17 research outputs found
Constrained Polynomial Zonotopes
We introduce constrained polynomial zonotopes, a novel non-convex set
representation that is closed under linear map, Minkowski sum, Cartesian
product, convex hull, intersection, union, and quadratic as well as
higher-order maps. We show that the computational complexity of the
above-mentioned set operations for constrained polynomial zonotopes is at most
polynomial in the representation size. The fact that constrained polynomial
zonotopes are generalizations of zonotopes, polytopes, polynomial zonotopes,
Taylor models, and ellipsoids, further substantiates the relevance of this new
set representation. The conversion from other set representations to
constrained polynomial zonotopes is at most polynomial with respect to the
dimension
Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets
Reachability analysis is a formal method to guarantee safety of dynamical
systems under the influence of uncertainties. A major bottleneck of all
reachability algorithms is the requirement to adequately tune certain algorithm
parameters such as the time step size, which requires expert knowledge. In this
work, we solve this issue with a fully-automated reachability algorithm that
tunes all algorithm parameters internally such that the reachable set enclosure
satisfies a user-defined accuracy in terms of distance to the exact reachable
set. Knowing the distance to the exact reachable set, an inner-approximation of
the reachable set can be efficiently extracted from the outer-approximation
using the Minkowski difference. Finally, we propose a novel verification
algorithm that automatically refines the accuracy of the outer- and
inner-approximation until specifications given by time-varying safe and unsafe
sets can either be verified or falsified. The numerical evaluation demonstrates
that our verification algorithm successfully verifies or falsifies benchmarks
from different domains without any requirement for manual tuning.Comment: 16 page
Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes
We present a novel approach to efficiently compute tight non-convex
enclosures of the image through neural networks with ReLU, sigmoid, or
hyperbolic tangent activation functions. In particular, we abstract the
input-output relation of each neuron by a polynomial approximation, which is
evaluated in a set-based manner using polynomial zonotopes. While our approach
can also can be beneficial for open-loop neural network verification, our main
application is reachability analysis of neural network controlled systems,
where polynomial zonotopes are able to capture the non-convexity caused by the
neural network as well as the system dynamics. This results in a superior
performance compared to other methods, as we demonstrate on various benchmarks
Provably Safe Reinforcement Learning via Action Projection using Reachability Analysis and Polynomial Zonotopes
While reinforcement learning produces very promising results for many
applications, its main disadvantage is the lack of safety guarantees, which
prevents its use in safety-critical systems. In this work, we address this
issue by a safety shield for nonlinear continuous systems that solve
reach-avoid tasks. Our safety shield prevents applying potentially unsafe
actions from a reinforcement learning agent by projecting the proposed action
to the closest safe action. This approach is called action projection and is
implemented via mixed-integer optimization. The safety constraints for action
projection are obtained by applying parameterized reachability analysis using
polynomial zonotopes, which enables to accurately capture the nonlinear effects
of the actions on the system. In contrast to other state-of-the-art approaches
for action projection, our safety shield can efficiently handle input
constraints and dynamic obstacles, eases incorporation of the spatial robot
dimensions into the safety constraints, guarantees robust safety despite
process noise and measurement errors, and is well suited for high-dimensional
systems, as we demonstrate on several challenging benchmark systems
ARCH-COMP19 Category Report: Continuous and hybrid systems with nonlinear dynamics
We present the results of a friendly competition for formal verification of continuous and hybrid systems with nonlinear continuous dynamics. The friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in 2019. In this year, 6 tools Ariadne, CORA, DynIbex, Flow*, Isabelle/HOL, and JuliaReach (in alphabetic order) participated. They are applied to solve reachability analysis problems on four benchmark problems, one of them with hybrid dynamics. We do not rank the tools based on the results, but show the current status and discover the potential advantages of different tools