26 research outputs found
A variational approach to modeling slow processes in stochastic dynamical systems
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
Sparse learning of stochastic dynamic equations
With the rapid increase of available data for complex systems, there is great
interest in the extraction of physically relevant information from massive
datasets. Recently, a framework called Sparse Identification of Nonlinear
Dynamics (SINDy) has been introduced to identify the governing equations of
dynamical systems from simulation data. In this study, we extend SINDy to
stochastic dynamical systems, which are frequently used to model biophysical
processes. We prove the asymptotic correctness of stochastics SINDy in the
infinite data limit, both in the original and projected variables. We discuss
algorithms to solve the sparse regression problem arising from the practical
implementation of SINDy, and show that cross validation is an essential tool to
determine the right level of sparsity. We demonstrate the proposed methodology
on two test systems, namely, the diffusion in a one-dimensional potential, and
the projected dynamics of a two-dimensional diffusion process
Koopman analysis of quantum systems
Koopman operator theory has been successfully applied to problems from
various research areas such as fluid dynamics, molecular dynamics, climate
science, engineering, and biology. Applications include detecting metastable or
coherent sets, coarse-graining, system identification, and control. There is an
intricate connection between dynamical systems driven by stochastic
differential equations and quantum mechanics. In this paper, we compare the
ground-state transformation and Nelson's stochastic mechanics and demonstrate
how data-driven methods developed for the approximation of the Koopman operator
can be used to analyze quantum physics problems. Moreover, we exploit the
relationship between Schr\"odinger operators and stochastic control problems to
show that modern data-driven methods for stochastic control can be used to
solve the stationary or imaginary-time Schr\"odinger equation. Our findings
open up a new avenue towards solving Schr\"odinger's equation using recently
developed tools from data science
Spectral Properties of Effective Dynamics from Conditional Expectations
The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting
Tensor-based computation of metastable and coherent sets
Recent years have seen rapid advances in the data-driven analysis of
dynamical systems based on Koopman operator theory -- with extended dynamic
mode decomposition (EDMD) being a cornerstone of the field. On the other hand,
low-rank tensor product approximations -- in particular the tensor train (TT)
format -- have become a valuable tool for the solution of large-scale problems
in a number of fields. In this work, we combine EDMD and the TT format,
enabling the application of EDMD to high-dimensional problems in conjunction
with a large set of features. We derive efficient algorithms to solve the EDMD
eigenvalue problem based on tensor representations of the data, and to project
the data into a low-dimensional representation defined by the eigenvectors. We
extend this method to perform canonical correlation analysis (CCA) of
non-reversible or time-dependent systems. We prove that there is a physical
interpretation of the procedure and demonstrate its capabilities by applying
the method to several benchmark data sets
Data-driven model reduction and transfer operator approximation
In this review paper, we will present different data-driven dimension
reduction techniques for dynamical systems that are based on transfer operator
theory as well as methods to approximate transfer operators and their
eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out
similarities and differences between methods developed independently by the
dynamical systems, fluid dynamics, and molecular dynamics communities such as
time-lagged independent component analysis (TICA), dynamic mode decomposition
(DMD), and their respective generalizations. As a result, extensions and best
practices developed for one particular method can be carried over to other
related methods
Error bounds for kernel-based approximations of the Koopman operator
We consider the data-driven approximation of the Koopman operator for
stochastic differential equations on reproducing kernel Hilbert spaces (RKHS).
Our focus is on the estimation error if the data are collected from long-term
ergodic simulations. We derive both an exact expression for the variance of the
kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and
probabilistic bounds for the finite-data estimation error. Moreover, we derive
a bound on the prediction error of observables in the RKHS using a finite
Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we
provide bounds on the full approximation error. Numerical experiments using the
Ornstein-Uhlenbeck process illustrate our results.Comment: 28 page
Slow collective variables and molecular kinetics from short off-equilibrium simulations
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 conformation dynamics/molecular kinetics (VAC) 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 VAC. Previous estimates from short trajectories have been strongly
biased and thus not variationally optimal. Here, we employ the Koopman
operator theory and the ideas from dynamic mode decomposition to extend the
VAC and TICA to non-equilibrium data. The main insight is that the VAC 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
provides estimates of relaxation time scales and slow collective variables for
dimension reduction. Koopman models are generalizations of Markov state
models, TICA, and the linear VAC and allow molecular kinetics to be described
without a cluster discretization
Partial observations, coarse graining and equivariance in Koopman operator theory for large-scale dynamical systems
The Koopman operator has become an essential tool for data-driven analysis,
prediction and control of complex systems, the main reason being the enormous
potential of identifying linear function space representations of nonlinear
dynamics from measurements. Until now, the situation where for large-scale
systems, we (i) only have access to partial observations (i.e., measurements,
as is very common for experimental data) or (ii) deliberately perform coarse
graining (for efficiency reasons) has not been treated to its full extent. In
this paper, we address the pitfall associated with this situation, that the
classical EDMD algorithm does not automatically provide a Koopman operator
approximation for the underlying system if we do not carefully select the
number of observables. Moreover, we show that symmetries in the system dynamics
can be carried over to the Koopman operator, which allows us to massively
increase the model efficiency. We also briefly draw a connection to domain
decomposition techniques for partial differential equations and present
numerical evidence using the Kuramoto--Sivashinsky equation
Variational Tensor Approach for Approximating the Rare-Event Kinetics of Macromolecular Systems
Essential information about the stationary and slow kinetic properties of macromolecules is contained in the eigenvalues and eigenfunctions of the dynamical operator of the molecular dynamics. A recent variational formulation allows to optimally approximate these eigenvalues and eigenfunctions when a basis set for the eigenfunctions is provided. In this study, we propose that a suitable choice of basis functions is given by products of one-coordinate basis functions, which describe changes along internal molecular coordinates, such as dihedral angles or distances. A sparse tensor product approach is employed in order to avoid a combinatorial explosion of products, i.e. of the basis-set size. Our results suggest that the high-dimensional eigenfunctions can be well approximated with relatively small basis set sizes