An Abstraction-Based Framework for Neural Network Verification

Deep neural networks are increasingly being used as controllers for safety-critical systems. Because neural networks are opaque, certifying their correctness is a significant challenge. To address this issue, several neural network verification approaches have recently been proposed. However, these approaches afford limited scalability, and applying them to large networks can be challenging. In this paper, we propose a framework that can enhance neural network verification techniques by using over-approximation to reduce the size of the network—thus making it more amenable to verification. We perform the approximation such that if the property holds for the smaller (abstract) network, it holds for the original as well. The over-approximation may be too coarse, in which case the underlying verification tool might return a spurious counterexample. Under such conditions, we perform counterexample-guided refinement to adjust the approximation, and then repeat the process. Our approach is orthogonal to, and can be integrated with, many existing verification techniques. For evaluation purposes, we integrate it with the recently proposed Marabou framework, and observe a significant improvement in Marabou’s performance. Our experiments demonstrate the great potential of our approach for verifying larger neural networks

Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming

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

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

Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks

In this paper, the output reachable estimation and safety verification problems for multi-layer perceptron neural networks are addressed. First, a conception called maximum sensitivity in introduced and, for a class of multi-layer perceptrons whose activation functions are monotonic functions, the maximum sensitivity can be computed via solving convex optimization problems. Then, using a simulation-based method, the output reachable set estimation problem for neural networks is formulated into a chain of optimization problems. Finally, an automated safety verification is developed based on the output reachable set estimation result. An application to the safety verification for a robotic arm model with two joints is presented to show the effectiveness of proposed approaches.Comment: 8 pages, 9 figures, to appear in TNNL