  We consider an optimal control problem where the average welfare of weakly
interacting agents is of interest. We examine the mean-field control problem as
the fluid approximation of the N-agent control problem with the setup of
finite-state space, continuous-time, and finite-horizon. The value function of
the mean-field control problem is characterized as the unique viscosity
solution of a Hamilton-Jacobi-Bellman equation in the simplex. We apply the DGM
to estimate the value function and the evolution of the distribution. We also
prove the numerical solution approximated by a neural network converges to the
analytical solution.
