  We introduce a new stochastic verification algorithm that formally quantifies
the behavioral robustness of any time-continuous process formulated as a
continuous-depth model. Our algorithm solves a set of global optimization (Go)
problems over a given time horizon to construct a tight enclosure (Tube) of the
set of all process executions starting from a ball of initial states. We call
our algorithm GoTube. Through its construction, GoTube ensures that the
bounding tube is conservative up to a desired probability and up to a desired
tightness. GoTube is implemented in JAX and optimized to scale to complex
continuous-depth neural network models. Compared to advanced reachability
analysis tools for time-continuous neural networks, GoTube does not accumulate
overapproximation errors between time steps and avoids the infamous wrapping
effect inherent in symbolic techniques. We show that GoTube substantially
outperforms state-of-the-art verification tools in terms of the size of the
initial ball, speed, time-horizon, task completion, and scalability on a large
set of experiments. GoTube is stable and sets the state-of-the-art in terms of
its ability to scale to time horizons well beyond what has been previously
possible.
