  Neural networks with synaptic weights constructed according to the weighted
Hebb rule, a variant of the familiar Hebb rule, are studied in the presence of
noise(finite temperature), when the number of stored patterns is finite and in
the limit that the number of neurons $N\rightarrow
  \infty$. The fact that different patterns enter the synaptic rule with
different weights changes the configuration of the free energy surface. For a
general choice of weights not all of the patterns are stored as {\sl global}
minima of the free energy function. However, as for the case of the usual Hebb
rule, there exists a temperature range in which only the stored patterns are
minima of the free energy. In particular, in the presence of a single extra
pattern stored with an appropriate weight in the synaptic rule, the temperature
at which the spurious minima of the free energy are eliminated is significantly
lower than for a similar network without this extra pattern. The convergence
time of the network, together with the overlaps of the equilibria of the
network with the stored patterns, can thereby be improved considerably.
