169 research outputs found
Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety
of engineering and scientific fields. Dynamic mode decomposition (DMD), which
is a numerical algorithm for the spectral analysis of Koopman operators, has
been attracting attention as a way of obtaining global modal descriptions of
NLDSs without requiring explicit prior knowledge. However, since existing DMD
algorithms are in principle formulated based on the concatenation of scalar
observables, it is not directly applicable to data with dependent structures
among observables, which take, for example, the form of a sequence of graphs.
In this paper, we formulate Koopman spectral analysis for NLDSs with structures
among observables and propose an estimation algorithm for this problem. This
method can extract and visualize the underlying low-dimensional global dynamics
of NLDSs with structures among observables from data, which can be useful in
understanding the underlying dynamics of such NLDSs. To this end, we first
formulate the problem of estimating spectra of the Koopman operator defined in
vector-valued reproducing kernel Hilbert spaces, and then develop an estimation
procedure for this problem by reformulating tensor-based DMD. As a special case
of our method, we propose the method named as Graph DMD, which is a numerical
algorithm for Koopman spectral analysis of graph dynamical systems, using a
sequence of adjacency matrices. We investigate the empirical performance of our
method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201
Analysis of Dynamic Mode Decomposition
In this master thesis, a study was conducted on a method known as Dynamic mode decomposition(DMD), an equation-free technique which does not require to know the underlying governing equations of the complex data. As a result of massive datasets from various resources, like experiments, simulation, historical records, etc. has led to an increasing demand for an efficient method for data mining and analysis techniques. The main goals of data mining are the description and prediction. Description involves finding patterns in the data and prediction involves predicting the system dynamics. An important aspect when analyzing an algorithm is testing. In this work, DMD-a data based technique is used to test different cases to find the underlying patterns, predict the system dynamics and for reconstruction of original data. Using real data for analyzing a new algorithm may not be appropriate due to lack of knowledge of the algorithm performance in various cases. So, testing is done on synthetic data for all the cases discussed in this work, as it is useful for visualization and to find the robustness of the new algorithm. Finally, this work makes an attempts to understand the DMD\u27s performance and limitations better for the future applications with real data
Analysis and synthesis of collective motion: from geometry to dynamics
The subject of this dissertation is collective motion, the coordinated motion of two or more individuals, in three-dimensional space.
Inspired by the problems of understanding collective motion in nature and designing artificial collectives that can produce complex behaviors, we introduce mathematical methods for the analysis of collective motion data, and biologically-inspired algorithms for generating collective motion in engineered systems.
We explore two complementary approaches to the analysis and synthesis of collective motion.
The first "top-down" approach consists in exploiting the geometry of n-body systems to identify certain elementary components of collective motion.
A main contribution of this thesis is to reveal a new geometrical structure (fiber bundle) of the translation-reduced configuration space and a corresponding classification of collective motions alternative to the classical one based on reduction to shape space.
We derive a mathematical framework for decomposing arbitrary collective motions into elementary components, which can help identify the main modes of an observed collective phenomenon.
We synthesize vector fields that implement some of the most interesting elementary collective motions, and suggest, whenever feasible, decentralized implementations.
The second "bottom-up" approach consists in starting from known biologically-plausible individual control laws and exploring how they can be used to generate collective behaviors. This approach is illustrated using the motion camouflage proportional guidance law as a building block.
We show that rich and coordinated motion patterns can be obtained when two individuals are engaged in mutual pursuit with this control law.
An extension of these dynamics yields coordinated motion for a collective of n individuals
Cooperative Motions and Topology-Driven Dynamical Arrest in Prime Knots
Knots are entangled structures that cannot be untangled without a cut.
Topological stability of knots is one of the many examples of their important
properties that can be used in information storage and transfer. Knot dynamics
is important for understanding general principles of entanglement as knots
provide an isolated system where tangles are highly controlled and easily
manipulated. To unravel the dynamics of these entangled topological objects,
the first step is to identify the dominant motions that are uniquely guided by
knot structure and its complexity. We identify and classify motions into three
main groups -- orthogonal, aligned, and mixed motions, which often act in
unison, orchestrating the complex dynamics of knots. The balance between these
motions is what creates an identifiable signature for every knot. As knot
complexity increases, the carefully orchestrated dynamics is gradually
silenced, eventually reaching a state of topologically driven dynamical arrest.
Depending on their complexity, knots undergo a transition from nearly
stochastic motions to either non-random or even quasiperiodic dynamics before
culminating in dynamical arrest. Here, we show for the first time that
connectivity alone can lead to a topology-driven dynamical arrest in knots of
high complexity. Unexpectedly, we noticed that some knots undergo cooperative
motions as they reach higher complexity, uniquely modulating conformational
patterns of a given knot. Together, these findings demonstrate a link between
topology and dynamics, presenting applications to nanoscale materials
Linearization and Identification of Multiple-Attractor Dynamical Systems through Laplacian Eigenmaps
Dynamical Systems (DS) are fundamental to the modeling and understanding time
evolving phenomena, and have application in physics, biology and control. As
determining an analytical description of the dynamics is often difficult,
data-driven approaches are preferred for identifying and controlling nonlinear
DS with multiple equilibrium points. Identification of such DS has been treated
largely as a supervised learning problem. Instead, we focus on an unsupervised
learning scenario where we know neither the number nor the type of dynamics. We
propose a Graph-based spectral clustering method that takes advantage of a
velocity-augmented kernel to connect data points belonging to the same
dynamics, while preserving the natural temporal evolution. We study the
eigenvectors and eigenvalues of the Graph Laplacian and show that they form a
set of orthogonal embedding spaces, one for each sub-dynamics. We prove that
there always exist a set of 2-dimensional embedding spaces in which the
sub-dynamics are linear and n-dimensional embedding spaces where they are
quasi-linear. We compare the clustering performance of our algorithm to Kernel
K-Means, Spectral Clustering and Gaussian Mixtures and show that, even when
these algorithms are provided with the correct number of sub-dynamics, they
fail to cluster them correctly. We learn a diffeomorphism from the Laplacian
embedding space to the original space and show that the Laplacian embedding
leads to good reconstruction accuracy and a faster training time through an
exponential decaying loss compared to the state-of-the-art diffeomorphism-based
approaches.Comment: Paper Accepted at Journal of Machine Learning Research 23 (2022
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