  We introduce a systematically improvable family of variational wave functions
for the simulation of strongly correlated fermionic systems. This family
consists of Slater determinants in an augmented Hilbert space involving
"hidden" additional fermionic degrees of freedom. These determinants are
projected onto the physical Hilbert space through a constraint which is
optimized, together with the single-particle orbitals, using a neural network
parametrization. This construction draws inspiration from the success of hidden
particle representations but overcomes the limitations associated with the
mean-field treatment of the constraint often used in this context. Our
construction provides an extremely expressive family of wave functions, which
is proven to be universal. We apply this construction to the ground state
properties of the Hubbard model on the square lattice, achieving levels of
accuracy which are competitive with state-of-the-art variational methods.
