Determining internal topological structures and running cost of mean field games with partial boundary measurement

This paper investigates the simultaneous reconstruction of the running cost function and the internal topological structure within the mean-field games (MFG) system utilizing partial boundary data. The inverse problem is notably challenging due to factors such as nonlinear coupling, the necessity for multi-parameter reconstruction, constraints on probability measures, and the limited availability of measurement information. To address these challenges, we propose an innovative approach grounded in a higher-order linearization method. This method is tailored for inverse problems in MFG systems that involve Dirichlet and Neumann boundary conditions. Initially, we present unique reconstruction results for the cost function and internal topological structure of the MFG system under various homogeneous boundary conditions. Subsequently, we extend these results to accommodate inhomogeneous boundary conditions. These findings greatly enhance our understanding of simultaneous reconstruction in complex MFG systems.