Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations
This addresses the challenge of predicting critical transitions in dynamical systems for applications in fields like ecology and physics, though it is incremental as it builds on existing Neural ODE methods.
The paper tackles the problem of forecasting system behavior near and across bifurcations in dynamical systems, using Neural Ordinary Differential Equations to learn parameter-dependent vector fields from time-series data, and demonstrates that this approach can recover bifurcation structures and forecast beyond training parameter regions, as shown in test cases like the Lorenz and Rössler systems.
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.