Physics-Informed Graph Neural Operator for Mean Field Games on Graph: A Scalable Learning Approach

Mean-field games (MFGs) are developed to model the decision-making processes of a large number of interacting agents in multi-agent systems. This paper studies mean-field games on graphs ($\mathcal{G}$-MFGs). The equilibria of $\mathcal{G}$-MFGs, namely, mean-field equilibria (MFE), are challenging to solve for their high-dimensional action space because each agent has to make decisions when they are at junction nodes or on edges. Furthermore, when the initial population state varies on graphs, we have to recompute MFE, which could be computationally challenging and memory-demanding. To improve the scalability and avoid repeatedly solving $\mathcal{G}$-MFGs every time their initial state changes, this paper proposes physics-informed graph neural operators (PIGNO). The PIGNO utilizes a graph neural operator to generate population dynamics, given initial population distributions. To better train the neural operator, it leverages physics knowledge to propagate population state transitions on graphs. A learning algorithm is developed, and its performance is evaluated on autonomous driving games on road networks. Our results demonstrate that the PIGNO is scalable and generalizable when tested under unseen initial conditions.