  In this paper, we propose a computationally efficient framework for interval
reachability of systems with neural network controllers. Our approach leverages
inclusion functions for the open-loop system and the neural network controller
to embed the closed-loop system into a larger-dimensional embedding system,
where a single trajectory over-approximates the original system's behavior
under uncertainty. We propose two methods for constructing closed-loop
embedding systems, which account for the interactions between the system and
the controller in different ways. The interconnection-based approach considers
the worst-case evolution of each coordinate separately by substituting the
neural network inclusion function into the open-loop inclusion function. The
interaction-based approach uses novel Jacobian-based inclusion functions to
capture the first-order interactions between the open-loop system and the
controller by leveraging state-of-the-art neural network verifiers. Finally, we
implement our approach in a Python framework called ReachMM to demonstrate its
efficiency and scalability on benchmarks and examples ranging to $200$ state
dimensions.
