(found 2 matches in 0.001054s)
A Topological Perspective on Regimes in Dynamical Systems
Kristian Strommen, Matthew Chantry, Joshua Dorrington, Nina Otter
The existence and behaviour of so-called `regimes' has been extensively studied in dynamical systems ranging from simple toy models to the atmosphere itself, due to their potential of drastically simplifying complex and chaotic dynamics. Nevertheless, no agreed-upon and clear-cut definition of a `regime' or a `regime system' exists in the literature. We argue here for a definition which equates the existence of regimes in a system with the existence of non-trivial topological structure. We show, using persistent homology, a tool in topological data analysis, that this definition is both computationally tractable, practically informative, and accounts for a variety of different examples. We further show that alternative, more strict definitions based on clustering and/or temporal persistence criteria fail to account for one or more examples of dynamical systems typically thought of as having regimes. We finally discuss how our methodology can shed light on regime behaviour in the atmosphere, and discuss future prospects.
Can Neural Networks Learn Persistent Homology Features?
Guido Montúfar, Nina Otter, Yuguang Wang
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data, and persistence diagrams describe the lifetime of topological invariants, such as connected components or holes, across the one-parameter family. In many applications, one is interested in working with features associated with persistence diagrams rather than the diagrams themselves. In our work, we explore the possibility of learning several types of features extracted from persistence diagrams using neural networks.