NECOLA: Towards a Universal Field-level Cosmological Emulator
  We train convolutional neural networks to correct the output of fast and
approximate N-body simulations at the field level. Our model, Neural Enhanced
COLA --NECOLA--, takes as input a snapshot generated by the computationally
efficient COLA code and corrects the positions of the cold dark matter
particles to match the results of full N-body Quijote simulations. We quantify
the accuracy of the network using several summary statistics, and find that
NECOLA can reproduce the results of the full N-body simulations with
sub-percent accuracy down to $k\simeq1~h{\rm Mpc}^{-1}$. Furthermore, the
model, that was trained on simulations with a fixed value of the cosmological
parameters, is also able to correct the output of COLA simulations with
different values of $\Omega_{\rm m}$, $\Omega_{\rm b}$, $h$, $n_s$, $\sigma_8$,
$w$, and $M_\nu$ with very high accuracy: the power spectrum and the
cross-correlation coefficients are within $\simeq1\%$ down to $k=1~h{\rm
Mpc}^{-1}$. Our results indicate that the correction to the power spectrum from
fast/approximate simulations or field-level perturbation theory is rather
universal. Our model represents a first step towards the development of a fast
field-level emulator to sample not only primordial mode amplitudes and phases,
but also the parameter space defined by the values of the cosmological
parameters.
