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Manifold learning for organizing unstructured sets of process observations.

Felix Dietrich1, Mahdi Kooshkbaghi2, Erik M Bollt3

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This study introduces manifold learning to organize complex system response data, enabling the reconstruction of underlying dynamics from partial observations. This data-driven approach reveals system dimensions and creates transport maps for improved understanding.

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Area of Science:

  • Dynamical Systems
  • Data Mining
  • Machine Learning

Background:

  • Dynamical systems are often studied using ensembles of temporal observations.
  • Organizing these observations, especially when unstructured or partial, is challenging for reconstructing system dynamics.

Purpose of the Study:

  • To develop a data-driven method using manifold learning to organize unstructured observation ensembles.
  • To reconstruct coherent response surfaces and systematically recover their dimension and parametrization.
  • To demonstrate the creation of informative transport maps between input and output spaces.

Main Methods:

  • Application of manifold learning techniques to unstructured observational data.
  • Utilizing Whitney and Takens embedding theorems for theoretical justification.
  • Reconstruction of system response surfaces from partial and disorganized observations.

Main Results:

  • Demonstrated successful organization of unstructured observation ensembles into coherent response surfaces.
  • Systematically recovered the dimension and parametrization of these surfaces.
  • Reconstructed a cusp bifurcation surface for hydrogen combustion in a continuous stirred tank reactor.
  • Generated informative transport maps linking input parameter and output/state variable spaces.

Conclusions:

  • Manifold learning provides a robust framework for reconstructing dynamical system behavior from complex observational data.
  • The developed approach facilitates a data-driven understanding of system response surfaces and their underlying manifolds.
  • This method enables the creation of valuable transport maps, bridging input and output spaces for enhanced system analysis.