  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.
