  Accurate estimate of long-term risk is critical for safe decision-making, but
sampling from rare risk events and long-term trajectories can be prohibitively
costly. Risk gradient can be used in many first-order techniques for learning
and control methods, but gradient estimate is difficult to obtain using Monte
Carlo (MC) methods because the infinitesimal divisor may significantly amplify
sampling noise. Motivated by this gap, we propose an efficient method to
evaluate long-term risk probabilities and their gradients using short-term
samples without sufficient risk events. We first derive that four types of
long-term risk probability are solutions of certain partial differential
equations (PDEs). Then, we propose a physics-informed learning technique that
integrates data and physics information (aforementioned PDEs). The physics
information helps propagate information beyond available data and obtain
provable generalization beyond available data, which in turn enables long-term
risk to be estimated using short-term samples of safe events. Finally, we
demonstrate in simulation that the proposed technique has improved sample
efficiency, generalizes well to unseen regions, and adapts to changing system
parameters.
