A variational approach to modeling slow processes in stochastic dynamical systems

Frank Noé; Feliks Nüske

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A variational approach to modeling slow processes in stochastic dynamical systems

Frank Noé, Feliks Nüske

arXiv:1211.7103v1Nov 29, 2012math-phmath.STphysics.chem-ph

TL;DR

Slow metastable dynamics are difficult to access by direct simulation because they require long sampling times. The paper approximates dominant propagator eigenfunctions and eigenvalues through a variational Rayleigh-coefficient principle, whose statistical evaluation supports adaptive modeling of Markov processes. The formulation also connects existing Markov state models to constrained variational solutions.

Problem
Slow processes in metastable stochastic dynamical systems are difficult to access by direct numerical simulation because they require the longest simulation times.
Method
The paper approximates dominant propagator eigenfunctions and eigenvalues by maximizing a Rayleigh coefficient that can be estimated from statistical observables.
Results
The Rayleigh coefficient is identical to the true eigenvalues in the exact-eigenfunction limit and can be estimated from simulation data.
Takeaways & Limitations
The formulation provides a basis for adaptive discretization and interprets existing Markov modeling approaches as constrained variational solutions.
Takeaways & Limitations
Finite-dimensional projection errors and statistical uncertainty in estimated Rayleigh coefficients remain subjects for subsequent studies.

Abstract

from arXiv · show

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.

1. Introduction

Metastable Markov processes contain scientifically important slow dynamics that are difficult to sample directly. The paper frames spectral approximation and a statistical variational principle as a route to modeling these processes and improving adaptive Markov models.

Motivation: Metastable Markov processes contain slow transitions between long-lived state-space sets, including molecular conformational dynamics.These slow processes often correspond to rare events such as folding, binding, or catalysis.
Motivation: Slow dynamics are difficult to simulate directly because they require the longest simulation times and can demand 10^4–10^6 particles for 10^7–10^10 time-steps.Such atomistic simulations may be intractable even with special-purpose supercomputers.
Motivation: The paper seeks accurate models of slow dynamics that can support efficient simulation without solving the full direct numerical simulation problem.The target processes are also often the most scientifically interesting because they change global structure or functional behavior.
Spectral formulation: The approach approximates dominant propagator eigenfunctions and eigenvalues, whose exponentially decaying eigenvalues characterize distinct dynamical time scales.For reversible systems, slow and fast eigenspaces are orthogonal, allowing a model parameter to separate selected slow processes from fast processes.
Existing approaches: Existing Markov state models approximate eigenfunctions with basis functions such as state-space characteristic functions, while related methods use committor functions or network coarse-graining.These approaches provide established approximations but typically rely on predefined or heuristic state-space representations.
Contribution: The proposed variational principle maximizes a Rayleigh coefficient and uses statistical observables, enabling constructive adaptive discretization without exact eigenfunctions.The formulation also interprets existing Markov modeling approaches as constrained optimal solutions with different basis sets.

2. Variational principle for conformation dynamics

The section develops a variational framework for estimating dominant propagator eigenfunctions and eigenvalues from correlation functions, including practical approximations through basis functions and Markov models.

Spectral properties: The propagator is a bounded, self-adjoint, compact operator under smoothness, boundedness, stationarity, and reversibility assumptions.Its eigenvalues are real, bounded in magnitude by one, and converge toward zero.
Spectral properties: Dominant eigenvalues decay exponentially with lag time, with rates κ_i, so the slow processes correspond to rates close to zero.After sufficiently long lag times, contributions from faster modes become small and the dominant modes approximate the dynamics.
Rayleigh variational principle: A Rayleigh variational principle estimates eigenfunctions by maximizing their autocorrelation under normalization and orthogonality constraints.For higher modes, the relevant previously identified dominant eigenfunctions must already be known or approximated.
Rayleigh variational principle: The autocorrelation of a weighted eigenfunction r_k = µ^-1l_k equals its corresponding eigenvalue λ_k(τ).This identity connects the variational objective to observables computed from trajectories rather than requiring explicit operator knowledge.
Rayleigh variational principle: The variational principle can be evaluated from correlation functions and therefore estimated directly from simulation data without a closed-form propagator.The approach uses statistically sufficient realizations or simulated time windows of length τ.
Ritz method: Using characteristic functions of disjoint state-space sets, the Ritz approximation reduces to eigenvalue and eigenvector computation for a Markov-model transition matrix.The resulting eigenfunctions are scaled to represent the transfer-operator eigenfunctions.

3. Modeling

The section develops practical representations of propagator eigenfunctions using basis functions, favoring local and analytically tractable choices. Half-weighted eigenfunctions support a Rayleigh-coefficient variational procedure that can be estimated from trajectory data.

The modeling problem is to choose basis-function representations for dominant propagator eigenfunctions in complex systems.
Local basis functions focus computation on specific state-space regions and can support adaptive refinement through added local functions.
Half-weighted eigenfunctions permit local basis functions, non-weighted normalization, and analytically computable expressions for several required quantities.
The transformed propagator P* is self-adjoint, with orthogonal eigenfunctions, enabling Rayleigh coefficients of exact eigenfunctions to equal the exact eigenvalues and approximate coefficients to lower-bound them.
The Rayleigh coefficient can be sampled from trajectories, allowing optimal approximate eigenfunctions to be obtained by maximizing it with Ritz or Roothaan-Hall methods.
Gaussian basis functions offer analytic practical benefits in Euclidean spaces, while other radial basis functions remain candidates for high-dimensional systems.

4. Numerical examples

Numerical examples compare characteristic, Hermite, Gaussian, and nonlinear basis approaches for approximating slow-process eigenfunctions. Smooth bases achieve close approximations in the one-dimensional tests, while the quartic-potential example shows comparable eigenvalue and timescale estimates using fewer Gaussian basis functions than MSM states.

4.1. Metastable potential from a Gaussian stationary density: The two-well example defines a double-Gaussian stationary density, its potential, Smoluchowski dynamics, and the slowest-process eigenfunction as a reference.
4.2. Ritz method with characteristic functions (Markov state model): With 20 characteristic functions, the Ritz/MSM approximation captures eigenvalues to two significant digits but retains significant eigenfunction discretization error.
4.3. Ritz method with a Hermite basis: With 20 Hermite functions, the Ritz method produces a nearly perfect approximation of both eigenvalues and eigenfunctions, though this basis is unsuitable for high-dimensional spaces.
4.4. Roothaan-Hall method with a Gaussian basis: With 11 Gaussian functions, the Roothaan-Hall method also achieves a nearly perfect approximation, while retaining potential suitability for high-dimensional problems.
4.5. Nonlinear optimization: A two-term nonlinear ansatz gives a good eigenvector approximation at the local optimum y2 = 1 and s2 = 0.8, but its high-dimensional usefulness remains unevaluated.
4.6. Quartic potential: In the quartic-potential example, Gaussian Roothaan-Hall and MSM methods comparably estimate the second eigenvalue and timescale, using 13 basis functions versus 100 MSM sets.

5. Conclusions and outlook

The study formulates a variational approach that approximates dominant propagator eigenfunctions by maximizing a Rayleigh coefficient, which can be estimated from distributed short simulations rather than a single long trajectory. Gaussian basis functions are highlighted as a possible option for high-dimensional processes, while basis-selection and statistical-error questions remain for future work.

Maximizing a Rayleigh coefficient approximates the dominant eigenfunctions, and in the exact-eigenfunction limit the coefficient equals the true eigenvalues.
The Rayleigh coefficient is equivalent to an autocorrelation function of an appropriately weighted test function, enabling estimates from many short simulations distributed across state space.
The usefulness of Gaussian and weakly coupled-coordinate basis constructions for complex molecular systems remains to be investigated.
Future work will address projection error from finite-dimensional basis choices and statistical issues in efficiently evaluating uncertainties of estimated Rayleigh coefficients.