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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Neural ordinary differential equations for learning and extrapolating system dynamics across bifurcations.

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Neural ordinary differential equations forecast system bifurcations using time series data. This machine learning approach predicts critical transitions beyond training data, advancing dynamical systems analysis.

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Area of Science:

  • Dynamical Systems and Machine Learning
  • Computational Physics
  • Nonlinear Dynamics

Background:

  • Forecasting critical transitions and bifurcation structures in dynamical systems is essential for understanding system shifts.
  • Existing machine learning methods for analyzing bifurcations are often limited to discrete-time models and local phenomena.
  • Neural Ordinary Differential Equations (NODEs) offer a data-driven approach to learn system dynamics from time series data.

Purpose of the Study:

  • To develop and demonstrate a machine learning framework using NODEs for learning and forecasting bifurcations from time series data.
  • To overcome limitations of discrete-time methods and local bifurcation analysis in current machine learning approaches.
  • To investigate the capability of NODEs in predicting bifurcations beyond the parameter range of the training data.

Main Methods:

  • Utilized neural ordinary differential equations (NODEs) as a data-driven framework to learn system dynamics.
  • Trained NODEs on time series data to learn parameter-dependent vector fields, thereby uncovering underlying bifurcation structures.
  • Applied the NODE approach to diverse test cases, including the Lorenz system, Rössler system, and a predator-prey model.

Main Results:

  • Demonstrated that NODEs can accurately recover bifurcation structures directly from time series data.
  • Showcased the ability of NODEs to forecast bifurcations effectively, even for parameter regions not included in the training dataset.
  • Successfully applied the method to model transitions from non-chaotic to chaotic behavior (Lorenz), chaos to period-doubling (Rössler), and global bifurcations leading to collapse (predator-prey).

Conclusions:

  • Neural ordinary differential equations provide a powerful tool for learning system dynamics and forecasting bifurcations from time series data.
  • The NODE framework extends machine learning capabilities for analyzing critical transitions in dynamical systems, including beyond training data parameters.
  • This approach offers a promising avenue for predicting abrupt shifts and understanding complex behaviors in various scientific domains.