88 research outputs found
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
Efficient Magnus-type integrators for solar energy conversion in Hubbard models
Strongly interacting electrons in solids are generically described by
Hubbardtype models, and the impact of solar light can be modeled by an
additional time-dependence. This yields a finite dimensional system of ordinary
differential equations (ODE)s of Schr\"odinger type, which can be solved
numerically by exponential time integrators of Magnus type. The efficiency may
be enhanced by combining these with operator splittings. We will discuss
several different approaches of employing exponential-based methods in
conjunction with an adaptive Lanczos method for the evaluation of matrix
exponentials and compare their accuracy and efficiency. For each integrator, we
use defect-based local error estimators to enable adaptive time-stepping. This
serves to reliably control the approximation error and reduce the computational
effor
A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics
We propose a new sensitivity analysis methodology for complex stochastic
dynamics based on the Relative Entropy Rate. The method becomes computationally
feasible at the stationary regime of the process and involves the calculation
of suitable observables in path space for the Relative Entropy Rate and the
corresponding Fisher Information Matrix. The stationary regime is crucial for
stochastic dynamics and here allows us to address the sensitivity analysis of
complex systems, including examples of processes with complex landscapes that
exhibit metastability, non-reversible systems from a statistical mechanics
perspective, and high-dimensional, spatially distributed models. All these
systems exhibit, typically non-gaussian stationary probability distributions,
while in the case of high-dimensionality, histograms are impossible to
construct directly. Our proposed methods bypass these challenges relying on the
direct Monte Carlo simulation of rigorously derived observables for the
Relative Entropy Rate and Fisher Information in path space rather than on the
stationary probability distribution itself. We demonstrate the capabilities of
the proposed methodology by focusing here on two classes of problems: (a)
Langevin particle systems with either reversible (gradient) or non-reversible
(non-gradient) forcing, highlighting the ability of the method to carry out
sensitivity analysis in non-equilibrium systems; and, (b) spatially extended
Kinetic Monte Carlo models, showing that the method can handle high-dimensional
problems
Graph similarity through entropic manifold alignment
In this paper we decouple the problem of measuring graph similarity into two sequential steps. The first step is the linearization of the quadratic assignment problem (QAP) in a low-dimensional space, given by the embedding trick. The second step is the evaluation of an information-theoretic distributional measure, which relies on deformable manifold alignment. The proposed measure is a normalized conditional entropy, which induces a positive definite kernel when symmetrized. We use bypass entropy estimation methods to compute an approximation of the normalized conditional entropy. Our approach, which is purely topological (i.e., it does not rely on node or edge attributes although it can potentially accommodate them as additional sources of information) is competitive with state-of-the-art graph matching algorithms as sources of correspondence-based graph similarity, but its complexity is linear instead of cubic (although the complexity of the similarity measure is quadratic). We also determine that the best embedding strategy for graph similarity is provided by commute time embedding, and we conjecture that this is related to its inversibility property, since the inverse of the embeddings obtained using our method can be used as a generative sampler of graph structure.The work of the first and third authors was supported by the projects TIN2012-32839 and TIN2015-69077-P of the Spanish Government. The work of the second author was supported by a Royal Society Wolfson Research Merit Award
Deep learning Markov and Koopman models with physical constraints
The long-timescale behavior of complex dynamical systems can be described by
linear Markov or Koopman models in a suitable latent space. Recent variational
approaches allow the latent space representation and the linear dynamical model
to be optimized via unsupervised machine learning methods. Incorporation of
physical constraints such as time-reversibility or stochasticity into the
dynamical model has been established for a linear, but not for arbitrarily
nonlinear (deep learning) representations of the latent space. Here we develop
theory and methods for deep learning Markov and Koopman models that can bear
such physical constraints. We prove that the model is an universal approximator
for reversible Markov processes and that it can be optimized with either
maximum likelihood or the variational approach of Markov processes (VAMP). We
demonstrate that the model performs equally well for equilibrium and
systematically better for biased data compared to existing approaches, thus
providing a tool to study the long-timescale processes of dynamical systems
Molecular Dynamics Simulation
Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...
Information Geometry
This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience
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