An Emergent Space for Distributed Data With Hidden Internal Order Through Manifold Learning

Abstract

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here on the case of time series data that can ultimately be modelled as a spatially distributed system (e.g. a partial differential equation, PDE), but where we do not know the space in which this PDE should be formulated. Hence, even the spatial coordinates for the distributed system themselves need to be identified - to “emerge from”-the data mining process. We will first validate this “emergent space” reconstruction for time series sampled without space labels in known PDEs; this brings up the issue of observability of physical space from temporal observation data, and the transition from spatially resolved to lumped (order-parameter-based) representations by tuning the scale of the data mining kernels. We will then present actual emergent space “discovery” illustrations. Our illustrative examples include chimera states (states of coexisting coherent and incoherent dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics, arising in partial differential equations and/or in heterogeneous networks. We also discuss how data-driven “spatial” coordinates can be extracted in ways invariant to the nature of the measuring instrument. Such gauge-invariant data mining can go beyond the fusion of heterogeneous observations of the same system, to the possible matching of apparently different systems. For an older version of this article, including other examples, see https://arxiv.org/abs/1708.05406.

Emergent Spaces for Coupled Oscillators

“…It has been shown that with a Chung-Lu network (30, 31) these oscillators are drawn to an attractive limit cycle along which their states can be described by two parameters: their applied current I app and their degree, κ. These two heterogeneities can also be described by two diffusion map coordinates, φ 1 and φ 2 (Kemeth et al, 2018). Plotting the potential, V, for a single period of the limit cycle produces the stack of surfaces shown in Figure 15B.…”

Particles to Partial Differential Equations Parsimoniously

“…This task requires a new notion of distribution distance, which is additionally informed by physics at the microscopic level. Finally, we note that one could even infer the independent variables (the appropriate parameterizations of space and even time) for coarse-grained PDEs in systems where there is no a priori notion of physical space or time for modeling [37][38][39].…”

Data-driven prediction of multistable systems from sparse measurements

“…In contrast, here we seek to learn an optimal metric given the available data. The importance of the metric has long been recognized in data science and machine learning communities [26,28,4,21,14]. In fact, our method draws on the work of Xing et al [28] who proposed a metric learning algorithm for semi-supervised clustering of data in finite dimensions.…”