  We discuss an implementation of a deep learning framework to gain insight
into dark matter (DM) structure formation. We investigate the impact of initial
velocity and density field information on the construction of halo mass
function (HMF) through cosmological $N$-body simulations. We train a
Convolutional Neural Network (CNN) on the initial snapshot of a DM-only
simulation to predict the HMF that individual particles fall into at $z=0$, in
the halo mass range of $10.5< \log(M / M_{\odot})<14$. Our results show a
negligible improvement from including the velocity in addition to the density
information when considering simulations based on ($\Lambda$CDM) with the
amplitude of initial scalar perturbations $A_s = 2\times10^{-9}$. To
investigate the effect of the initial velocity field in constructing the halo
mass function, we increase the initial power spectrum such that we see the
effect of velocities in larger halos visible to the resolution of our
simulations. The CNN model trained on the simulation snapshots with large $A_s$
shows a considerable improvement in the HMF prediction when adding the velocity
field information. Eventually, for the simulation with $A_s = 8 \times
10^{-8}$, the model trained with only density information shows at least $80\%$
increase in the mean squared error relative to the model with both velocity and
density information, which indicates the failure of the density-only model to
predict the halo mass function in this case. Our work shows the
interpretability and ability of CNNs to read higher-order information from
simple images, making them an excellent tool for cosmological studies.
