Large deviations in safety-critical systems with probabilistic initial conditions

We often rely on probabilistic measures -- e.g. event probability or expected time -- to characterize systems' safety. However, determining these quantities for extremely low-probability events is generally challenging, as standard safety methods usually struggle due to conservativeness, high-dimension scalability, tractability or numerical limitations. We address these issues by leveraging rigorous approximations grounded in the principles of Large Deviations theory. By assuming deterministic initial conditions, Large Deviations identifies a single dominant path in the low-noise limit as the most significant contributor to the rare-event probability: the instanton. We extend this result to incorporate stochastic uncertainty in the initial states, which is a common assumption in many applications. To that end, we determine an expression for the probability density of the initial states, conditioned on the unsafe rare event being observed. This expression gives access to the most probable initial conditions, as well as the most probable hitting time and path deviations, leading to the realization of the unsafe event. We demonstrate it's effectiveness by solving a high-dimensional and non-linear problem: a space collision.