A Mathematical Theory of Inverse Problems for Mean Field Games

Mean Field Games (MFGs) are non-atomic differential games which study the limiting behaviours of large systems with symmetric players/agents. The MFG theory offers power tools towards the quantitative modelling and understanding of the macroscopic Nash equilibrium behaviours. It has called great attentions and grows rapidly in recent years due to its practical significance in many cutting-edge applications. The study of MFGs from the perspective of inverse problems is an emerging new topic and still in its beginning stage. The aim of this project is to conduct a systematic and comprehensive mathematical study on MFG inverse problems. We shall mainly study the MFG inverse problems from an analysis of PDE (Partial Differential Equation) perspective associated with the MFG system. The MFG system is a coupled system of nonlinear PDEs with peculiar structures. Associated with the MFG system, we propose and formulate a variety of inverse problems. Based on the type of measurement/observation data used for the inverse problems, we consider both single-event or multiple-event measurements, and both internal and boundary measurements, as well as the cases with mixed-type measurements. Based on the type of unknowns to be recovered or determined, we consider both coefficient inverse problems (CIPs) and geometrical inverse problems (GIPs). The CIPs are concerned with determining unknowns given in terms of certain function parameters within the MFG system, whereas the GIPs are concerned with identifying certain geometrical or topological structures within the MFG system. It is emphasised that the studies of CIPs and GIPs are not sharply separated and they may connect to each other in certain cases. Moreover, we shall consider both single-population and multi-population MFGs for the proposed inverse problem study. Though the modelling issue is addressed mainly from a theoretical perspective, we shall associate the proposed inverse problems with applications of practical interest in some cases. After addressing the modelling issue, we shall study two fundamental issues for the proposed inverse problems including the unique identifiability and the stability/sensitivity. Some of our analysis is constructive and hence can also provide reconstruction formulae for certain cases. There are several salient technical features of the MFG inverse problems that make the corresponding study highly intriguing and challenging. First, the MFG systems are coupled nonlinear PDE systems with peculiar structures. In particular, they usually posses a forward-backward nature in time. Second, there is a probability density constraint which poses significant difficulties for constructing proper "probing modes" for the inverse problems, especially for the case with multiple-event measurements. We shall develop several effective mathematical methods in tackling the inverse problems including high-order linearisation and high-order variation, construction of proper CGO (Complex Geometrical Optics) and Gaussian beam solutions, Carleman estimates and microlocal analysis of singularity formation by applying advanced tools from the theory of nonlinear PDEs, nonlinear analysis, microlocal and harmonic analysis, as well as asymptotic analysis. Our study contributes to lay a solid and comprehensive mathematical framework for this emerging field of research on inverse problems for MFGs, though we cannot be complete and thorough since even the MFG theory is still being actively and rapidly developed in many aspects.