  In this paper, we propose several approaches to learn the optimal
population-dependent controls in order to solve mean field control problems
(MFC). Such policies enable us to solve MFC problems with forms of common
noises at a level of generality that was not covered by existing methods. We
analyze rigorously the theoretical convergence of the proposed approximation
algorithms. Of particular interest for its simplicity of implementation is the
$N$-particle approximation. The effectiveness and the flexibility of our
algorithms is supported by numerical experiments comparing several combinations
of distribution approximation techniques and neural network architectures. We
use three different benchmark problems from the literature: a systemic risk
model, a price impact model, and a crowd motion model. We first show that our
proposed algorithms converge to the correct solution in an explicitly solvable
MFC problem. Then, we show that population-dependent controls outperform
state-dependent controls. Along the way, we show that specific neural network
architectures can improve the learning further.
