  Solving Hamilton-Jacobi-Isaacs (HJI) PDEs numerically enables equilibrial
feedback control in two-player differential games, yet faces the curse of
dimensionality (CoD). While physics-informed neural networks (PINNs) have shown
promise in alleviating CoD in solving PDEs, vanilla PINNs fall short in
learning discontinuous solutions due to their sampling nature, leading to poor
safety performance of the resulting policies when values are discontinuous due
to state or temporal logic constraints. In this study, we explore three
potential solutions to this challenge: (1) a hybrid learning method that is
guided by both supervisory equilibria and the HJI PDE, (2) a value-hardening
method where a sequence of HJIs are solved with increasing Lipschitz constant
on the constraint violation penalty, and (3) the epigraphical technique that
lifts the value to a higher dimensional state space where it becomes
continuous. Evaluations through 5D and 9D vehicle and 13D drone simulations
reveal that the hybrid method outperforms others in terms of generalization and
safety performance by taking advantage of both the supervisory equilibrium
values and costates, and the low cost of PINN loss gradients.
