  Variational wave functions have enabled exceptional scientific breakthroughs
related to the understanding of novel phases of matter. Examples include the
Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the
fractional quantum Hall effect through the Laughlin state, and Feynman's
variational understanding of large-scale quantum effects in liquid Helium. More
recently, Gutzwiller-projected wave functions, typically constructed from
fermionic degrees of freedom, have been employed to examine quantum spin models
in the presence of competing interactions, where exotic phases with no
spontaneous symmetry breaking and fractional excitations may exist. In this
work, we investigate the aforementioned fermionic wave functions supplemented
with neural networks, specifically with the so-called restricted Boltzmann
machine (RBM), to boost their accuracy and obtain reliable approximations to
the ground state of generic spin models. In particular, we apply our neural
augmented fermionic construction to the description of both magnetically
ordered and disordered phases of increasing complexity, including cases where
the ground state displays a non-trivial sign structure. Even though the RBM
state is by far more effective for N\'eel states endowed with a particularly
simple sign structure, it provides a significant improvement over the original
fermionic state in highly frustrated regimes where a complex sign structure is
anticipated, thus marking the path to an understanding of strongly-correlated
spin models on the lattice via neural Gutzwiller-projected variational wave
functions.
