369 research outputs found
Cluster-based reduced-order modelling of a mixing layer
We propose a novel cluster-based reduced-order modelling (CROM) strategy of
unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's
group (Burkardt et al. 2006) and and transition matrix models introduced in
fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a
potential alternative to POD models and generalises the Ulam-Galerkin method
classically used in dynamical systems to determine a finite-rank approximation
of the Perron-Frobenius operator. The proposed strategy processes a
time-resolved sequence of flow snapshots in two steps. First, the snapshot data
are clustered into a small number of representative states, called centroids,
in the state space. These centroids partition the state space in complementary
non-overlapping regions (centroidal Voronoi cells). Departing from the standard
algorithm, the probabilities of the clusters are determined, and the states are
sorted by analysis of the transition matrix. Secondly, the transitions between
the states are dynamically modelled using a Markov process. Physical mechanisms
are then distilled by a refined analysis of the Markov process, e.g. using
finite-time Lyapunov exponent and entropic methods. This CROM framework is
applied to the Lorenz attractor (as illustrative example), to velocity fields
of the spatially evolving incompressible mixing layer and the three-dimensional
turbulent wake of a bluff body. For these examples, CROM is shown to identify
non-trivial quasi-attractors and transition processes in an unsupervised
manner. CROM has numerous potential applications for the systematic
identification of physical mechanisms of complex dynamics, for comparison of
flow evolution models, for the identification of precursors to desirable and
undesirable events, and for flow control applications exploiting nonlinear
actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material.
Accepted for publication in Journal of Fluid Mechanic
On the Geometry of Message Passing Algorithms for Gaussian Reciprocal Processes
Reciprocal processes are acausal generalizations of Markov processes
introduced by Bernstein in 1932. In the literature, a significant amount of
attention has been focused on developing dynamical models for reciprocal
processes. Recently, probabilistic graphical models for reciprocal processes
have been provided. This opens the way to the application of efficient
inference algorithms in the machine learning literature to solve the smoothing
problem for reciprocal processes. Such algorithms are known to converge if the
underlying graph is a tree. This is not the case for a reciprocal process,
whose associated graphical model is a single loop network. The contribution of
this paper is twofold. First, we introduce belief propagation for Gaussian
reciprocal processes. Second, we establish a link between convergence analysis
of belief propagation for Gaussian reciprocal processes and stability theory
for differentially positive systems.Comment: 15 pages; Typos corrected; This paper introduces belief propagation
for Gaussian reciprocal processes and extends the convergence analysis in
arXiv:1603.04419 to the Gaussian cas
A matrix-based approach to solving the inverse Frobenius–Perron problem using sequences of density functions of stochastically perturbed dynamical systems
The paper introduces a matrix-based approach to estimate the unique one-dimensional discrete-time dynamical system that generated a given sequence of probability density functions whilst subjected to an additive stochastic perturbation with known density
Reconstruction of one-dimensional chaotic maps from sequences of probability density functions
In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations
Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
Simultaneous Coherent Structure Coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity
The clustering of data into physically meaningful subsets often requires
assumptions regarding the number, size, or shape of the subgroups. Here, we
present a new method, simultaneous coherent structure coloring (sCSC), which
accomplishes the task of unsupervised clustering without a priori guidance
regarding the underlying structure of the data. sCSC performs a sequence of
binary splittings on the dataset such that the most dissimilar data points are
required to be in separate clusters. To achieve this, we obtain a set of
orthogonal coordinates along which dissimilarity in the dataset is maximized
from a generalized eigenvalue problem based on the pairwise dissimilarity
between the data points to be clustered. This sequence of bifurcations produces
a binary tree representation of the system, from which the number of clusters
in the data and their interrelationships naturally emerge. To illustrate the
effectiveness of the method in the absence of a priori assumptions, we apply it
to three exemplary problems in fluid dynamics. Then, we illustrate its capacity
for interpretability using a high-dimensional protein folding simulation
dataset. While we restrict our examples to dynamical physical systems in this
work, we anticipate straightforward translation to other fields where existing
analysis tools require ad hoc assumptions on the data structure, lack the
interpretability of the present method, or in which the underlying processes
are less accessible, such as genomics and neuroscience
Learning dynamics on invariant measures using PDE-constrained optimization
We extend the methodology in [Yang et al., 2023] to learn autonomous
continuous-time dynamical systems from invariant measures. The highlight of our
approach is to reformulate the inverse problem of learning ODEs or SDEs from
data as a PDE-constrained optimization problem. This shift in perspective
allows us to learn from slowly sampled inference trajectories and perform
uncertainty quantification for the forecasted dynamics. Our approach also
yields a forward model with better stability than direct trajectory simulation
in certain situations. We present numerical results for the Van der Pol
oscillator and the Lorenz-63 system, together with real-world applications to
Hall-effect thruster dynamics and temperature prediction, to demonstrate the
effectiveness of the proposed approach.Comment: This article may be downloaded for personal use only. Any other use
requires prior permission of the author and AIP Publishing. This article
appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume
33, Issue 6, June 2023, and may be found at https://doi.org/10.1063/5.014967
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