Simultaneous Synthesis and Verification of Neural Control Barrier Functions through Branch-and-Bound Verification-in-the-loop Training

Control Barrier Functions (CBFs) that provide formal safety guarantees have been widely used for safety-critical systems. However, it is non-trivial to design a CBF. Utilizing neural networks as CBFs has shown great success, but it necessitates their certification as CBFs. In this work, we leverage bound propagation techniques and the Branch-and-Bound scheme to efficiently verify that a neural network satisfies the conditions to be a CBF over the continuous state space. To accelerate training, we further present a framework that embeds the verification scheme into the training loop to synthesize and verify a neural CBF simultaneously. In particular, we employ the verification scheme to identify partitions of the state space that are not guaranteed to satisfy the CBF conditions and expand the training dataset by incorporating additional data from these partitions. The neural network is then optimized using the augmented dataset to meet the CBF conditions. We show that for a non-linear control-affine system, our framework can efficiently certify a neural network as a CBF and render a larger safe set than state-of-the-art neural CBF works. We further employ our learned neural CBF to derive a safe controller to illustrate the practical use of our framework.Comment: 8 pages, 6 figures, under review for ECC 202

Inner approximations of stochastic programs for data-driven stochastic barrier function design

This paper studies finite-horizon safety guarantees for discrete-time piece-wise affine systems with stochastic noise of unknown distributions. Our approach is based on a novel approach to synthesise a stochastic barrier function from noise data. In particular, we first build a chance-constraint tightening to obtain an inner approximation of a stochastic program. Then, we apply this methodology for stochastic barrier function design, yielding a robust linear program to which the scenario approach theory applies. In contrast to existing approaches, our method is data efficient as it only requires the number of data to be proportional to the logarithm in the negative inverse of the confidence level and is computationally efficient due to its reduction to linear programming. Furthermore, while state-of-the-art methods assume known statistics on the noise distribution, our approach does not require any information about it. We empirically evaluate the efficacy of our method on various verification benchmarks. Experiments show a significant improvement with respect to state-of-the-art, obtaining tighter certificates with a confidence that is several orders of magnitude higher

Safety Certification for Stochastic Systems via Neural Barrier Functions

Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem that limits the wider adoption of autonomous systems in safety-critical applications. One promising solution to address this problem is barrier functions. The composition of a barrier function with a stochastic system forms a supermartingale, thus enabling the computation of the probability that the system stays in a safe set over a finite time horizon via martingale inequalities. However, existing approaches to find barrier functions for stochastic systems generally rely on convex optimization programs that restrict the search of a barrier to a small class of functions such as low degree SoS polynomials and can be computationally expensive. In this paper, we parameterize a barrier function as a neural network and show that techniques for robust training of neural networks can be successfully employed to find neural barrier functions. Specifically, we leverage bound propagation techniques to certify that a neural network satisfies the conditions to be a barrier function via linear programming and then employ the resulting bounds at training time to enforce the satisfaction of these conditions. We also present a branch-and-bound scheme that makes the certification framework scalable. We show that our approach outperforms existing methods in several case studies and often returns certificates of safety that are orders of magnitude larger

Safety Certification for Stochastic Systems via Neural Barrier Functions

Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem. One promising solution to address this problem is the use of barrier functions. Barrier functions are functions whose composition with the system forms a Martingale and enable the computation of the probability that the system stays within a safe set over a finite time horizon. However, existing approaches to find barrier functions generally restrict the search to a small class of functions, often leading to conservatism. To address this problem, in this letter, we parameterize barrier functions as neural networks and show that bound propagation techniques and linear programming can be successfully employed to find Neural Barrier Functions. Further, we develop a branch-and-bound scheme based on linear relaxations that improves the scalability of the proposed framework. On several case studies we show that our approach scales to neural networks of hundreds of neurons and multiple hidden layers and often produces certificates of safety that are tighter than state-of-the-art methods.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Team Luca LaurentiTransport and Plannin