Pontryagin Neural Operator for Solving Parametric General-Sum Differential Games

The values of two-player general-sum differential games are viscosity solutions to Hamilton-Jacobi-Isaacs (HJI) equations. Value and policy approximations for such games suffer from the curse of dimensionality (CoD). Alleviating CoD through physics-informed neural networks (PINN) encounters convergence issues when differentiable values with large Lipschitz constants are present due to state constraints. On top of these challenges, it is often necessary to learn generalizable values and policies across a parametric space of games, e.g., for game parameter inference when information is incomplete. To address these challenges, we propose in this paper a Pontryagin-mode neural operator that outperforms the current state-of-the-art hybrid PINN model on safety performance across games with parametric state constraints. Our key contribution is the introduction of a costate loss defined on the discrepancy between forward and backward costate rollouts, which are computationally cheap. We show that the costate dynamics, which can reflect state constraint violation, effectively enables the learning of differentiable values with large Lipschitz constants, without requiring manually supervised data as suggested by the hybrid PINN model. More importantly, we show that the close relationship between costates and policies makes the former critical in learning feedback control policies with generalizable safety performance.