  This paper develops a new deep learning algorithm to solve a class of
finite-horizon mean-field games. The proposed hybrid algorithm uses Markov
chain approximation method combined with a stochastic approximation-based
iterative deep learning algorithm. Under the framework of finite-horizon
mean-field games, the induced measure and Monte-Carlo algorithm are adopted to
establish the iterative mean-field interaction in Markov chain approximation
method and deep learning, respectively. The Markov chain approximation method
plays a key role in constructing the iterative algorithm and estimating an
initial value of a neural network, whereas stochastic approximation is used to
find accurate parameters in a bounded region. The convergence of the hybrid
algorithm is proved; two numerical examples are provided to illustrate the
results.
