Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations

Abstract

Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data, most studies remain limited as they exclusively focus on discrete-time methods and local bifurcations. To address these limitations, we use Neural Ordinary Differential Equations which provide a data-driven framework for learning system dynamics. Our results show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from time-series data by learning parameter-dependent vector fields. Notably, we demonstrate that Neural Ordinary Differential Equations can forecast bifurcations even beyond the parameter regions represented in the training data. We demonstrate our approach on three test cases: the Lorenz system transitioning from non-chaotic to chaotic behaviour, the Rössler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.

Added two additional test cases; major restructuring of text accordingly

References in corpus (12)

• Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network
• Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems
• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
• Stochastic Physics-Informed Neural Ordinary Differential Equations
• Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems
• Using Machine Learning to Anticipate Tipping Points and Extrapolate to Post-Tipping Dynamics of Non-Stationary Dynamical Systems
• Predicting discrete-time bifurcations with deep learning
• Interpretable Polynomial Neural Ordinary Differential Equations
• Predicting Critical Transitions in Multiscale Dynamical Systems Using Reservoir Computing
• Templates and subtemplates of Rössler attractors from a bifurcation diagram
• Modelling of physical systems with a Hopf bifurcation using mechanistic models and machine learning
• Neural Chronos ODE: Unveiling Temporal Patterns and Forecasting Future and Past Trends in Time Series Data