Affine Invariant Langevin Dynamics for rare-event sampling

TL;DR

This paper introduces ALDI, an affine invariant Langevin dynamics framework, for efficient rare-event sampling in nonlinear dynamical systems, especially suited for complex, anisotropic, and nonsmooth problems.

Contribution

The paper presents ALDI, a derivative-free, affine invariant particle system that effectively samples rare events in complex, nonsmooth, and anisotropic Bayesian inverse problems.

Findings

ALDI accurately concentrates near critical sets in benchmark problems.

ALDI provides effective proposals for importance sampling in complex models.

The method is robust and potentially gradient-free, suitable for high-dimensional systems.

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.