Deep learning reconstruction of the large scale structure of the
  Universe from luminosity distance observations
  Supernovae Ia (SN) are among the brightest objects we can observe and can
provide a unique window on the large scale structure of the Universe at
redshifts where other observations are not available. The photons emitted by
SNe are in fact affected by the density field between the source and the
observer, and from the observed luminosity distance it is possible to solve the
inversion problem (IP), i.e. to reconstruct the density field which produced
those effects.
  So far the IP was only solved assuming some restrictions about the geometry
of the problem, such as spherical symmetry for example, and the approach was
based on solving complicated systems of differential equations which required
smooth function as inputs, while observational data is not smooth, due to its
discrete nature. In order to overcome these limitations we develop for the
first time an inversion method which is not assuming any symmetry, and can be
applied directly to observational data, without the need of any data smoothing
procedure.
  The method is based on the use of convolutional neural networks (CNN) trained
on simulated data, and it shows quite accurate results. The training data set
is obtained by first generating random density and velocity profiles, and then
computing their effects on the luminosity distance. The CNN is then trained to
reconstruct the density field from the luminosity distance. The CNN is a
modified version of U-Net to account for the tridimensionality of the data, and
can reconstruct the density and velocity fields with a good level of accuracy.
  The use of neural networks to analyze observational data from future SNe
catalogues will allow to reconstruct the large scale structure of the Universe
to an unprecedented level of accuracy, at a redshift at which few other
observations are available.
