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RINO combines a zonotopic set representation with generalized mean-value AE extensions to compute under and over-approximations of the robust range of differentiable functions, and applies these techniques to the particular case of learning-enabled dynamical systems. The AE extensions require an efficient and accurate evaluation of the function and its Jacobian with respect to the inputs and initial conditions. For continuous-time systems, possibly controlled by neural networks, the function to evaluate is the solution of the dynamical system. It is over-approximated in RINO using Taylor methods in time coupled with a set-based evaluation with zonotopes. We demonstrate the good performances of RINO compared to state-of-the art tools Verisig 2.0 and ReachNN* on a set of classical benchmark examples of neural network controlled closed loop systems. For generally comparable precision to Verisig 2.0 and higher precision than ReachNN*, RINO is always at least one order of magnitude faster, while also computing the more involved inner-approximations that the other tools do not compute.<\/jats:p>","DOI":"10.1007\/978-3-031-13185-1_25","type":"book-chapter","created":{"date-parts":[[2022,8,6]],"date-time":"2022-08-06T19:29:09Z","timestamp":1659814149000},"page":"511-523","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["RINO: Robust INner and\u00a0Outer Approximated Reachability of\u00a0Neural Networks Controlled Systems"],"prefix":"10.1007","author":[{"given":"Eric","family":"Goubault","sequence":"first","affiliation":[]},{"given":"Sylvie","family":"Putot","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,7]]},"reference":[{"key":"25_CR1","unstructured":"Althoff, M.: An introduction to CORA 2015. 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