  We propose two algorithms for the solution of the optimal control of ergodic
McKean-Vlasov dynamics. Both algorithms are based on approximations of the
theoretical solutions by neural networks, the latter being characterized by
their architecture and a set of parameters. This allows the use of modern
machine learning tools, and efficient implementations of stochastic gradient
descent.The first algorithm is based on the idiosyncrasies of the ergodic
optimal control problem. We provide a mathematical proof of the convergence of
the approximation scheme, and we analyze rigorously the approximation by
controlling the different sources of error. The second method is an adaptation
of the deep Galerkin method to the system of partial differential equations
issued from the optimality condition. We demonstrate the efficiency of these
algorithms on several numerical examples, some of them being chosen to show
that our algorithms succeed where existing ones failed. We also argue that both
methods can easily be applied to problems in dimensions larger than what can be
found in the existing literature. Finally, we illustrate the fact that,
although the first algorithm is specifically designed for mean field control
problems, the second one is more general and can also be applied to the partial
differential equation systems arising in the theory of mean field games.
