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We consider an optimal control problem where the average welfare of weaklyinteracting agents is of interest. We examine the mean-field control problem asthe fluid approximation of the N-agent control problem with the setup offinite-state space, continuous-time, and finite-horizon. The value function ofthe mean-field control problem is characterized as the unique viscositysolution of a Hamilton-Jacobi-Bellman equation in the simplex. We apply the DGMto estimate the value function and the evolution of the distribution. We alsoprove the numerical solution approximated by a neural network converges to theanalytical solution.

Deep Galerkin Method for Mean Field Control Problem