  This paper proposes a computationally efficient framework, based on interval
analysis, for rigorous verification of nonlinear continuous-time dynamical
systems with neural network controllers. Given a neural network, we use an
existing verification algorithm to construct inclusion functions for its
input-output behavior. Inspired by mixed monotone theory, we embed the
closed-loop dynamics into a larger system using an inclusion function of the
neural network and a decomposition function of the open-loop system. This
embedding provides a scalable approach for safety analysis of the neural
control loop while preserving the nonlinear structure of the system.
  We show that one can efficiently compute hyper-rectangular
over-approximations of the reachable sets using a single trajectory of the
embedding system. We design an algorithm to leverage this computational
advantage through partitioning strategies, improving our reachable set
estimates while balancing its runtime with tunable parameters. We demonstrate
the performance of this algorithm through two case studies. First, we
demonstrate this method's strength in complex nonlinear environments. Then, we
show that our approach matches the performance of the state-of-the art
verification algorithm for linear discretized systems.
