Pontryagin Neural Operator for Solving Parametric General-Sum
  Differential Games
Lei Zhang Mukesh Ghimire Zhe Xu Wenlong Zhang Yi Ren
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 value discontinuity is 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 existing state-of-the-art (SOTA) 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 discontinuity of
costate dynamics (in the presence of state constraints) effectively enables the
learning of discontinuous values, without requiring manually supervised data as
suggested by the current SOTA. 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.