  Neural networks (NNs) playing the role of controllers have demonstrated
impressive empirical performances on challenging control problems. However, the
potential adoption of NN controllers in real-life applications also gives rise
to a growing concern over the safety of these neural-network controlled systems
(NNCSs), especially when used in safety-critical applications. In this work, we
present POLAR-Express, an efficient and precise formal reachability analysis
tool for verifying the safety of NNCSs. POLAR-Express uses Taylor model
arithmetic to propagate Taylor models (TMs) across a neural network
layer-by-layer to compute an overapproximation of the neural-network function.
It can be applied to analyze any feed-forward neural network with continuous
activation functions. We also present a novel approach to propagate TMs more
efficiently and precisely across ReLU activation functions. In addition,
POLAR-Express provides parallel computation support for the layer-by-layer
propagation of TMs, thus significantly improving the efficiency and scalability
over its earlier prototype POLAR. Across the comparison with six other
state-of-the-art tools on a diverse set of benchmarks, POLAR-Express achieves
the best verification efficiency and tightness in the reachable set analysis.
