  We describe the application of tools from statistical mechanics to analyse
the dynamics of various classes of supervised learning rules in perceptrons.
The character of this paper is mostly that of a cross between a biased
non-encyclopedic review and lecture notes: we try to present a coherent and
self-contained picture of the basics of this field, to explain the ideas and
tricks, to show how the predictions of the theory compare with (simulation)
experiments, and to bring together scattered results. Technical details are
given explicitly in an appendix. In order to avoid distraction we concentrate
the references in a final section. In addition this paper contains some new
results: (i) explicit solutions of the macroscopic equations that describe the
error evolution for on-line and batch learning rules, (ii) an analysis of the
dynamics of arbitrary macroscopic observables (for complete and incomplete
trainingsets), leading to a general Fokker-Planck equation, and (iii) the
macroscopic laws describing batch learning with complete training sets. We
close the paper with a preliminary expose of ongoing research on the dynamics
of learning for the case where the training set is incomplete (i.e. where the
number of examples scales linearly with the network size).
