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Area of Science:

  • Complex Systems Dynamics
  • Computational Science
  • Mathematical Modeling

Background:

  • Disorganized, spatially and temporally varying data from systems with varying parameters pose challenges for traditional modeling.
  • Deriving governing equations from such complex, high-dimensional datasets requires novel data-driven approaches.

Purpose of the Study:

  • To develop a generalizable, data-driven framework for deriving parameter-dependent evolutionary partial differential equation (PDE) models from disorganized observations.
  • To demonstrate the emergent nature of space, time, and parameters within these derived models.
  • To apply the framework to diverse systems, including biological development and network dynamics.

Main Methods:

  • Utilizing a diffusion map-based questionnaire approach to create a smooth parametrization of emergent space, time, and parameter dimensions.
  • Iteratively organizing tensor data by observing its behavior across different axes.
  • Employing machine learning, specifically neural networks, to approximate the operators governing the emergent evolutionary equations.

Main Results:

  • Successfully derived parameter-dependent PDE models from disorganized data across various complex systems.
  • Demonstrated the emergent properties of space, time, and parameters, which are determined directly from the data.
  • Validated the approach on advection-diffusion, Drosophila development, neuronal networks, and coupled oscillator models.

Conclusions:

  • The developed data-driven method effectively reconstructs governing evolutionary PDEs from complex, disorganized observational data.
  • The emergent properties of the derived models, including symmetry breaking and translational invariance, offer insights into system dynamics.
  • This framework provides a powerful tool for understanding and modeling complex systems where underlying equations are unknown or difficult to ascertain.