Safety Certification is Classification

Barrier certificates are scalar functions over the state space of dynamical systems that separate all unsafe states from all reachable states. The existence of a barrier certificate formally verifies the safety of the dynamical system. Recent approaches synthesize barrier certificates by iteratively training a neural network. In each iteration, the candidate is formally verified - if successful, the barrier certificate is found. Instead, we propose a set-based training approach that tightly integrates verification into training via a set-based loss function that soundly encodes all barrier certificate properties. A loss of zero formally proves the validity of the barrier certificate, collapsing the iterative training and verification into a single training procedure. Our experiments demonstrate that our set-based training approach scales well with the system dimension and naturally handles complex nonlinear dynamics.

Safety of stochastic dynamic systems in environments with dynamic obstacles is studied in this paper through the lens of stochastic barrier functions. We introduce both time-invariant and time-varying barrier certificates for discrete-time, continuous-space systems subject to uncertainty, which provide certified lower bounds on the probability of remaining within a safe set over a finite horizon. These certificates explicitly account for time-varying unsafe regions induced by obstacle dynamics. By leveraging Bellman's optimality perspective, the time-varying formulation directly captures temporal structure and yields less conservative bounds than state-of-the-art approaches. By restricting certificates to polynomial functions, we show that time-varying barrier synthesis can be formulated as a convex sum-of-squares program, enabling tractable optimization. Empirical evaluations on nonlinear systems with dynamic obstacles show that time-varying certificates consistently achieve tight guarantees, demonstrating improved accuracy and scalability over state-of-the-art methods.

In this work, we study how to ensure probabilistic safety for nonlinear systems under distributional ambiguity. Our approach builds on a backup-based safety filtering framework that switches between a high-performance nominal policy and a certified backup policy to ensure safety. To handle arbitrary uncertainties from ambiguous distributions, i.e., where the distribution is not of specific structure and the true distribution is unknown, we adopt a distributionally robust (DR) formulation using Wasserstein ambiguity sets. Rather than solving a high-dimensional DR trajectory optimization problem online, we exploit the structure of backup-based safety filtering to reduce safety certification to a one-dimensional search over the switching time between nominal and backup policies. We then develop a sampling-based certification procedure with finite-sample guarantees, where empirical failure probabilities are compared against a Wasserstein-inflated threshold. We validate our method through simulations across three systems, from a Dubins vehicle to a high-speed racing car and a fighter jet, demonstrating the broad applicability and computational efficiency.