  Supernovae Ia (SNe) can provide a unique window on the large scale structure
(LSS) of the Universe at redshifts where few other observations are available,
by solving the inversion problem (IP) consisting in reconstructing the LSS from
its effects on the observed luminosity distance. So far the IP was solved
assuming some restrictions about space-time, such as spherical symmetry for
example, while we obtain for the first time solutions of the IP problem for
arbitrary space-time geometries using deep learning. The method is based on the
use of convolutional neural networks (CNN) trained on simulated data. The
training data set is obtained by first generating random density and velocity
fields, and then computing their effects on the luminosity distance. The CNN,
based on an appriately modified version of U-Net to account for the
tridimensionality of the data, is then trained to reconstruct the density and
velocity fields from the luminosity distance.
  We find that the velocity field inversion is more accurate than the density
field, because the effects of the velocity on the luminosity distance only
depend on the source velocity, while in the case of the density it is an
integrated effect along the line of sight, giving rise to more degeneracy in
the solution of the IP. Improved versions of these neural networks, modified to
accommodate the non uniform distribution of the SNe, can be applied to
observational data to reconstruct the large scale structure of the Universe at
redshifts at which few other observations are available.
