  We use a convolutional restricted Boltzmann machine (CRBM) neural network to
construct a variational wave function (WF) for the Hubbard model on a square
lattice and study it using the variational Monte Carlo (VMC) method. In the
wave function, the CRBM acts as a correlation factor to a mean-field BCS state.
The number of variational parameters in the WF does not grow automatically with
the lattice size and it is computationally much more efficient compared to
other neural network based WFs. We find that in the intermediate to strong
coupling regime of the model at half-filling, the wave function outperforms
even the highly accurate long range backflow-Jastrow correlated wave function.
Using the WF, we study the ground state of the half-filled model as a function
of onsite Coulomb repulsion $U$. We consider two cases for the
next-nearest-neighbor hopping parameter, e.g., $t'=0$ as well as a frustrated
model case with $t'\neq 0$. By examining several quantities, e.g., double
occupancy, charge gap, momentum distribution, and spin-spin correlations, we
find that the weekly correlated phase in both cases is paramagnetic metallic
(PM). As $U$ is increased, the system undergoes a first-order Mott transition
to an insulating state at a critical $U_c$, the value of which depends upon
$t'$. The Mott state in both cases is spin gapped with long range
antiferromagnetic (AF) order. Remarkably, the AF order emerges spontaneously
from the wave function which does not have any explicitly broken symmetry in
it. Apart from some quantitative differences in the results for the two values
of $t'$, we find some interesting qualitative differences in the way the Mott
transition takes place in the two cases.
