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We present POLAR (The source code can be found at https://github.com/ChaoHuang2018/POLAR_Tool. The full version of this paper can be found at https://arxiv.org/abs/2106.13867.), a POLynomial ARithmetic-based framework for efficient time-bounded reachability analysis of neural-network controlled systems. Existing approaches leveraging 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 TMs. POLAR overcomes these shortcomings by integrating TM arithmetic with Bernstein polynomial interpolation and symbolic remainders. 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 neural network …