Solving mean field game based on physics-informed operator learning and deep reinforcement learning

The theory of mean field game (MFG) aims to characterize the evolution of optimal strategies as the number of players tends to infinity. The dynamics of MFG are governed by a coupled system of the Hamilton-Jacobi-Bellman (HJB) equation and the Fokker-Planck-Kolmogorov (FPK) equation, which describes the interaction between individual rational decision-making and the evolution of population density. Currently, the numerical solution of MFG equilibria has become a research focus. This paper addresses the problem of congestion avoidance for autonomous vehicles (AVs) and proposes two numerical methods for solving MFG based on physics-informed operator learning (PIOL). The first is a 3-phase solution framework based on PIOL, and the second is a 2-phase solution strategy combining PIOL with deep reinforcement learning (DRL). Both methods construct operator networks based on the deep operator networks (DeepONets) and the variational autoencoders (VAEs). Based on the AVs congestion avoidance scenario, experiments are designed to evaluate the performance of the proposed methods compared to existing approaches in predicting equilibrium solutions. Experimental results show that the proposed algorithms can effectively predict the equilibrium, with higher prediction accuracy and model stability than baseline methods. Moreover, the control strategies generated by the proposed algorithms exhibit better physical fidelity according to the simulations.