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

TL;DR

This paper introduces Neural Ordinary Differential Equations as a data-driven method to learn and predict bifurcation structures in dynamical systems, enabling extrapolation beyond training data.

Contribution

It demonstrates that Neural ODEs can recover bifurcation structures from time-series data and forecast system behavior across bifurcations, including regions not seen during training.

Findings

Successfully recovered bifurcation structures from data

Predicted bifurcations beyond training parameter regions

Applied method to Lorenz, R"ossler, and predator-prey systems

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 R\"ossler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.