Importance sampling of unbounded random stopping times: computing committor functions and exit rates without reweighting

TL;DR

The work addresses the challenge of sampling rare events in stochastic dynamics without the burdens of trajectory reweighting by casting importance sampling as a stochastic-control problem with indefinite time horizons. It develops two zero-variance representations based on convex transformations: a log-transform of the moment generating function and a square-root transform of the second moment, each yielding a different, tractable HJB formulation and a corresponding optimal control. An approximate policy iteration framework is proposed to compute the optimal feedback controls in high dimensions, with convergence properties established for the log-transform case and regularization requirements for the square-root case; the methods are demonstrated on committor problems and high-dimensional toy models. The paper also discusses the exit problem, highlighting how the log-transform approach provides robust, low-variance estimators for mean first exit times, while the second-moment approach can lead to pathological infinite-running trajectories, and it connects these ideas to control variates for robust estimation in complex systems.

Abstract

Rare events in molecular dynamics are often related to noise-induced transitions between different macroscopic states (e.g., in protein folding). A common feature of these rare transitions is that they happen on timescales that are on average exponentially long compared to the characteristic timescale of the system, with waiting time distributions that have (sub)exponential tails and infinite support. As a result, sampling such rare events can lead to trajectories that can be become arbitrarily long, with not too low probability, which makes the reweighting of such trajectories a real challenge. Here, we discuss rare event simulation by importance sampling from a variational perspective, with a focus on applications in molecular dynamics, in particular the computation of committor functions. The idea is to design importance sampling schemes that (a) reduce the variance of a rare event estimator while controlling the average length of the trajectories and (b) that do not require the reweighting of possibly very long trajectories. In doing so, we study different stochastic control formulations for committor and mean first exit times, which we compare both from a theoretical and a computational point of view, including numerical studies of some benchmark examples.

Figures (7)

• Figure 1: Committor function $\phi$ (orange curve) and the corresponding biased potentials $V^+_\epsilon$ (dashed red curve) and $V^-_\epsilon$ (dashed blue curve) for the symmetric double-well potential (solid blue curve). Unlike the committor, the shape of the bias potential depends on the regularization parameter, $\epsilon$, where here $\epsilon=0.2$.
• Figure 2: Forward committor (\ref{['committorBM2']}) for $R_1=5$ and $R_2=10$ and the resulting optimal control $u^*=u^*(r)$ as a function of the radius (left panel) and in the 2-dimensional Cartesian representation (right panel). The vector field for $|x|\le 5$ is outside the range of physically relevant initial conditions.
• Figure 3: RBF approximation of the radial committor functions and their derivatives for $d=2$ (upper panel) and $d=10$ (lower panel).
• Figure 4: API approximation of the committor based on the MGF representation of the committor.
• Figure 5: API approximation of the committor based on the second moment representation of the committor.

Theorems & Definitions (45)

• Lemma 2.1: Gibbs variational principle, cf. daipra1996anum
• Definition 2.2
• Lemma 2.3: Zero variance
• proof
• Theorem 3.1: Girsanov's Theorem
• Remark 3.2
• Lemma 3.3: Generalized stochastic optimal control problem
• proof
• Definition 3.4: Stochastic optimal control problem no. 1
• Lemma 3.5: Value function and Gibbs variational principle