  We introduce a method to efficiently study the dynamical properties of
many-body localized systems in the regime of strong disorder and weak
interactions. Our method reproduces qualitatively and quantitatively the
real-time evolution with a polynomial effort in system size and independent of
the desired time scales. We use our method to study quantum information
propagation, correlation functions, and temporal fluctuations in one- and
two-dimensional MBL systems. Moreover, we outline strategies for a further
systematic improvement of the accuracy and we point out relations of our method
to recent attempts to simulate the real-time dynamics of quantum many-body
systems in classical or artificial neural networks.
