Stochastic Reachability of Uncontrolled Systems via Probability Measures: Approximation via Deep Neural Networks

This paper poses a theoretical characterization of the stochastic reachability problem in terms of probability measures, capturing the probability measure of the state of the system that satisfies the reachability specification for all probabilities over a finite horizon. We achieve this by constructing the level sets of the probability measure for all probability values and, since our approach is only for autonomous systems, we can determine the level sets via forward simulations of the system from a point in the state space at some time step in the finite horizon to estimate the reach probability. We devise a training procedure which exploits this forward simulation and employ it to design a deep neural network (DNN) to predict the reach probability provided the current state and time step. We validate the effectiveness of our approach through three examples.