# Learning collective variables that preserve transition rates

arXiv: https://arxiv.org/abs/2506.01222 · HTML: https://mlsanity.com/paper/2506.01222/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 74 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 12 of 12)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration](https://mlsanity.com/paper/1801.00203/) (2017)
- [Variational Encoding of Complex Dynamics](https://mlsanity.com/paper/1711.08576/) (2017)
- [Diffusion Nets](https://mlsanity.com/paper/1506.07840/) (2015)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Selecting the independent coordinates of manifolds with large aspect ratios](https://mlsanity.com/paper/1907.01651/) (2019)
- [On finding optimal collective variables for complex systems by minimizing the deviation between effective and full dynamics](https://mlsanity.com/paper/2405.02001/) (2024)
- [Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles](https://mlsanity.com/paper/2409.06428/) (2024)
- [Understanding recent deep-learning techniques for identifying collective variables of molecular dynamics](https://mlsanity.com/paper/2307.00365/) (2023)
- [Sharp error estimates for target measure diffusion maps with applications to the committor problem](https://mlsanity.com/paper/2312.14418/) (2023)
- [G-invariant diffusion maps](https://mlsanity.com/paper/2306.07350/) (2023)

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

- [A Deep Generative Approach to Stratified Learning](https://mlsanity.com/paper/2604.10650/) (2026)

## Abstract (arXiv)

Collective variables (CVs) play a crucial role in capturing rare events in high-dimensional systems, motivating the continual search for principled approaches to their design. In this work, we revisit the framework of quantitative coarse graining and identify the orthogonality condition from Legoll and Lelievre (2010) as a key criterion for constructing CVs that accurately preserve the statistical properties of the original process. We establish that satisfaction of the orthogonality condition enables error estimates for both relative entropy and pathwise distance to scale proportionally with the degree of scale separation. Building on this foundation, we introduce a general numerical method for designing neural network-based CVs that integrates tools from manifold learning with group-invariant featurization. To demonstrate the efficacy of our approach, we construct CVs for butane and achieve a CV that reproduces the anti-gauche transition rate with less than ten percent relative error. Additionally, we provide empirical evidence challenging the necessity of uniform positive definiteness in diffusion tensors for transition rate reproduction and highlight the critical role of light atoms in CV design for molecular dynamics.