On-the-Fly Lifting of Coarse Reaction-Coordinate Paths to Full-Dimensional Transition Path Ensembles

Abstract

Effective dynamics on a low-dimensional collective-variable (CV) or latent space can be simulated far more cheaply than the underlying high-dimensional stochastic system, but exploiting such coarse predictions requires lifting: turning a coarse CV trajectory into dynamically consistent full-dimensional states and path ensembles, without relying on global sampling of invariant or conditional fiber measures. We present a local, on-the-fly lifting strategy based on guided full-system trajectories. First an effective model in CV space is used to obtain a coarse reference trajectory. Then, an ensemble of full-dimensional trajectories is generated from a guided version of the original dynamics, where the guidance steers the trajectory to track the CV reference path. Because guidance biases the path distribution, we correct it via pathwise Girsanov reweighting, yielding a correct-by-construction importance-sampling approximation of the conditional law of the uncontrolled dynamics. We further connect the approach to stochastic optimal control, clarifying how coarse models can inform variance-reducing guidance for rare-event quantities. Numerical experiments demonstrate that inexpensive coarse transition paths can be converted into realistic full-system transition pathways (including barrier crossings and detours) and can accelerate estimation of transition pathways and statistics while providing minimal bias through weighted ensembles.

Figures (16)

• Figure 1: Illustration of potential function $V=V_{dw}$ with two main wells around $(x_1,x_2)=(-1,-1)$ as well as $(x_1,x_2)=(1,1)$, and associated $\chi$-function, associated with a dominant second eigenvalue $\lambda=-2.4\cdot 10^{-3}$ of $\mathcal{L}$ (with third eigenvalue $-6.6\cdot 10^{-3}$).
• Figure 2: Full system committor function $q$ (left), reactive density $\mu_{AB}$ (middle), and reactive flux $j_{AB}$ for sets $A$ and $B$ as given in the text.
• Figure 3: Statistics of transition times of the 2d test system for $\alpha=\beta=1$, $\gamma=2$ and $\sigma=0.7$. The statistics results from an ensemble of more than 600 reactive trajectories starting in $A=\{x: \chi(x)\le 0.1\}$ and going next to $B=\{x: \chi(x)\ge 0.9\}$ that were cut out a long-term simulation of the system.
• Figure 4: Effective diffusion coefficient $D_{\text{eff}}$ and effective potential $V_{\text{eff}}$ of the effective dynamics for the simple test system.
• Figure 5: Typical trajectory of the effective dynamics, started in $z_0=0.05$.