In experimentally realistic situations, quantum systems are never perfectly
isolated and the coupling to their environment needs to be taken into account.
Often, the effect of the environment can be well approximated by a Markovian
master equation. However, solving this master equation for quantum many-body
systems, becomes exceedingly hard due to the high dimension of the Hilbert
space. Here we present an approach to the effective simulation of the dynamics
of open quantum many-body systems based on machine learning techniques. We
represent the mixed many-body quantum states with neural networks in the form
of restricted Boltzmann machines and derive a variational Monte-Carlo algorithm
for their time evolution and stationary states. We document the accuracy of the
approach with numerical examples for a dissipative spin lattice system
