Decoding a mean field game by the Cauchy data around its unknown stationary states

In recent years, mean field games (MFGs) have garnered considerable attention and emerged as a dynamic and actively researched field across various domains, including economics, social sciences, finance, and transportation. The inverse design and decoding of MFGs offer valuable means to extract information from observed data and gain insights into the intricate underlying dynamics and strategies of these complex physical systems. This paper presents a novel approach to the study of inverse problems in MFGs by analyzing the Cauchy data around their unknown stationary states. This study distinguishes itself from existing inverse problem investigations in three key significant aspects: First, we consider MFG problems in a highly general form. Second, we address the technical challenge of the probability measure constraint by utilizing Cauchy data in our inverse problem study. Third, we enhance existing high-order linearization methods by introducing a novel approach that involves conducting linearization around non-trivial stationary states of the MFG system, which are not a priori known. These contributions provide new insights and offer promising avenues for studying inverse problems for MFGs. By unraveling the hidden structure of MFGs, researchers and practitioners can make informed decisions, optimize system performance, and address real-world challenges more effectively. © 2025 The Author(s). Journal of the London Mathematical Society is copyright © London Mathematical Society.