  Obtaining an accurate ground state wave function is one of the great
challenges in the quantum many-body problem. In this paper, we propose a new
class of wave functions, neural network backflow (NNB). The backflow approach,
pioneered originally by Feynman, adds correlation to a mean-field ground state
by transforming the single-particle orbitals in a configuration-dependent way.
NNB uses a feed-forward neural network to find the optimal transformation. NNB
directly dresses a mean-field state, can be systematically improved and
directly alters the sign structure of the wave-function. It generalizes the
standard backflow which we show how to explicitly represent as a NNB. We
benchmark the NNB on a Hubbard model at intermediate doping finding that it
significantly decreases the relative error, restores the symmetry of both
observables and single-particle orbitals, and decreases the double-occupancy
density. Finally, we illustrate interesting patterns in the weights and bias of
the optimized neural network.
