  Neural Networks (NNs) can provide major empirical performance improvements
for robotic systems, but they also introduce challenges in formally analyzing
those systems' safety properties. In particular, this work focuses on
estimating the forward reachable set of closed-loop systems with NN
controllers. Recent work provides bounds on these reachable sets, yet the
computationally efficient approaches provide overly conservative bounds (thus
cannot be used to verify useful properties), whereas tighter methods are too
intensive for online computation. This work bridges the gap by formulating a
convex optimization problem for reachability analysis for closed-loop systems
with NN controllers. While the solutions are less tight than prior semidefinite
program-based methods, they are substantially faster to compute, and some of
the available computation time can be used to refine the bounds through input
set partitioning, which more than overcomes the tightness gap. The proposed
framework further considers systems with measurement and process noise, thus
being applicable to realistic systems with uncertainty. Finally, numerical
comparisons show $10\times$ reduction in conservatism in $\frac{1}{2}$ of the
computation time compared to the state-of-the-art, and the ability to handle
various sources of uncertainty is highlighted on a quadrotor model.
