Affine Invariant Langevin Dynamics for rare-event sampling

Abstract

We introduce an affine invariant Langevin dynamics (ALDI) framework for the efficient estimation of rare events in nonlinear dynamical systems. Rare events are formulated as Bayesian inverse problems through a nonsmooth limit-state function whose zero level set characterises the event of interest. To overcome the nondifferentiability of this function, we propose a smooth approximation that preserves the failure set and yields a posterior distribution satisfying the small-noise limit. The resulting potential is sampled by ALDI, a (derivative-free) interacting particle system whose affine invariance allows it to adapt to the local anisotropy of the posterior. We demonstrate the performance of the method across a hierarchy of benchmarks, namely two low-dimensional examples (an algebraic problem with convex geometry and a dynamical problem of saddle-type instability) and a point-vortex model for atmospheric blockings. In all cases, ALDI concentrates near the relevant near-critical sets and provides accurate proposal distributions for self-normalised importance sampling. The framework is computationally robust, potentially gradient-free, and well-suited for complex forward models with strong geometric anisotropy. These results highlight ALDI as a promising tool for rare-event estimation in unstable regimes of dynamical systems.