An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning

TL;DR

This paper explores how manifold learning can discover hidden spatial structures in time series data from complex systems, enabling the reconstruction of emergent spaces without prior spatial labels, with applications to PDEs and network dynamics.

Contribution

It introduces a method for emergent space discovery from temporal data, validated on known PDEs and demonstrated on complex spatiotemporal phenomena, including chimera states and chaotic dynamics.

Findings

Successful reconstruction of spatial coordinates from time series data.

Demonstrated invariance of emergent space to measurement instruments.

Applicable to heterogeneous and multi-system data fusion.

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.