  We present a technique for the estimation of photometric redshifts based on
feed-forward neural networks. The Multilayer Perceptron (MLP) Artificial Neural
Network is used to predict photometric redshifts in the HDF-S from an ultra
deep multicolor catalog. Various possible approaches for the training of the
neural network are explored, including the deepest and most complete
spectroscopic redshift catalog currently available (the Hubble Deep Field North
dataset) and models of the spectral energy distribution of galaxies available
in the literature. The MLP can be trained on observed data, theoretical data
and mixed samples. The prediction of the method is tested on the spectroscopic
sample in the HDF-S (44 galaxies). Over the entire redshift range, $0.1<z<3.5$,
the agreement between the photometric and spectroscopic redshifts in the HDF-S
is good: the training on mixed data produces sigma_z(test) ~ 0.11, showing that
model libraries together with observed data provide a sufficiently complete
description of the galaxy population. The neural system capability is also
tested in a low redshift regime, 0<z<0.4, using the Sloan Digital Sky Survey
Data Release One (DR1) spectroscopic sample. The resulting accuracy on 88108
galaxies is sigma_z(test) ~ 0.022. Inputs other than galaxy colors - such as
morphology, angular size and surface brightness - may be easily incorporated in
the neural network technique. An important feature, in view of the application
of the technique to large databases, is the computational speed: in the
evaluation phase, redshifts of 10^5 galaxies are estimated in few seconds.
