# Nonlinear Discovery of Slow Molecular Modes using State-Free Reversible VAMPnets

arXiv: https://arxiv.org/abs/1902.03336 · HTML: https://mlsanity.com/paper/1902.03336/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2019.

## Built on by (most cited first, 14 of 14)

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- [Flow Matching for Optimal Reaction Coordinates of Biomolecular System](https://mlsanity.com/paper/2408.17139/) (2024)
- [Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques](https://mlsanity.com/paper/2412.20868/) (2024)
- [Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds](https://mlsanity.com/paper/2204.10413/) (2022)
- [Hierarchical geometric deep learning enables scalable analysis of molecular dynamics](https://mlsanity.com/paper/2512.06520/) (2025)
- [MD-GAN with multi-particle input: the machine learning of long-time molecular behavior from short-time MD data](https://mlsanity.com/paper/2202.00995/) (2022)
- [Learning Geometrically Disentangled Representations of Protein Folding Simulations](https://mlsanity.com/paper/2205.10423/) (2022)
- [Random functions as data compressors for machine learning of molecular processes](https://mlsanity.com/paper/2509.17937/) (2025)

## Abstract (arXiv)

The success of enhanced sampling molecular simulations that accelerate along collective variables (CVs) is predicated on the availability of variables coincident with the slow collective motions governing the long-time conformational dynamics of a system. It is challenging to intuit these slow CVs for all but the simplest molecular systems, and their data-driven discovery directly from molecular simulation trajectories has been a central focus of the molecular simulation community to both unveil the important physical mechanisms and to drive enhanced sampling. In this work, we introduce state-free reversible VAMPnets (SRV) as a deep learning architecture that learns nonlinear CV approximants to the leading slow eigenfunctions of the spectral decomposition of the transfer operator that evolves equilibrium-scaled probability distributions through time. Orthogonality of the learned CVs is naturally imposed within network training without added regularization. The CVs are inherently explicit and differentiable functions of the input coordinates making them well-suited to use in enhanced sampling calculations. We demonstrate the utility of SRVs in capturing parsimonious nonlinear representations of complex system dynamics in applications to 1D and 2D toy systems where the true eigenfunctions are exactly calculable and to molecular dynamics simulations of alanine dipeptide and the WW domain protein.