Interpretable Functional Koopman Learning with Non-Markovian Closure for Spatiotemporal Systems

Wanfeng Lu, He Ma, Wei Lin, Qunxi Zhu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:81278-81322, 2026.

Abstract

Precise prediction of spatiotemporal dynamics over predictive horizons is constrained by the computational cost of high-fidelity solvers and the sparsity, noise, and irregularity of data. We introduce MERLIN, a Koopman-based framework that lifts dynamics to the evolution of learned observation functionals with near-linear progression, enabling full-field reconstruction at arbitrary resolutions. Theoretically, we develop a functional Koopman theory for PDEs and compensate for the loss of finite-dimensional linear invariance via the Mori–Zwanzig formalism, which augments the linear backbone with non-Markovian memory terms to improve predictive accuracy. Practically, MERLIN employs discretization-invariant function encoders that map partial, irregular observations to observables, and resolution-free function decoders that reconstruct states at arbitrary query points. Training under linear constraints yields an interpretable, low-dimensional model that captures principal modes and supports reduced-order modeling, while memory correction further enables stable long-horizon rollouts even in ultra-low-dimensional latent spaces. Our code is available at: https://github.com/RobinLufdu/MERLIN.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-lu26u, title = {Interpretable Functional Koopman Learning with Non-{M}arkovian Closure for Spatiotemporal Systems}, author = {Lu, Wanfeng and Ma, He and Lin, Wei and Zhu, Qunxi}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {81278--81322}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/lu26u/lu26u.pdf}, url = {https://proceedings.mlr.press/v306/lu26u.html}, abstract = {Precise prediction of spatiotemporal dynamics over predictive horizons is constrained by the computational cost of high-fidelity solvers and the sparsity, noise, and irregularity of data. We introduce MERLIN, a Koopman-based framework that lifts dynamics to the evolution of learned observation functionals with near-linear progression, enabling full-field reconstruction at arbitrary resolutions. Theoretically, we develop a functional Koopman theory for PDEs and compensate for the loss of finite-dimensional linear invariance via the Mori–Zwanzig formalism, which augments the linear backbone with non-Markovian memory terms to improve predictive accuracy. Practically, MERLIN employs discretization-invariant function encoders that map partial, irregular observations to observables, and resolution-free function decoders that reconstruct states at arbitrary query points. Training under linear constraints yields an interpretable, low-dimensional model that captures principal modes and supports reduced-order modeling, while memory correction further enables stable long-horizon rollouts even in ultra-low-dimensional latent spaces. Our code is available at: https://github.com/RobinLufdu/MERLIN.} }
Endnote
%0 Conference Paper %T Interpretable Functional Koopman Learning with Non-Markovian Closure for Spatiotemporal Systems %A Wanfeng Lu %A He Ma %A Wei Lin %A Qunxi Zhu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-lu26u %I PMLR %P 81278--81322 %U https://proceedings.mlr.press/v306/lu26u.html %V 306 %X Precise prediction of spatiotemporal dynamics over predictive horizons is constrained by the computational cost of high-fidelity solvers and the sparsity, noise, and irregularity of data. We introduce MERLIN, a Koopman-based framework that lifts dynamics to the evolution of learned observation functionals with near-linear progression, enabling full-field reconstruction at arbitrary resolutions. Theoretically, we develop a functional Koopman theory for PDEs and compensate for the loss of finite-dimensional linear invariance via the Mori–Zwanzig formalism, which augments the linear backbone with non-Markovian memory terms to improve predictive accuracy. Practically, MERLIN employs discretization-invariant function encoders that map partial, irregular observations to observables, and resolution-free function decoders that reconstruct states at arbitrary query points. Training under linear constraints yields an interpretable, low-dimensional model that captures principal modes and supports reduced-order modeling, while memory correction further enables stable long-horizon rollouts even in ultra-low-dimensional latent spaces. Our code is available at: https://github.com/RobinLufdu/MERLIN.
APA
Lu, W., Ma, H., Lin, W. & Zhu, Q.. (2026). Interpretable Functional Koopman Learning with Non-Markovian Closure for Spatiotemporal Systems. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:81278-81322 Available from https://proceedings.mlr.press/v306/lu26u.html.

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