31 research outputs found
Variational approach for learning Markov processes from time series data
Inference, prediction and control of complex dynamical systems from time
series is important in many areas, including financial markets, power grid
management, climate and weather modeling, or molecular dynamics. The analysis
of such highly nonlinear dynamical systems is facilitated by the fact that we
can often find a (generally nonlinear) transformation of the system coordinates
to features in which the dynamics can be excellently approximated by a linear
Markovian model. Moreover, the large number of system variables often change
collectively on large time- and length-scales, facilitating a low-dimensional
analysis in feature space. In this paper, we introduce a variational approach
for Markov processes (VAMP) that allows us to find optimal feature mappings and
optimal Markovian models of the dynamics from given time series data. The key
insight is that the best linear model can be obtained from the top singular
components of the Koopman operator. This leads to the definition of a family of
score functions called VAMP-r which can be calculated from data, and can be
employed to optimize a Markovian model. In addition, based on the relationship
between the variational scores and approximation errors of Koopman operators,
we propose a new VAMP-E score, which can be applied to cross-validation for
hyper-parameter optimization and model selection in VAMP. VAMP is valid for
both reversible and nonreversible processes and for stationary and
non-stationary processes or realizations
Graph Dynamical Networks for Unsupervised Learning of Atomic Scale Dynamics in Materials
Understanding the dynamical processes that govern the performance of
functional materials is essential for the design of next generation materials
to tackle global energy and environmental challenges. Many of these processes
involve the dynamics of individual atoms or small molecules in condensed
phases, e.g. lithium ions in electrolytes, water molecules in membranes, molten
atoms at interfaces, etc., which are difficult to understand due to the
complexity of local environments. In this work, we develop graph dynamical
networks, an unsupervised learning approach for understanding atomic scale
dynamics in arbitrary phases and environments from molecular dynamics
simulations. We show that important dynamical information can be learned for
various multi-component amorphous material systems, which is difficult to
obtain otherwise. With the large amounts of molecular dynamics data generated
everyday in nearly every aspect of materials design, this approach provides a
broadly useful, automated tool to understand atomic scale dynamics in material
systems.Comment: 25 + 7 pages, 5 + 3 figure
Optimal reaction coordinates and kinetic rates from the projected dynamics of transition paths
Finding optimal reaction coordinates and predicting accurate kinetic rates
for activated processes are two of the foremost challenges of molecular
simulations. We introduce an algorithm that tackles the two problems at once:
starting from a limited number of reactive molecular dynamics trajectories
(transition paths), we automatically generate with a Monte Carlo approach a
sequence of different reaction coordinates that progressively reduce the
kinetic rate of their projected effective dynamics. Based on a variational
principle, the minimal rate accurately approximates the exact one, and it
corresponds to the optimal reaction coordinate. After benchmarking the method
on an analytic double-well system, we apply it to complex atomistic systems:
the interaction of carbon nanoparticles of different sizes in water.Comment: 19 pages, 10 figure