Information Technology Reference
In-Depth Information
FNN that use Hermite basis functions demonstrate the particle-wave nature of
information as described by Schrödinger's diffusion equation. It is considered that
the input variable x of the neural network can be described not only by crisp
values (particle equivalent) but also by the normal modes of a wave function (wave
equivalent). Since the basis functions are the eigenstates of the quantum harmonic
oscillator, this means that the output of the proposed neural network is the weighted
sum .x/ D
P
kD1
w
k
k
.x/ where j .x/j
2
is the probability that the input of the
neural network (quantum particle equivalent) is found between x and x C x.The
weight
w
k
provides a measure of the probability to find the input on the neural
network in the region of the patterns space associated with the eigenfunction
k
.x/.
Issues related to the uncertainty principle have been studied for the case of the
QHO-based neural networks. The uncertainty principle was presented first through
the Balian-Low theorem and next an expression of the uncertainty principle for
Hermite basis functions has been given. It was shown that the Gauss-Hermite basis
functions as well as their Fourier transforms cannot be uniformly concentrated in
the time-frequency plane.
In the simulation tests, FNN with Hermite basis functions were evaluated in
image compression problems. In this task the performance of FNN with Hermite
basis functions was satisfactory and comparable to the compression succeeded by
other neural structures, such as MLP or RBF networks. The fact that the basis
functions of the QHO-based neural networks contain various frequencies makes
them suitable for multi-resolution analysis.
Additionally, the use of FNN comprising as basis functions the eigenstates of
the QHO was tested in the problem of fault diagnosis. Faults could be diagnosed
by monitoring changes in the spectral content of a system and these in turn were
associated with the values of the weights of the neural network.
