This paper proposes a physics-informed neural operator (PINO) framework to solve a system of coupled forward-backward partial differential equations (PDEs) arising from mean field games (MFGs). The MFG system incorporates a forward PDE to model the propagation of population dynamics and a backward PDE for a representative agent's optimal control. The PINO is developed to tackle the forward PDE efficiently, particularly when the initial population density varies. A learning algorithm is devised and its performance is evaluated on one application domain, which is autonomous driving velocity control. The PINO exhibits both memory efficiency and generalization capabilities, compared to physics-informed neural networks (PINN).