ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

Yuanchao Xu, Kaidi Shao, Nikos K. Logothetis, Zhongwei Shen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:69647-69674, 2025.

Abstract

Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the spectral residual to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator’s complete spectral information, a limitation known as the ‘spectral inclusion’ problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the spectral residual to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-xu25y, title = {{R}es{K}oop{N}et: Learning Koopman Representations for Complex Dynamics with Spectral Residuals}, author = {Xu, Yuanchao and Shao, Kaidi and Logothetis, Nikos K. and Shen, Zhongwei}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {69647--69674}, 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/xu25y/xu25y.pdf}, url = {https://proceedings.mlr.press/v267/xu25y.html}, abstract = {Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the spectral residual to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator’s complete spectral information, a limitation known as the ‘spectral inclusion’ problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the spectral residual to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.} }
Endnote
%0 Conference Paper %T ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals %A Yuanchao Xu %A Kaidi Shao %A Nikos K. Logothetis %A Zhongwei Shen %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-xu25y %I PMLR %P 69647--69674 %U https://proceedings.mlr.press/v267/xu25y.html %V 267 %X Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the spectral residual to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator’s complete spectral information, a limitation known as the ‘spectral inclusion’ problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the spectral residual to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.
APA
Xu, Y., Shao, K., Logothetis, N.K. & Shen, Z.. (2025). ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:69647-69674 Available from https://proceedings.mlr.press/v267/xu25y.html.

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