  Deep neural networks have been extremely successful as highly accurate wave
function ans\"atze for variational Monte Carlo calculations of molecular ground
states. We present an extension of one such ansatz, FermiNet, to calculations
of the ground states of periodic Hamiltonians, and study the homogeneous
electron gas. FermiNet calculations of the ground-state energies of small
electron gas systems are in excellent agreement with previous initiator full
configuration interaction quantum Monte Carlo and diffusion Monte Carlo
calculations. We investigate the spin-polarized homogeneous electron gas and
demonstrate that the same neural network architecture is capable of accurately
representing both the delocalized Fermi liquid state and the localized Wigner
crystal state. The network is given no \emph{a priori} knowledge that a phase
transition exists, but converges on the translationally invariant ground state
at high density and spontaneously breaks the symmetry to produce the
crystalline ground state at low density.
