  Over-approximating the reachable sets of dynamical systems is a fundamental
problem in safety verification and robust control synthesis. The representation
of these sets is a key factor that affects the computational complexity and the
approximation error. In this paper, we develop a new approach for
over-approximating the reachable sets of neural network dynamical systems using
adaptive template polytopes. We use the singular value decomposition of linear
layers along with the shape of the activation functions to adapt the geometry
of the polytopes at each time step to the geometry of the true reachable sets.
We then propose a branch-and-bound method to compute accurate
over-approximations of the reachable sets by the inferred templates. We
illustrate the utility of the proposed approach in the reachability analysis of
linear systems driven by neural network controllers.
