Solving Hamilton-Jacobi-Isaacs (HJI) PDEs enables equilibrial feedback
control in two-player differential games, yet faces the curse of dimensionality
(CoD). While physics-informed machine learning has been adopted to address CoD
in solving PDEs, this method falls short in learning discontinuous solutions
due to its sampling nature, leading to poor safety performance of the resulting
controllers in robotics applications where values are discontinuous due to
state or other temporal logic constraints. In this study, we explore three
potential solutions to this problem: (1) a hybrid learning method that uses
both equilibrium demonstrations 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 auxiliary state space where the value becomes
continuous. Evaluations through 5D and 9D vehicle simulations and 13D drone
simulations reveal that the hybrid method outperforms others in terms of
generalization and safety performance.

使用三种方法解决了物理信息机器学习方法在机器人应用中由于采样性质而产生的不连续解的问题，并在 5D、9D 车辆模拟和 13D 无人机模拟中证明了混合方法在泛化和安全性能方面的优越性。