RECORD
Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
Machine Learning · Submitted 20 Oct 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variational approach (VA) of molecular kinetics and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VA. Previous estimates from short trajectories, have been strongly biased and thus not variationally optimal. Here, we employ Koopman operator theory and ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to non-equilibrium data. The main insight is that the VA and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. The Koopman model can be used to propagate densities, and its eigenvalue decomposition provide estimates of relaxation timescales and slow collective variables for dimension reduction. Koopman models are generalizations of Markov state models, TICA and the linear VA and allow molecular kinetics to be described without a cluster discretization.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | Recently, a powerful generalization of MSMs has been introduced, the variational approach (VA) of molecular kinetics and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. | background 1.00 | 0.05 | 0.04 | asserted 0.99 |
| s2 | While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VA. | gap 1.00 | 0.06 | 0.05 | asserted 1.00 |
| s3 | Previous estimates from short trajectories, have been strongly biased and thus not variationally optimal. | background 0.55 | 0.09 | 0.03 | asserted 1.00 |
| s4 | Here, we employ Koopman operator theory and ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to non-equilibrium data. | contribution 0.57 | 0.92 | 0.09 | asserted 1.00 |
| s5 | The main insight is that the VA and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. | method 0.41 | 0.69 | 0.16 | asserted 1.00 |
| s6 | This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. | method 0.89 | 0.68 | 0.11 | asserted 0.96 |
| s7 | From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. | method 0.35 | 0.80 | 0.42 | asserted 0.96 |
| s8 | The Koopman model can be used to propagate densities, and its eigenvalue decomposition provide estimates of relaxation timescales and slow collective variables for dimension reduction. | method 0.48 | 0.62 | 0.10 | asserted 0.99 |
| s9 | Koopman models are generalizations of Markov state models, TICA and the linear VA and allow molecular kinetics to be described without a cluster discretization. | interpretation 0.54 | 0.50 | 0.18 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variational approach (VA) of molecular kinetics and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VA. Previous estimates from short trajectories, have been strongly biased and thus not variationally optimal. Here, we employ Koopman operator theory and ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to non-equilibrium data. The main insight is that the VA and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. The Koopman model can be used to propagate densities, and its eigenvalue decomposition provide estimates of relaxation timescales and slow collective variables for dimension reduction. Koopman models are generalizations of Markov state models, TICA and the linear VA and allow molecular kinetics to be described without a cluster discretization.
Human review
How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 2 sentences without a confident label
Uses the method or tool of 1 paper
-
For all covariance estimates in this paper we can employ the shrinkage approach [ref] in order to reduce the sensitivity of estimated covariances to statistical noise [ref] and improve the robustness of eigenvalues and eigenvectors computed from ([ref]-[ref]).
Concise Probability Distributions of Eigenvalues of Real-Valued Wishart Matrices · Estimation of covariance matrices · 0.99
Cites as background 1 paper
-
EDMD and DMD approximate components of the Koopman operator which is a generalization of the backward operator usually used in molecular kinetics [ref] .
On the numerical approximation of the Perron-Frobenius and Koopman operator · Koopman models of equilibrium kinetics · 0.94
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.