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Neural Networks (NNs) have been successfully employed to represent the stateevolution of complex dynamical systems. Such models, referred to as NN dynamicmodels (NNDMs), use iterative noisy predictions of NN to estimate adistribution of system trajectories over time. Despite their accuracy, safetyanalysis of NNDMs is known to be a challenging problem and remains largelyunexplored. To address this issue, in this paper, we introduce a method ofproviding safety guarantees for NNDMs. Our approach is based on stochasticbarrier functions, whose relation with safety are analogous to that of Lyapunovfunctions with stability. We first show a method of synthesizing stochasticbarrier functions for NNDMs via a convex optimization problem, which in turnprovides a lower bound on the system's safety probability. A key step in ourmethod is the employment of the recent convex approximation results for NNs tofind piece-wise linear bounds, which allow the formulation of the barrierfunction synthesis problem as a sum-of-squares optimization program. If theobtained safety probability is above the desired threshold, the system iscertified. Otherwise, we introduce a method of generating controls for thesystem that robustly maximizes the safety probability in a minimally-invasivemanner. We exploit the convexity property of the barrier function to formulatethe optimal control synthesis problem as a linear program. Experimental resultsillustrate the efficacy of the method. Namely, they show that the method canscale to multi-dimensional NNDMs with multiple layers and hundreds of neuronsper layer, and that the controller can significantly improve the safetyprobability.

Safety Guarantees for Neural Network Dynamic Systems via Stochastic Barrier Functions