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The equation of neuron (
i,j
) is easily derived, for instance in the case of
binary neurons. At time
t
+1, one gets
N
N
∆
v
i,j
(
t
+1)=
w
ij,kl
y
k,l
(
t
)+
I
i,j
.
k
=1
k
=
i
l
=1
l
=
j
In the case of analog neurons, the motion equations are given by
⎛
⎞
N
N
d
v
i,j
d
t
⎝
−
⎠
=
µ
i,j
α
i,j
v
i,j
+
w
ij,kl
y
k,l
+
I
i,j
,
k
=1
k
=
i
l
=1
l
=
j
where
µ
i,j
and
α
i,j
are real positive numbers.
Step 4: Starting the Dynamics
From a valid random solution, the initial values of the neuron inputs and
outputs can be defined. Then, a random asynchronous dynamics, based on
the motion equations of the neurons, makes the network converge to a local
minimum of the energy, in which the inputs and the outputs of the neurons
do not evolve any longer. After convergence, reading out the values of the
neuron outputs provides the encoding of a possible solution to the problem.
The validity of the solution must nevertheless be checked. If the solution is not
valid (i.e., if one or more constraints are violated), the network can be started
again from another random initial state, and/or the weighting constants can
be adjusted, for instance by increasing the constant associated to the violated
constraint.
8.6.4.8 Limitations of Hopfield Neural Networks
Solving optimization problems with Hopfield neural networks raises some
problems.
The main di
culty stems from the fact that the dynamics drives often the
network into a local minimum of the energy, which is not necessarily close to
the optimum, or which does not correspond to a valid solution. That is due
to the fact that the constraints are combined with the cost function in the
network energy.
Moreover, the Hopfield neural network makes no difference between the
strict constraints and the preference constraints, except by different weights
in the energy function.
Finally, the values of the parameters that weight the different terms of the
network energy have an influence on the number of iterations to convergence.
Those di
culties have given rise to numerous investigations aiming at
overcoming those limitations.
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