Lagrangian Reachtubes: The Next Generation

Abstract

We introduce LRT-NG, a set of techniques and an associated toolset that computes a reachtube (an overapproximation of the set of reachable states over a given time horizon) of a nonlinear dynamical system. LRT-NG significantly advances the state-of-the-art Langrangian Reachability and its associated tool LRT. From a theoretical perspective, LRT-NG is superior to LRT in three ways. First, it uses for the first time an analytically computed metric for the propagated ball which is proven to minimize the ball's volume. We emphasize that the metric computation is the centerpiece of all bloatingbased techniques. Secondly, it computes the next reachset as the intersection of two balls: one based on the Cartesian metric and the other on the new metric. While the two metrics were previously considered opposing approaches, their joint use considerably tightens the reachtubes. Thirdly, it avoids the "wrapping effect" associated with the validated integration of the center of the reachset, by optimally absorbing the interval approximation in the radius of the next ball. From a tool-development perspective, LRT-NG is superior to LRT in two ways. First, it is a standalone tool that no longer relies on CAPD. This required the implementation of the Lohner method and a Runge-Kutta time-propagation method. Secondly, it has an improved interface, allowing the input model and initial conditions to be provided as external input files. Our experiments on a comprehensive set of benchmarks, including two Neural ODEs, demonstrates LRT-NG's superior performance compared to LRT, CAPD, and Flow*.

On The Verification of Neural ODEs with Stochastic Guarantees

“…We capture the reachsets of B by ellipsoids B j = B Mj (χ tj t0 (x 0 ), δ j ) with center χ tj t0 (x 0 ), radius δ j , and metric M j . At every time t j , we use as the center χ tj t0 (x 0 ), the numerical integration of x 0 , and as the metric M j , the optimal metric in χ tj t0 (x 0 ) minimizing the volume of the ellipsoid, as proposed in (Gruenbacher et al 2020).…”

“…with A j from Def. 3 and M j as the metric in χ tj t0 (x 0 ) minimizing the volume of the ellipsoid (Gruenbacher et al 2020).…”