“…It has wide applications in various fields, such as economics [1,23], engineering [18,50] as well as machine learning and reinforcement learning [16,49,51,19]. Recently, mean field control problems have been extended into chemistry, biology , pandemic control, traffic flow models and social dynamics [35,36,37,38,22]. An important task in mean-field games is to study the flow of all the agents in the state space and to understand behavior of mean-field Nash equilibrium.…”