KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling

Shimin Zhang, Ziyuan Ye, Yinsong Yan, Zeyang Song, Yujie Wu, Jibin Wu
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:76563-76582, 2025.

Abstract

Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at link.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-zhang25cp, title = {{K}oop{STD}: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling}, author = {Zhang, Shimin and Ye, Ziyuan and Yan, Yinsong and Song, Zeyang and Wu, Yujie and Wu, Jibin}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {76563--76582}, 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/zhang25cp/zhang25cp.pdf}, url = {https://proceedings.mlr.press/v267/zhang25cp.html}, abstract = {Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at link.} }
Endnote
%0 Conference Paper %T KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling %A Shimin Zhang %A Ziyuan Ye %A Yinsong Yan %A Zeyang Song %A Yujie Wu %A Jibin Wu %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-zhang25cp %I PMLR %P 76563--76582 %U https://proceedings.mlr.press/v267/zhang25cp.html %V 267 %X Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at link.
APA
Zhang, S., Ye, Z., Yan, Y., Song, Z., Wu, Y. & Wu, J.. (2025). KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:76563-76582 Available from https://proceedings.mlr.press/v267/zhang25cp.html.

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