Inverse Problems for Mean Field Games
Hongyu Liu, Catharine W. K. Lo, Shen Zhang
TL;DR
This work surveys and develops inverse-problem theory for Mean Field Games (MFGs), focusing on identifiability, stability, and reconstruction of unknown running costs $F$, terminal costs $G$, and Hamiltonians $H$ from boundary and Cauchy data. It combines forward–backward MFG analysis with high-order linearization, CGO-type constructions, and Runge approximation to achieve unique recoverability in periodic and general domains, including multi-population models. Key methods include time-space boundary measurements, local well-posedness, and probing-mode construction to excite specific PDE components, enabling sequential recovery of coefficients order by order. The results extend to static and time-dependent problems, and to internal topological anomalies, providing a versatile toolkit for calibrating and validating MFG models against real-world data with theoretical guarantees.
Abstract
In this book, we present a curated collection of existing results on inverse problems for Mean Field Games (MFGs), a cutting-edge and rapidly evolving field of research. Our aim is to provide fresh insights, novel perspectives, and a comprehensive foundation for future investigations into this fascinating area. MFGs, a class of differential games involving a continuum of non-atomic players, offer a powerful framework for analyzing the collective behavior of large populations of symmetric agents as the number of agents approaches infinity. This framework has proven to be an invaluable tool for quantitatively modeling the macroscopic dynamics of agents striving to minimize specific costs in complex systems, such as crowd dynamics, financial markets, traffic flows, and social networks. The study of MFGs has traditionally focused on forward problems, where the goal is to determine the equilibrium behavior of agents given a set of model parameters, such as cost functions, interaction mechanisms, and initial conditions. However, the inverse problems for MFGs -- which seek to infer these underlying parameters from observed data -- have received comparatively less attention in the literature. This book seeks to address this gap by delving into the fundamental aspects of MFG inverse problems, with a particular emphasis on issues of unique identifiability, stability, and reconstruction of unknown parameters. These problems are not only mathematically challenging but also of immense practical significance, as they enable the calibration and validation of MFG models using real-world data. This book is intended to serve as a valuable resource for researchers interested in the theory and applications of MFGs, particularly in inverse problems. Through this book, we hope to inspire further exploration and innovation in this dynamic and interdisciplinary area of study.