One-Shot Reachability Analysis of Neural Network Dynamical Systems

The arising application of neural networks (NN) in robotic systems has driven the development of safety verification methods for neural network dynamical systems (NNDS). Recursive techniques for reachability analysis of dynamical systems in closed-loop with a NN controller, planner, or perception can over-approximate the reachable sets of the NNDS by bounding the outputs of the NN and propagating these NN output bounds forward. However, this recursive reachability analysis may suffer from compounding errors, rapidly becoming overly conservative over a longer horizon. In this work, we prove that an alternative one-shot reachability analysis framework which directly verifies the unrolled NNDS can significantly mitigate the compounding errors for a general class of NN verification methods built on layerwise abstraction. Our analysis is motivated by the fact that certain NN verification methods give rise to looser bounds when applied in one shot than recursively. In our analysis, we characterize the performance gap between the recursive and one-shot frameworks for NNDS with general computational graphs. The applicability of one-shot analysis is demonstrated through numerical examples on a cart-pole system.