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.

This paper addresses the design of safety certificates for stochastic systems, with a focus on ensuring long-term safety through fast real-time control. In stochastic environments, set invariance-based methods that restrict the probability of risk events in infinitesimal time intervals may exhibit significant long-term risks due to cumulative uncertainties/risks. On the other hand, reachability-based approaches that account for the long-term future may require prohibitive computation in real-time decision making. To overcome this challenge involving stringent long-term safety vs. computation tradeoffs, we first introduce a novel technique termed `probabilistic invariance'. This technique characterizes the invariance conditions of the probability of interest. When the target probability is defined using long-term trajectories, this technique can be used to design myopic conditions/controllers with assured long-term safe probability. Then, we integrate this technique into safe control and learning. The proposed control methods efficiently assure long-term safety using neural networks or model predictive controllers with short outlook horizons. The proposed learning methods can be used to guarantee long-term safety during and after training. Finally, we demonstrate the performance of the proposed techniques in numerical simulations.