提出精确的多时间步变分方法，提升过渡概率与速率估计的稳定性。
An exact multiple-time-step variational formulation for the committor and the transition rate
- 用停止时间替代滞后时间，实现任意滞后时间下的精确解
- 数值实验显示对滞后时间选择不敏感，误差显著降低
- 适合研究复杂动力系统中罕见事件的统计特性
在两个稳定态之间的转变中，共谋函数（committor）表示动力学在到达另一状态前先到达某一状态的概率。通过最小化依赖于滞后时间的过渡速率表达式，可从轨迹数据中估计该概率。我们证明，现有表达式仅在滞后时间为单个时间步时被精确共谋函数最小化，导致实际应用中的偏差。本文提出一种新表达式，在任意滞后时间下均被精确共谋函数最小化。核心思想是：当轨迹进入稳定态时，应使用其进入时间（停止时间）而非滞后时间来估计共谋函数与过渡速率。基准系统上的数值测试表明，本方法对滞后时间的选择不敏感，估计结果更稳定。我们还展示了如何结合两个不同滞后时间的结果进一步提高过渡速率精度，并将该速率表达式与基于均方残差的动能统计变分方法联系起来，通过动态模式分解分析误差构成。
原文摘要 · Abstract (English)
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.
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