  We present herein a new approach based on the simultaneous application of the
deep learning and statistical physics methods to solve the combinatorial
optimization problems. The recent modern advanced techniques, such as an
artificial neural network, demonstrate their efficiency for solving various
physical tasks for the quantum many-body systems, which may be related directly
to the problems of combinatorial optimization. One of them is a classical
Maximum Cut (MaxCut) problem, which we ascribe here to the search of the ground
state of a quantum many-body system using an artificial neural network and deep
learning. We found that the exact solution received for a quantum system
corresponds to its counterpart in a classical MaxCut problem. As a proof, we
have realized our approach for two random graphs of different size, containing
60 vertices (885 edges) and 100 vertices (2475 edges), and achieved for them
the total performances p60~0.99 and $p100~0.97 of known maximal cuts,
respectively.
