Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops
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In this paper, we propose a computationally efficient framework for interval reachability of neural network controlled systems. Our approach builds upon inclusion functions for the neural network controller and the open-loop system. We observe that many state-of-the-art neural network verifiers can produce inclusion functions for neural networks. We introduce and analyze a new class of inclusion functions for the open-loop dynamics based on bounds of the function Jacobian that is particularly suitable for capturing the interactions between systems and neural network controllers. Next, for any dynamical system, we use inclusion functions to construct an embedding system with twice the number of states as the original system. We show that a single trajectory of this embedding system provides hyper-rectangular over-approximations of reachable sets. We then propose two approaches for constructing a closed-loop embedding system for a neural network controlled dynamical system that accounts for the interaction between the system and the controller in different ways. The interconnection-based approach accounts for the worst-case evolution of each coordinate separately by substituting the neural network inclusion function into the open-loop embedding system. The interaction-based approach uses the newly introduced class of Jacobian-based inclusion functions to fully capture first-order interactions between the system and the controller. Finally, we implement our approach in a Python framework called and show that on several existing benchmarks, our methods outperform the existing approaches in the literature. We also demonstrate the scalability of our method on a vehicle platooning example with up to 200 states.
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