The paper presents the formulation of p-adic CNNs as limits of hierarchical discrete CNNs, extending classical models with infinite hierarchical and hidden layers.

Findings

p-adic CNNs can be approximated by hierarchical discrete CNNs

The networks are modeled by integro-differential equations with p-adic variables

Numerical methods for solving these equations are developed

Abstract

In this article we introduce the p-adic cellular neural networks which are mathematical generalizations of the classical cellular neural networks (CNNs) introduced by Chua and Yang. The new networks have infinitely many cells which are organized hierarchically in rooted trees, and also they have infinitely many hidden layers. Intuitively, the p-adic CNNs occur as limits of large hierarchical discrete CNNs. More precisely, the new networks can be very well approximated by hierarchical discrete CNNs. Mathematically speaking, each of the new networks is modeled by one integro-differential equation depending on several p-adic spatial variables and the time. We study the Cauchy problem associated to these integro-differential equations and also provide numerical methods for solving them.