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Kathleen Champion1, Bethany Lusch2, J Nathan Kutz3

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This study introduces a new method to discover governing equations from data by learning both the equations and the best coordinate system simultaneously. This approach enhances model interpretability and generalizability for complex dynamical systems.

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Area of Science:

  • Dynamical systems theory
  • Machine learning
  • Scientific modeling

Background:

  • Discovering governing equations from data is crucial for scientific fields lacking quantitative descriptions.
  • Sparse regression methods like sparse identification of nonlinear dynamics (SINDy) enable tractable model discovery but require effective coordinate systems.
  • Current methods often struggle with high-dimensional data and finding optimal representations.

Purpose of the Study:

  • To develop a novel framework for simultaneous discovery of governing equations and coordinate transformations.
  • To address the limitation of existing methods that rely on pre-defined coordinate systems.
  • To create parsimonious and interpretable models for complex dynamical systems.

Main Methods:

  • Designed a custom deep autoencoder network to learn coordinate transformations.
  • Integrated the autoencoder with sparse regression techniques for equation discovery.
  • Applied the framework to high-dimensional systems exhibiting low-dimensional dynamics.

Main Results:

  • Successfully identified governing equations and learned coordinate systems for example systems.
  • Demonstrated the ability to represent complex dynamics in a reduced, sparsely representable space.
  • Achieved a balance between model complexity and descriptive accuracy, enhancing interpretability.

Conclusions:

  • The developed framework effectively combines deep learning and sparse identification for robust model discovery.
  • Simultaneously learning coordinates and dynamics offers a powerful approach to uncovering hidden scientific laws from data.
  • This method advances the algorithmic discovery of parsimonious models, embodying Occam's razor for scientific modeling.