  In this paper, we introduce a bilevel optimization framework for addressing
inverse mean-field games, alongside an exploration of numerical methods
tailored for this bilevel problem. The primary benefit of our bilevel
formulation lies in maintaining the convexity of the objective function and the
linearity of constraints in the forward problem. Our paper focuses on inverse
mean-field games characterized by unknown obstacles and metrics. We show
numerical stability for these two types of inverse problems. More importantly,
we, for the first time, establish the identifiability of the inverse mean-field
game with unknown obstacles via the solution of the resultant bilevel problem.
The bilevel approach enables us to employ an alternating gradient-based
optimization algorithm with a provable convergence guarantee. To validate the
effectiveness of our methods in solving the inverse problems, we have designed
comprehensive numerical experiments, providing empirical evidence of its
efficacy.
