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Affine Invariant Langevin Dynamics for rare-event sampling

Deepyaman Chakraborty, Ruben Harris, Rupert Klein, Guillermo Olicón-Méndez, Sebastian Reich, Claudia Schillings

TL;DR

The paper tackles the challenge of estimating rare events in nonlinear dynamical systems by recasting the problem as Bayesian inverse inference on a limit-state function G. It introduces affine-invariant Langevin dynamics (ALDI) to sample the resulting smoothed posterior ρ_*(x) ∝ exp(- tilde{G}(x)^2 /(2R)) ρ_0(x), and proves that in the small-noise limit the posterior concentrates on the failure set F, yielding μ_F(dx) = (1/μ_0(F)) 1_F(x) ρ_0(x) dx. A smoothing procedure tilde{G}_δ preserves the failure set while enabling the gradient-free ALDI sampler to adapt to posterior geometry; consistency results show independence from δ in the R→0 limit. The authors develop two importance-sampling strategies based on ALDI, including a mixture-based IS that fits a proposal to ALDI samples, which delivers robust, variance-controlled estimates of P_f in a hierarchy of test problems: a convex limit-state, a hyperbolic saddle, and a point-vortex atmospheric-blocking model. Overall, ALDI provides a gradient-free, geometry-aware framework for rare-event estimation in complex, anisotropic regimes with practical implications for atmospheric and geophysical applications, and the paper outlines avenues for extensions to time-dependent PDEs and surrogate-enabled high-dimensional settings.

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.

Affine Invariant Langevin Dynamics for rare-event sampling

TL;DR

The paper tackles the challenge of estimating rare events in nonlinear dynamical systems by recasting the problem as Bayesian inverse inference on a limit-state function G. It introduces affine-invariant Langevin dynamics (ALDI) to sample the resulting smoothed posterior ρ_*(x) ∝ exp(- tilde{G}(x)^2 /(2R)) ρ_0(x), and proves that in the small-noise limit the posterior concentrates on the failure set F, yielding μ_F(dx) = (1/μ_0(F)) 1_F(x) ρ_0(x) dx. A smoothing procedure tilde{G}_δ preserves the failure set while enabling the gradient-free ALDI sampler to adapt to posterior geometry; consistency results show independence from δ in the R→0 limit. The authors develop two importance-sampling strategies based on ALDI, including a mixture-based IS that fits a proposal to ALDI samples, which delivers robust, variance-controlled estimates of P_f in a hierarchy of test problems: a convex limit-state, a hyperbolic saddle, and a point-vortex atmospheric-blocking model. Overall, ALDI provides a gradient-free, geometry-aware framework for rare-event estimation in complex, anisotropic regimes with practical implications for atmospheric and geophysical applications, and the paper outlines avenues for extensions to time-dependent PDEs and surrogate-enabled high-dimensional settings.

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.
Paper Structure (24 sections, 5 theorems, 69 equations, 15 figures, 2 algorithms)

This paper contains 24 sections, 5 theorems, 69 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and structure
  3. Preliminaries
  4. Rare events as Bayesian inverse problems
  5. Affine-Invariant Interacting Langevin Dynamics
  6. A smoothing of the limit-state function
  7. Consistency of the smoothed Posterior and ALDI
  8. Importance sampling for the probability of failure
  9. Variant 1: Product estimator based on ALDI samples.
  10. Variant 2: Mixture-based IS using a fitted proposal.
  11. Gradient-free ALDI method
  12. Illustrative Test Cases
  13. A convex limit-state function
  14. Numerical results
  15. A rarer-event regime beyond Monte Carlo.
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $\mathbb X=\mathbb{R}^d$. Assume that the initial covariance $\mathcal{C}(X_0)$ is positive definite, the potential $\Phi \in C^2(\mathbb{R}^d)\cap L^1(\rho_*)$, and that there exist a compact set $K \subset \mathbb{R}^d$ and constants $0 < c_1 < c_2$ such that for all $x \in \mathbb{R}^d \setmi Then the ALDI system eq:ALDI_smooth satisfies the following properties: 1. (Affine invariance) The

Figures (15)

  • Figure 1: Final ALDI ensembles for the convex example. Columns correspond to ensemble sizes $J\in\{100,1000,10000\}$. Colours indicate gradient-based and gradient-free ALDI for $R=0.1$ and $R=0.01$. The red curve denotes the failure boundary $G=0$.
  • Figure 2: Dependence of the ALDI estimator on the ensemble size $J$. In (a), the results of Alg. \ref{['alg:aldi_product']} are shown, in (b), the results of Alg. \ref{['alg:aldi_mixture_is']} are shown for the corresponding number of ALDI samples $J$ in the IS estimator.
  • Figure 3: Influence of the smoothing parameter $\delta$ (a) and the observational noise level $R$ (b) on the ALDI estimator for the convex example. The estimate is computed based on \ref{['alg:aldi_mixture_is']} with $10^3$ samples and $8$ mixture components fitted to each ALDI output.
  • Figure 4: In (a), the number of particles lying inside of the failure domain is shown. In (b), the absolute error of the estimator $\hat{P}_f$ as a function of the number of Gaussian mixture components used in the importance-sampling proposal Alg. \ref{['alg:aldi_mixture_is']} with $10^3$ samples.
  • Figure 5: Absolute error of the IS estimator in Alg. \ref{['alg:aldi_mixture_is']} as a function of the number of IS samples for different ALDI ensemble sizes. Larger ensembles yield more accurate proposals and faster error decay.
  • ...and 10 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Theorem 2.2: Consistency of the smoothed posterior
  • proof
  • Corollary 2.3: Consistency of ALDI
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Theorem 3.2
  • ...and 3 more