  Projected variational wavefunctions such as the Gutzwiller, many-body
correlator and Jastrow ansatzes have provided crucial insight into the nature
of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two
or more spatial dimensions. However, these ansatzes have no obvious tractable
and systematic way of being improved. A promising alternative is to use
Neural-network quantum states (NQS) based on Restricted Boltzmann Machines
(RBMs). With binary visible and hidden units NQS have proven to be a highly
effective at describing quantum states of interacting spin-1/2 lattice systems.
The application of NQS to bosonic systems has so far been based on one-hot
encoding from machine learning where the multi-valued site occupation is
distributed across several binary-valued visible units of an RBM. Compared to
spin-1/2 systems one-hot encoding greatly increases the number of variational
parameters whilst also making their physical interpretation opaque. Here we
revisit the construction of NQS for bosonic systems by reformulating a one-hot
encoded RBM into a correlation operator applied to a reference state, analogous
to the structure of the projected variational ansatzes. In this form we then
propose a number of specialisations of the RBM motivated by the physics of the
BHM and the ability to capture exactly the projected variational ansatzes. We
analyse in detail the variational performance of these new RBM variants for a
10 x 10 BHM, using both a standard Bose condensate state and a pre-optimised
Jastrow + many-body correlator state as the reference state of the calculation.
Several of our new ansatzes give robust results as nearly good as one-hot
encoding across the regimes of the BHM, but at a substantially reduced cost.
Such specialised NQS are thus primed tackle bosonic lattice problems beyond the
accuracy of classic variational wavefunctions.
