A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems

By Frank No'e, Feliks Nuske. 2012

323 citations recorded by the source corpus.

The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due the sampling problem. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena.

Subfields: Molecular Biology, Statistical and Nonlinear Physics

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Citations in the corpus

Deep Generative Markov State Models (2018, neutral): “2c), as expected from theory [22, 18].”
Variational Approach for Learning Markov Processes from Time Series Data (2017, methodology): “…operator is equivalent to the eigenvalue decomposition when the Markov process is time-reversible and stationary with ρ0 = ρ1, and therefore the variational principle presented here is a generalization of that developed for reversible conformation dynamics (Noé and Nüske 2013; Nüske et al. 2014).”
On the numerical approximation of the Perron-Frobenius and Koopman operator (2015, methodology): “This results in a method similar to the variational approach presented in [42] for reversible processes.”
Variational Encoding of Complex Dynamics (2017, methodology): “First, when training deep autoencoders using a autocorrelation loss (i.e. as inspired by the VAC), there is a noticeable dependence on batch size that arises during training.”