  We design a neural network Ansatz for variationally finding the ground-state
wave function of the Homogeneous Electron Gas, a fundamental model in the
physics of extended systems of interacting fermions. We study the
spin-polarised and paramagnetic phases with 7, 14 and 19 electrons over a broad
range of densities from $r_s=1$ to $r_s=100$, obtaining similar or higher
accuracy compared to a state-of-the-art iterative backflow baseline even in the
challenging regime of very strong correlation. Our work extends previous
applications of neural network Ans\"{a}tze to molecular systems with methods
for handling periodic boundary conditions, and makes two notable changes to
improve performance: splitting the pairwise streams by spin alignment and
generating backflow coordinates for the orbitals from the network. We
illustrate the advantage of our high quality wave functions in computing the
reduced single particle density matrix. This contribution establishes neural
network models as flexible and high precision Ans\"{a}tze for periodic
electronic systems, an important step towards applications to crystalline
solids.
