Information Technology Reference
In-Depth Information
Chapter 12
Neural Networks Based on the Eigenstates
of the Quantum Harmonic Oscillator
Abstract
The chapter introduces feed-forward neural networks where the hidden
units employ orthogonal Hermite polynomials for their activation functions. These
neural networks have some interesting properties: (a) the basis functions are
invariant under the Fourier transform, subject only to a change of scale, (b) the
basis functions are the eigenstates of the quantum harmonic oscillator (QHO),
and stem from the solution of Schrödinger's harmonic equation. The proposed
neural networks have performance equivalent to wavelet networks and belong to
the general category of nonparametric estimators. They can be used for function
approximation, system modelling, image processing and fault diagnosis. These
neural networks demonstrate the particle-wave nature of information and give the
incentive to analyze significant issues related to this dualism, such as the principle
of uncertainty and Balian-Low's theorem.
12.1
Overview
Feed-forward neural networks (FNN) are the most popular artificial neural struc-
tures due to their structural flexibility, good representational capabilities, and
availability of a large number of training algorithms. The hidden units in an FNN
usually have the same activation functions and are often selected as sigmoidal
functions or Gaussians. This chapter presents FNN that use orthogonal Hermite
polynomials as basis functions. The proposed neural networks have some interesting
properties: (a) the basis functions are invariant under the Fourier transform, subject
only to a change of scale (b) the basis functions are the eigenstates of the quantum
harmonic oscillator (QHO), and stem from the solution of Schrödinger's diffusion
equation. The proposed neural networks belong to the general category of nonpara-
metric estimators and are suitable for function approximation, image processing,
and system fault diagnosis. Two-dimensional QHO-based neural networks can be
also constructed by taking products of the one-dimensional basis functions.
