  Producing thousands of simulations of the dark matter distribution in the
Universe with increasing precision is a challenging but critical task to
facilitate the exploitation of current and forthcoming cosmological surveys.
Many inexpensive substitutes to full $N$-body simulations have been proposed,
even though they often fail to reproduce the statistics of the smaller,
non-linear scales. Among these alternatives, a common approximation is
represented by the lognormal distribution, which comes with its own limitations
as well, while being extremely fast to compute even for high-resolution density
fields. In this work, we train a generative deep learning model, mainly made of
convolutional layers, to transform projected lognormal dark matter density
fields to more realistic dark matter maps, as obtained from full $N$-body
simulations. We detail the procedure that we follow to generate highly
correlated pairs of lognormal and simulated maps, which we use as our training
data, exploiting the information of the Fourier phases. We demonstrate the
performance of our model comparing various statistical tests with different
field resolutions, redshifts and cosmological parameters, proving its
robustness and explaining its current limitations. When evaluated on 100 test
maps, the augmented lognormal random fields reproduce the power spectrum up to
wavenumbers of $1 \ h \ \rm{Mpc}^{-1}$, and the bispectrum within 10%, and
always within the error bars, of the fiducial target simulations. Finally, we
describe how we plan to integrate our proposed model with existing tools to
yield more accurate spherical random fields for weak lensing analysis.
