Neural ordinary differential equations for learning and extrapolating system dynamics across bifurcations

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.