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

TL;DR

This work tackles the challenge of forecasting system behavior near and across bifurcations when governing equations are unknown. It introduces Neural Ordinary Differential Equations augmented with a bifurcation parameter as input, learning a parameter-dependent vector field $f_\theta(z,\alpha)$ and integrating it in continuous time to generate trajectories and bifurcation diagrams. The approach is demonstrated on three canonical systems—the Lorenz, Roessler, and a predator–prey model—showing the ability to extrapolate across bifurcations and reveal global regime changes; a physics-informed loss further improves generalization in the predator–prey case. Overall, the study provides a data-driven, continuous-time framework for reconstructing and forecasting critical transitions, with potential applications to real-world dynamical 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 Rössler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.

Figures (11)

• Figure 1: Computing trajectories using a bifurcation parameter-dependent Neural ODE. Top: Zoomed in forward step. The value of the bifurcation parameter and the current state variable are used as input for the neural network. The derivative that is outputted by the neural network together with the previous state is used as input for the ODE solver, which calculates the next state. Bottom: Starting from an initial point ($z_{t_0}$), the Neural ODE iteratively computes the state at next time points using forward steps. The solver can take intermediate time steps between data points. The loss is only calculated at the data points.
• Figure 2: From trajectory data to bifurcation diagram. (a) Trajectory data is used as input to train the Neural ODE. (b) The Neural ODE defines a learned vector field, which can be directly extracted from the model. (c) We determine the bifurcation structure from the vector field (Appendix \ref{['Ap:biffigures']}).
• Figure 3: Bifurcation diagrams and sampling locations of training data for different systems and experiments. Top: Lorenz system. Middle: Rössler system. Bottom: Predator-prey system; Experiment 1: Data is sampled from different regimes; Experiment 2: Amount of data is limited, either by decreasing the number of sampled $\alpha$-values or by reducing the number of initial conditions from two to one; Experiment 3: Different levels of measurement noise are added to the data. Line style indicates amount of initial conditions (striped = two initial conditions, dotted = one initial condition).
• Figure 4: Lorenz system trajectories. Top: True system. Bottom: Dynamics learned by the Neural ODE.
• Figure 5: Bifurcation diagram for Lorenz system. Top: True system. Bottom: Dynamics learned by the Neural ODE.