Joint Inference of Trajectory and Obstacle in Mean-Field Games via Bilevel Optimization

Mean field game (MFG) is an expressive modeling framework for systems with a continuum of interacting agents. While many approaches exist for solving the forward MFG, few have studied its \textit{inverse} problem. In this work, we seek to recover optimal agent trajectories and the unseen spatial obstacle given partial observation on the former. To this end, we use a special type of generative models, normalizing flow, to represent the trajectories and propose a novel formulation of inverse MFG as a bilevel optimization (BLO) problem. We demonstrate the effectiveness of our approach across various MFG scenarios, including those involving multi-modal and disjoint obstacles, highlighting its robustness with respect to obstacle complexity and dimensionality. Alternatively, our formulation can be interpreted as regularizing maximum likelihood trajectory learning with MFG assumptions, which improves generalization performance especially with scarce training data. Impressively, our method also recovers the hidden obstacle with high fidelity in this low-data regime.