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

Area of Science:

• Complex Systems Science
• Data Mining
• Dynamical Systems Theory

Background:

• Manifold learning is vital for analyzing complex spatiotemporal data and nonlinear model identification.
• Identifying the underlying spatial manifold for time series data modeled by partial differential equations (PDEs) remains a challenge.

Purpose of the Study:

• To develop and validate a data-driven method for reconstructing the "emergent space" of spatiotemporal systems from time series data without explicit spatial labels.
• To demonstrate the identification of unknown spatial coordinates and the transition from resolved to lumped representations.

Main Methods:

• Utilizing manifold-learning techniques on time series data sampled from known PDEs without space labels.
• Investigating the observability of physical space from temporal data and the effect of kernel scale on data representation.
• Applying gauge-invariant data mining for extracting spatial coordinates invariant to measurement instruments.

Main Results:

• Successfully reconstructed emergent spatial coordinates for time series data from known PDEs.
• Demonstrated the method's applicability to diverse spatiotemporal dynamics, including chimera states, chaotic, and quasiperiodic behaviors in PDEs and networks.
• Showcased gauge-invariant data mining for fusing heterogeneous observations and matching different systems.

Conclusions:

• The developed emergent space reconstruction technique effectively identifies hidden spatial structures in complex spatiotemporal data.
• This approach offers a powerful tool for nonlinear model identification and understanding diverse dynamical phenomena.
• Gauge-invariant methods enhance data fusion and system comparison capabilities in complex systems analysis.