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Accurate estimates of long-term risk probabilities and their gradients arecritical for many stochastic safe control methods. However, computing such riskprobabilities in real-time and in unseen or changing environments ischallenging. Monte Carlo (MC) methods cannot accurately evaluate theprobabilities and their gradients as an infinitesimal devisor can amplify thesampling noise. In this paper, we develop an efficient method to evaluate theprobabilities of long-term risk and their gradients. The proposed methodexploits the fact that long-term risk probability satisfies certain partialdifferential equations (PDEs), which characterize the neighboring relationsbetween the probabilities, to integrate MC methods and physics-informed neuralnetworks. We provide theoretical guarantees of the estimation error givencertain choices of training configurations. Numerical results show the proposedmethod has better sample efficiency, generalizes well to unseen regions, andcan adapt to systems with changing parameters. The proposed method can alsoaccurately estimate the gradients of risk probabilities, which enables first-and second-order techniques on risk probabilities to be used for learning andcontrol.

A Generalizable Physics-informed Learning Framework for Risk Probability Estimation