  Complex dynamical systems rely on the correct deployment and operation of
numerous components, with state-of-the-art methods relying on learning-enabled
components in various stages of modeling, sensing, and control at both offline
and online levels. This paper addresses the run-time safety monitoring problem
of dynamical systems embedded with neural network components. A run-time safety
state estimator in the form of an interval observer is developed to construct
lower-bound and upper-bound of system state trajectories in run time. The
developed run-time safety state estimator consists of two auxiliary neural
networks derived from the neural network embedded in dynamical systems, and
observer gains to ensure the positivity, namely the ability of estimator to
bound the system state in run time, and the convergence of the corresponding
error dynamics. The design procedure is formulated in terms of a family of
linear programming feasibility problems. The developed method is illustrated by
a numerical example and is validated with evaluations on an adaptive cruise
control system.
