  We propose and study several inverse problems for the mean field games (MFG)
system in a bounded domain. Our focus is on simultaneously recovering the
running cost and the Hamiltonian within the MFG system by the associated
boundary observation. There are several technical novelties that make the study
intriguing and challenging. First, the MFG system couples two nonlinear
parabolic PDEs with one moving forward and the other one moving backward in
time. Second, there is a probability density constraint on the population
distribution of the agents. Third, the simultaneous recovery of two coupling
factors within the MFG system is technically far from being trivial. Fourth, we
consider both cases that the running cost depends on the population density
locally and non-locally, and the two cases present different technical
challenges for the inverse problem study. We develop two mathematical
strategies that can ensure the probability constraint as well as effectively
tackle the inverse problems, which are respectively termed as high-order
variation and successive linearisation. In particular, the high-order variation
method is new to the literature, which demonstrates a novel concept to examine
the inverse problems by non-negative inputs only. We believe the methods
developed can find applications to inverse problems in other contexts.
