  In this work we study the treatment of asymmetric open quantum systems with
neural networks based on the restricted Boltzmann machine. In particular, we
are interested in the non-equilibrium steady state current in the
boundary-driven (anisotropic) Heisenberg spin chain. We address previously
published difficulties in treating asymmetric dissipative systems with
neural-network quantum states and Monte-Carlo sampling and present an
optimization method and a sampling technique that can be used to obtain
high-fidelity steady state approximations of such systems. We point out some
inherent symmetries of the Lindblad operator under consideration and exploit
them during sampling. We show that local observables are not always a good
indicator of the quality of the approximation and finally present results for
the spin current that are in agreement with known results of simple open
Heisenberg chains.
