A Mean Field Game Inverse Problem

Mean-field games arise in various fields, including economics, engineering, and machine learning. They study strategic decision-making in large populations where the individuals interact via specific mean-field quantities. The games' ground metrics and running costs are of essential importance but are often unknown or only partially known. This paper proposes mean-field game inverse-problem models to reconstruct the ground metrics and interaction kernels in the running costs. The observations are the macro motions, to be specific, the density distribution and the velocity field of the agents. They can be corrupted by noise to some extent. Our models are PDE constrained optimization problems, solvable by first-order primal-dual methods. We apply the Bregman iteration method to improve the parameter reconstruction. We numerically demonstrate that our model is both efficient and robust to the noise.