  Accurate estimates of long-term risk probabilities and their gradients are
critical for many stochastic safe control methods. However, computing such risk
probabilities in real-time and in unseen or changing environments is
challenging. Monte Carlo (MC) methods cannot accurately evaluate the
probabilities and their gradients as an infinitesimal devisor can amplify the
sampling noise. In this paper, we develop an efficient method to evaluate the
probabilities of long-term risk and their gradients. The proposed method
exploits the fact that long-term risk probability satisfies certain partial
differential equations (PDEs), which characterize the neighboring relations
between the probabilities, to integrate MC methods and physics-informed neural
networks. We provide theoretical guarantees of the estimation error given
certain choices of training configurations. Numerical results show the proposed
method has better sample efficiency, generalizes well to unseen regions, and
can adapt to systems with changing parameters. The proposed method can also
accurately estimate the gradients of risk probabilities, which enables first-
and second-order techniques on risk probabilities to be used for learning and
control.
