An exact multiple-time-step variational formulation for the committor and the transition rate

TL;DR

The paper introduces an exact multiple-time-step variational framework to estimate the committor and transition rate for transitions between stable states, using stopping times at A and B to handle arbitrary lag times. It defines EV-CN and VCN loss functions and demonstrates that the exact committor is recovered for any lag time, with a rigorous connection to transition-path theory and MSR perspectives. Through tests on AIB9, Trp-cage, and villin, the authors show that the EVCN loss yields lag-time-robust committors and that a two-lag extrapolation improves transition-rate estimates, while VCN becomes unreliable at longer lags. The work provides practical guidance for learning committors from trajectory data, highlights limitations of equilibrium-data-driven approaches for rare events, and outlines future directions involving lag-time integration and short-trajectory data. Overall, the approach offers a principled, variational pathway to accurate kinetic statistics from finite-time trajectories.

Abstract

For a transition between two stable states, the committor is the probability that the dynamics leads to one stable state before the other. It can be estimated from trajectory data by minimizing an expression for the transition rate that depends on a lag time. We show that an existing such expression is minimized by the exact committor only when the lag time is a single time step, resulting in a biased estimate in practical applications. We introduce an alternative expression that is minimized by the exact committor at any lag time. The key idea is that, when trajectories enter the stable states, the times that they enter (stopping times) must be used for estimating the committor and transition rate instead of the lag time. Numerical tests on benchmark systems demonstrate that our committor and transition rate estimates are much less sensitive to the choice of lag time. We show how further accuracy for the transition rate can be achieved by combining results from two lag times. We also relate the transition rate expression to a variational approach for kinetic statistics based on the mean-squared residual and discuss further numerical considerations with the aid of a decomposition of the error into dynamic modes.

Figures (9)

• Figure 1: CV projections and reliability diagrams for AIB9. (top left) CV projection of the stationary distribution. (top right) CV projection of the empirical committor. (left columns of VCN and EVCN) CV projections of the predicted committors. (right columns of VCN and EVCN) Reliability diagrams for the predicted committors. All results shown are obtained with the dihedral neural-network inputs.
• Figure 2: CV projections and reliability diagrams for Trp-cage. (top left) CV projection of the stationary distribution. (top right) CV projection of the empirical committor. (left columns of VCN and EVCN) CV projections of the predicted committors. (right columns of VCN and EVCN) Reliability diagrams for the predicted committors. All results shown are obtained with the dihedral neural-network inputs.
• Figure 3: CV projections and reliability diagrams for villin. (top left) CV projection of the stationary distribution. (top right) CV projection of the empirical committor. (left columns of VCN and EVCN) CV projections of the predicted committors. (right columns of VCN and EVCN) Reliability diagrams for the predicted committors. All results shown are obtained with the dihedral neural-network inputs.
• Figure 4: Ratio of predicted to empirical transition rates for AIB9, for different committor and transition rate estimator hyperparameters. The empirical transition rate is $9.1 \times 10^{-3}$.
• Figure 5: Ratio of predicted to empirical transition rates for Trp-cage, for different committor and transition rate estimator hyperparameters. The empirical transition rate is $7.7 \times 10^{-2}$.