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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.
