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. Comment: 17 pages. Submitted to A&A in Feb. 2003, revised in Oct. 2003