  We present POLAR, a polynomial arithmetic-based framework for efficient
bounded-time reachability analysis of neural-network controlled systems
(NNCSs). Existing approaches that leverage the standard Taylor Model (TM)
arithmetic for approximating the neural-network controller cannot deal with
non-differentiable activation functions and suffer from rapid explosion of the
remainder when propagating the TMs. POLAR overcomes these shortcomings by
integrating TM arithmetic with \textbf{Bernstein B{\'e}zier Form} and
\textbf{symbolic remainder}. The former enables TM propagation across
non-differentiable activation functions and local refinement of TMs, and the
latter reduces error accumulation in the TM remainder for linear mappings in
the network. Experimental results show that POLAR significantly outperforms the
current state-of-the-art tools in terms of both efficiency and tightness of the
reachable set overapproximation.
  The source code can be found in https://github.com/ChaoHuang2018/POLAR_Tool
