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

Forecasting system behavior 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 behavior, the Rössler system moving from chaos to period-doubling, and a predator-prey model exhibiting collapse via a global bifurcation.