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

arXiv: https://arxiv.org/abs/2507.19036 · HTML: https://mlsanity.com/paper/2507.19036/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 66 references that are in the map.
- First posted on arXiv 25 Jul 2025.

## Builds on (most cited first, 23 of 23)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Generalizing to Unseen Domains: A Survey on Domain Generalization](https://mlsanity.com/paper/2103.03097/) (2021)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Time Series Data Augmentation for Deep Learning: A Survey](https://mlsanity.com/paper/2002.12478/) (2020)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Machine learning prediction of critical transition and system collapse](https://mlsanity.com/paper/2012.01545/) (2020)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Bayesian Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.07244/) (2020)
- [Learning from Irregularly-Sampled Time Series: A Missing Data Perspective](https://mlsanity.com/paper/2008.07599/) (2020)
- [Using Machine Learning to Anticipate Tipping Points and Extrapolate to Post-Tipping Dynamics of Non-Stationary Dynamical Systems](https://mlsanity.com/paper/2207.00521/) (2022)
- [Interpretable Polynomial Neural Ordinary Differential Equations](https://mlsanity.com/paper/2208.05072/) (2022)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [Predicting discrete-time bifurcations with deep learning](https://mlsanity.com/paper/2303.09669/) (2023)
- [Unsupervised learning for anticipating critical transitions](https://mlsanity.com/paper/2501.01579/) (2025)
- [Using scientific machine learning for experimental bifurcation analysis of dynamic systems](https://mlsanity.com/paper/2110.11854/) (2021)
- [Modelling of physical systems with a Hopf bifurcation using mechanistic models and machine learning](https://mlsanity.com/paper/2209.06910/) (2022)
- [Scientific machine learning in ecological systems: A study on the predator-prey dynamics](https://mlsanity.com/paper/2411.06858/) (2024)
- [Neural Chronos ODE: Unveiling Temporal Patterns and Forecasting Future and Past Trends in Time Series Data](https://mlsanity.com/paper/2307.01023/) (2023)

## Built on by (most cited first, 4 of 4)

- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [Topological Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2606.22969/) (2026)
- [Extrapolating the emergence of Hamiltonian chaos with random-feature Hamiltonian neural networks](https://mlsanity.com/paper/2607.28977/) (2026)
- [Learning Chaos Without Seeing Chaos: Extrapolation of Global Dynamics in Autoregressive Transformers](https://mlsanity.com/paper/2609.38814/) (2026)

## Abstract (arXiv)

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.