KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis

Hanru Bai, Weiyang Ding
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:2484-2519, 2025.

Abstract

Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-bai25b, title = {{K}o{NODE}: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis}, author = {Bai, Hanru and Ding, Weiyang}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {2484--2519}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/bai25b/bai25b.pdf}, url = {https://proceedings.mlr.press/v267/bai25b.html}, abstract = {Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.} }
Endnote
%0 Conference Paper %T KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis %A Hanru Bai %A Weiyang Ding %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-bai25b %I PMLR %P 2484--2519 %U https://proceedings.mlr.press/v267/bai25b.html %V 267 %X Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.
APA
Bai, H. & Ding, W.. (2025). KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:2484-2519 Available from https://proceedings.mlr.press/v267/bai25b.html.

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