7 research outputs found
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many
statistics and machine learning applications ranging from support vector
machines to Gaussian processes and kernel embeddings of distributions.
Operators acting on such spaces are, for instance, required to embed
conditional probability distributions in order to implement the kernel Bayes
rule and build sequential data models. It was recently shown that transfer
operators such as the Perron-Frobenius or Koopman operator can also be
approximated in a similar fashion using covariance and cross-covariance
operators and that eigenfunctions of these operators can be obtained by solving
associated matrix eigenvalue problems. The goal of this paper is to provide a
solid functional analytic foundation for the eigenvalue decomposition of RKHS
operators and to extend the approach to the singular value decomposition. The
results are illustrated with simple guiding examples
The Augmented Jump Chain -- a sparse representation of time-dependent Markov jump processes
Modern methods of simulating molecular systems are based on the mathematical
theory of Markov operators with a focus on autonomous equilibrated systems.
However, non-autonomous physical systems or non-autonomous simulation processes
are becoming more and more important. We present a representation of
non-autonomous Markov jump processes as autonomous Markov chains on space-time.
Augmenting the spatial information of the embedded Markov chain by the temporal
information of the associated jump times, we derive the so-called augmented
jump chain. The augmented jump chain inherits the sparseness of the
infinitesimal generator of the original process and therefore provides a useful
tool for studying time-dependent dynamics even in high dimensions. We
furthermore discuss possible generalizations and applications to the
computation of committor functions and coherent sets in the non-autonomous
setting. After deriving the theoretical foundations we illustrate the concepts
with a proof-of-concept Galerkin discretization of the transfer operator of the
augmented jump chain applied to simple examples.Comment: 22 pages, 8 figure
Transfer Operators from Optimal Transport Plans for Coherent Set Detection
The topic of this study lies in the intersection of two fields. One is
related with analyzing transport phenomena in complicated flows.For this
purpose, we use so-called coherent sets: non-dispersing, possibly moving
regions in the flow's domain. The other is concerned with reconstructing a flow
field from observing its action on a measure, which we address by optimal
transport. We show that the framework of optimal transport is well suited for
delivering the formal requirements on which a coherent-set analysis can be
based on. The necessary noise-robustness requirement of coherence can be
matched by the computationally efficient concept of unbalanced regularized
optimal transport. Moreover, the applied regularization can be interpreted as
an optimal way of retrieving the full dynamics given the extremely restricted
information of an initial and a final distribution of particles moving
according to Brownian motion
Machine learning for molecular simulation
Machine learning (ML) is transforming all areas of science. The complex and
time-consuming calculations in molecular simulations are particularly suitable
for a machine learning revolution and have already been profoundly impacted by
the application of existing ML methods. Here we review recent ML methods for
molecular simulation, with particular focus on (deep) neural networks for the
prediction of quantum-mechanical energies and forces, coarse-grained molecular
dynamics, the extraction of free energy surfaces and kinetics and generative
network approaches to sample molecular equilibrium structures and compute
thermodynamics. To explain these methods and illustrate open methodological
problems, we review some important principles of molecular physics and describe
how they can be incorporated into machine learning structures. Finally, we
identify and describe a list of open challenges for the interface between ML
and molecular simulation
Optimal Data-Driven Estimation of Generalized Markov State Models for Non-Equilibrium Dynamics
There are multiple ways in which a stochastic system can be out of statistical equilibrium. It might be subject to time-varying forcing; or be in a transient phase on its way towards equilibrium; it might even be in equilibrium without us noticing it, due to insufficient observations; and it even might be a system failing to admit an equilibrium distribution at all. We review some of the approaches that model the effective statistical behavior of equilibrium and non-equilibrium dynamical systems, and show that both cases can be considered under the unified framework of optimal low-rank approximation of so-called transfer operators. Particular attention is given to the connection between these methods, Markov state models, and the concept of metastability, further to the estimation of such reduced order models from finite simulation data. All these topics bear an important role in, e.g., molecular dynamics, where Markov state models are often and successfully utilized, and which is the main motivating application in this paper. We illustrate our considerations by numerical examples