  We present a neural flow wavefunction, Gauge-Fermion FlowNet, and use it to
simulate 2+1D lattice compact quantum electrodynamics with finite density
dynamical fermions. The gauge field is represented by a neural network which
parameterizes a discretized flow-based transformation of the amplitude while
the fermionic sign structure is represented by a neural net backflow. This
approach directly represents the $U(1)$ degree of freedom without any
truncation, obeys Guass's law by construction, samples autoregressively
avoiding any equilibration time, and variationally simulates Gauge-Fermion
systems with sign problems accurately. In this model, we investigate
confinement and string breaking phenomena in different fermion density and
hopping regimes. We study the phase transition from the charge crystal phase to
the vacuum phase at zero density, and observe the phase seperation and the net
charge penetration blocking effect under magnetic interaction at finite
density. In addition, we investigate a magnetic phase transition due to the
competition effect between the kinetic energy of fermions and the magnetic
energy of the gauge field. With our method, we further note potential
differences on the order of the phase transitions between a continuous $U(1)$
system and one with finite truncation. Our state-of-the-art neural network
approach opens up new possibilities to study different gauge theories coupled
to dynamical matter in higher dimensions.
