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.
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