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Pattern Formation in Quantum Hierarchical Cellular Neural Networks

W. A. Zúñiga-Galindo, B. A. Zambrano-Luna, Chayapuntika Indoung

Abstract

We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of $p$-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired in the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new $p$-adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new $p$ -adic Schrödinger equations.

Pattern Formation in Quantum Hierarchical Cellular Neural Networks

Abstract

We present a new class of quantum neural networks (QNNs) whose states are solutions of -adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of -adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired in the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new -adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new -adic Schrödinger equations.

Paper Structure

This paper contains 18 sections, 3 theorems, 73 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Basic concepts of p-adic analysis
  3. $p$-Adic numbers
  4. $p-$Adic balls and spheres
  5. Test functions
  6. Lebesgue spaces
  7. CNNs and QNNs
  8. Discretization of $p$-adic QNNs
  9. $p$-adic QNNs on graphs
  10. Numerical simulations
  11. Numerical simulation 1
  12. Numerical simulation 2
  13. Numerical simulation 3
  14. Numerical simulations 4
  15. Numerical simulations 5
  16. ...and 3 more sections

Key Result

Lemma 8.1

If $\phi\in L^{\infty}\left( \mathbb{R}\right)$, then there is a constant $C(F)$ independent of $t$ such that $\Vert F \Vert\leq C(F)$. Then $\int_0^T \Vert F(\Psi(x,t)) \Vert_2^2 dt\leq TC(F)$ for all $\Psi(x,t)\in \mathcal{C}([0,T], L^2(\mathbb Z_p))$.

Figures (15)

  • Figure 1:
  • Figure 2: Results of the numerical simulation 1
  • Figure 3: Numerical simulation 2
  • Figure 4: Pulse use in Simulation 2
  • Figure 5: Results for Simulation 3.1
  • ...and 10 more figures

Theorems & Definitions (6)

  • Lemma 8.1
  • proof
  • Proposition 8.1
  • proof
  • Theorem 8.1
  • proof