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Neural networks have become increasingly effective at many difficult machine learning tasks. However, the nonlinear and large-scale nature of neural networks makes them hard to analyze, and, therefore, they are mostly used as blackbox models without formal guarantees. This issue becomes even more complicated when neural networks are used in learning-enabled closed-loop systems, where a small perturbation can substantially impact the system being controlled. Therefore, it is of utmost importance to develop tools that can provide useful certificates of stability, safety, and robustness for neural network-driven systems.In this overview, we present a convex optimization framework for the analysis of neural networks. The main idea is to abstract hard-to-analyze components of a neural network (e.g., the nonlinear activation functions) with the formalism of quadratic constraints. This abstraction allows us to reason about various properties of neural networks (safety, robustness, generalization, stability in closed-loop settings, etc.) via semidefinite programming. 

This paper presents Verisig 2.0, a verification tool for closed-loop systems with neural network (NN) controllers. We focus on NNs with tanh/sigmoid activations and develop a Taylor-model-based reachability algorithm through Taylor model preconditioning and shrink wrapping. Furthermore, we provide a parallelized implementation that allows Verisig 2.0 to efficiently handle larger NNs than existing tools can. We provide an extensive evaluation over 10 benchmarks and compare Verisig 2.0 against three state-of-the-art verification tools. We show that Verisig 2.0 is both more accurate and faster, achieving speed-ups of up to 21x and 268x against different tools, respectively. 

This article addresses the problem of verifying the safety of autonomous systems with neural network (NN) controllers. We focus on NNs with sigmoid/tanh activations and use the fact that the sigmoid/tanh is the solution to a quadratic differential equation. This allows us to convert the NN into an equivalent hybrid system and cast the problem as a hybrid system verification problem, which can be solved by existing tools. Furthermore, we improve the scalability of the proposed method by approximating the sigmoid with a Taylor series with worst-case error bounds. Finally, we provide an evaluation over four benchmarks, including comparisons with alternative approaches based on mixed integer linear programming as well as on star sets. 

There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that we can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system. 

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant suffer from either lack of accuracy or poor scalability. In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently. Our main idea is to interpret activation functions as gradients of convex potential functions. Hence, they satisfy certain properties that can be described by quadratic constraints. This particular description allows us to pose the Lipschitz constant estimation problem as a semidefinite program (SDP). The resulting SDP can be adapted to increase either the estimation accuracy (by capturing the interaction between activation functions of different layers) or scalability (by decomposition and parallel implementation). We illustrate the utility of our approach with a variety of experiments on randomly generated networks and on classifiers trained on the MNIST and Iris datasets. In particular, we experimentally demonstrate that our Lipschitz bounds are the most accurate compared to those in the literature. We also study the impact of adversarial training methods on the Lipschitz bounds of the resulting classifiers and show that our bounds can be used to efficiently provide robustness guarantees. 

This paper presents Verisig, a hybrid system approach to verifying safety properties of closed-loop systems using neural networks as controllers. We focus on sigmoid-based networks and exploit the fact that the sigmoid is the solution to a quadratic differential equation, which allows us to transform the neural network into an equivalent hybrid system. By composing the network's hybrid system with the plant's, we transform the problem into a hybrid system verification problem which can be solved using state-of-the-art reachability tools. We show that reachability is decidable for networks with one hidden layer and decidable for general networks if Schanuel's conjecture is true. We evaluate the applicability and scalability of Verisig in two case studies, one from reinforcement learning and one in which the neural network is used to approximate a model predictive controller.