Learning collective variables that preserve transition rates

2025/06/02 by Shashank Sule, Arnav Mehta, Sule, Shashank +3
Decision Sciences · #58-08 #60G25 #68T07 #70-08 #Chemical Physics (physics.chem-ph) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #demographic modeling and climate adaptation

paper · pdf · doi:10.48550/arxiv.2506.01222

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• On finding optimal collective variables for complex systems by minimizing the deviation between effective and full dynamics
• An Information Bottleneck Approach for Markov Model Construction
• Sharp error estimates for target measure diffusion maps with applications to the committor problem
• Understanding recent deep-learning techniques for identifying collective variables of molecular dynamics
• G-invariant diffusion maps
• Collective Variables for Crystallization Simulations -- from Early Developments to Recent Advances
• Collective variables for the study of crystallisation
• Coarse-graining of non-reversible stochastic differential equations:\n quantitative results and connections to averaging
• Selecting the independent coordinates of manifolds with large aspect ratios
• Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry\n Suffices
• Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates
• Reweighted Autoencoded Variational Bayes for Enhanced Sampling (RAVE)
• Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint
• Pathwise estimates for an effective dynamics
• Diffusion Nets
• Diffusion nets
• Estimation of reactive fluxes in gradient stochastic systems using an analogy with electric circuits
• Non-linear dimensionality reduction: Riemannian metric estimation and\n the problem of geometric discovery
• A variational approach to modeling slow processes in stochastic\n dynamical systems
• Towards a Theory of Transition Paths
• Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders
• Diffusion maps

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