  In this paper, we are concerned with the inverse problem of determining
anomalies in the state space associated with the stationary mean field game
(MFG) system. We establish novel unique identifiability results for the
intrinsic structure of these anomalies in mean field games systems, including
their topological structure and parameter configurations, in several general
scenarios of practical interest, including traffic flow, market economics and
epidemics. To the best of our knowledge, this is the first work that considers
anomalies in the state space for the nonlinear coupled MFG system.
