Information Technology Reference
In-Depth Information
where x
k
D Œx
1
;x
2
; ;x
n
is the k-th input vector of the neural network. The
activation function is usually a sigmoidal function h.x/ D
1
1Ce
x
while in the case
of radial basis functions networks it is a Gaussian [
78
]. Several learning algorithms
for neural networks have been studied. The objective of all these algorithms is to
find numerical values for the network's weights so as to minimize the mean square
error E
RMS
of Eq. (
12.4
). The algorithms are usually based on first and second order
gradient techniques. These algorithms belong to: (a) batch-mode learning, where to
perform parameters update the outputs of a large training set are accumulated and
the mean square error is calculated (back-propagation algorithm, Gauss-Newton
method, Levenberg-Marquardt method, etc.), (b) pattern-mode learning, in which
training examples are run in cycles and the parameters update is carried out each
time a new datum appears (Extended Kalman Filter algorithm).
12.3
Eigenstates of the Quantum Harmonic Oscillator
FNN with Hermite basis functions (see Fig.
12.1
b) show the particle-wave nature of
information, as described by Schrödinger's diffusion equation, i.e.
@ .x;t/
@t
D
„
2
@ .x;t/
@t
2
.x;t/ C V.x/ .x;t/) i„
D H .x;t/
(12.5)
i„
2m
r
where „ is Planck's constant, H is the Hamiltonian, i.e. the sum of the potential
V.x/ and of the Laplacian
2
2m
r
2
2m
@
2
„
2
„
D
@x
2
. The probability density function
2
gives the probability at time instant t the input x of the neural network
(quantum particle equivalent) to have a value between x and x C x. The general
solution of the quantum harmonic oscillator, i.e. of Eq. (
12.5
) with V.x/ being a
parabolic potential, is [
34
,
186
]:
j .x;t/j
k
.x;t/ D H
k
.x/e
x
2
=2
e
i.2kC1/t
k D 0;1;2;
(12.6)
where H
k
.x/ are the associated Hermite polynomials. In case of the quantum har-
monic oscillator, the spatial component X.x/ of the solution .x;t/ D X.x/T.t/ is
X
k
.x/ D H
k
.x/e
x
2
=2
k D 0;1;2;
(12.7)
Eq. (
12.7
) satisfies the boundary condition lim
x!˙1
X.x/ D 0. From the above
it can be noticed that the Hermite basis functions H
k
.x/e
x
2
are the eigenstates of
the quantum harmonic oscillator (see Fig.
12.2
). The general relation for the Hermite
polynomials is
H
k
.x/ D .1/
k
e
x
2
d
.k/
dx
.k/
e
x
2
(12.8)
