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

arXiv: https://arxiv.org/abs/2509.03539 · HTML: https://mlsanity.com/paper/2509.03539/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 56 references that are in the map.
- First posted on arXiv 29 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Scaling Exponents Across Parameterizations and Optimizers](https://mlsanity.com/paper/2407.05872/) (2024)
- [Maintaining Plasticity in Continual Learning via Regenerative Regularization](https://mlsanity.com/paper/2308.11958/) (2023)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Computing Committor Functions for the Study of Rare Events Using Deep Learning](https://mlsanity.com/paper/1906.06285/) (2019)
- [Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction](https://mlsanity.com/paper/2303.12534/) (2023)
- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [The surprising efficiency of temporal difference learning for rare event prediction](https://mlsanity.com/paper/2405.17638/) (2024)
- [The fast committor machine: Interpretable prediction with kernels](https://mlsanity.com/paper/2405.10410/) (2024)
- [Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics](https://mlsanity.com/paper/2409.19838/) (2024)

## Built on by (most cited first, 1 of 1)

- [Machine learning kinetics from molecular dynamics data](https://mlsanity.com/paper/2609.17736/) (2026)

## Abstract (arXiv)

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.