This paper presents, in a unified fashion, deterministic as well as statistical Lagrangian-verification techniques. They formally quantify the behavioral robustness of any time-continuous process, formulated as a continuous-depth model. To this end, we review LRT-NG, SLR, and GoTube, algorithms for constructing a tight reachtube, that is, an over-approximation of the set of states reachable within a given time-horizon, and provide guarantees for the reachtube bounds. We compare the usage of the variational equations, associated to the system equations, the mean value theorem, and the Lipschitz constants, in achieving deterministic and statistical guarantees. In LRT-NG, the Lipschitz constant is used as a bloating factor of the initial perturbation, to compute the radius of an ellipsoid in an optimal metric, which over-approximates the set of reachable states. In SLR and GoTube, we get statistical guarantees, by using the Lipschitz constants to compute local balls around samples. These are needed to calculate the probability of having found an upper bound, of the true maximum perturbation at every timestep. Our experiments demonstrate the superior performance of Lagrangian techniques, when compared to LRT, Flow*, and CAPD, and illustrate their use in the robustness analysis of various continuous-depth models.

We introduce a new stochastic verification algorithm that formally quantifies the behavioral robustness of any time-continuous process formulated as a continuous-depth model. The 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. GoTube is implemented in JAX and optimized to scale to complex continuous-depth models. Compared to advanced reachability analysis tools for time-continuous neural networks, GoTube provably does not accumulate over-approximation 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 for its ability to scale up to time horizons well beyond what has been possible before.