Consistent Long-Term Forecasting of Ergodic Dynamical Systems

Vladimir R Kostic, Karim Lounici, Prune Inzerilli, Pietro Novelli, Massimiliano Pontil
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:25370-25395, 2024.

Abstract

We study the problem of forecasting the evolution of a function of the state (observable) of a discrete ergodic dynamical system over multiple time steps. The elegant theory of Koopman and transfer operators can be used to evolve any such function forward in time. However, their estimators are usually unreliable in long-term forecasting. We show how classical techniques of eigenvalue deflation from operator theory and feature centering from statistics can be exploited to enhance standard estimators. We develop a novel technique to derive high probability bounds on powers of empirical estimators. Our approach, rooted in the stability theory of non-normal operators, allows us to establish uniform in time bounds for the forecasting error, which hold even on infinite time horizons. We further show that our approach can be seamlessly employed to forecast future state distributions from an initial one, with provably uniform error bounds. Numerical experiments illustrate the advantages of our approach in practice.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-kostic24a, title = {Consistent Long-Term Forecasting of Ergodic Dynamical Systems}, author = {Kostic, Vladimir R and Lounici, Karim and Inzerilli, Prune and Novelli, Pietro and Pontil, Massimiliano}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {25370--25395}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/kostic24a/kostic24a.pdf}, url = {https://proceedings.mlr.press/v235/kostic24a.html}, abstract = {We study the problem of forecasting the evolution of a function of the state (observable) of a discrete ergodic dynamical system over multiple time steps. The elegant theory of Koopman and transfer operators can be used to evolve any such function forward in time. However, their estimators are usually unreliable in long-term forecasting. We show how classical techniques of eigenvalue deflation from operator theory and feature centering from statistics can be exploited to enhance standard estimators. We develop a novel technique to derive high probability bounds on powers of empirical estimators. Our approach, rooted in the stability theory of non-normal operators, allows us to establish uniform in time bounds for the forecasting error, which hold even on infinite time horizons. We further show that our approach can be seamlessly employed to forecast future state distributions from an initial one, with provably uniform error bounds. Numerical experiments illustrate the advantages of our approach in practice.} }
Endnote
%0 Conference Paper %T Consistent Long-Term Forecasting of Ergodic Dynamical Systems %A Vladimir R Kostic %A Karim Lounici %A Prune Inzerilli %A Pietro Novelli %A Massimiliano Pontil %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-kostic24a %I PMLR %P 25370--25395 %U https://proceedings.mlr.press/v235/kostic24a.html %V 235 %X We study the problem of forecasting the evolution of a function of the state (observable) of a discrete ergodic dynamical system over multiple time steps. The elegant theory of Koopman and transfer operators can be used to evolve any such function forward in time. However, their estimators are usually unreliable in long-term forecasting. We show how classical techniques of eigenvalue deflation from operator theory and feature centering from statistics can be exploited to enhance standard estimators. We develop a novel technique to derive high probability bounds on powers of empirical estimators. Our approach, rooted in the stability theory of non-normal operators, allows us to establish uniform in time bounds for the forecasting error, which hold even on infinite time horizons. We further show that our approach can be seamlessly employed to forecast future state distributions from an initial one, with provably uniform error bounds. Numerical experiments illustrate the advantages of our approach in practice.
APA
Kostic, V.R., Lounici, K., Inzerilli, P., Novelli, P. & Pontil, M.. (2024). Consistent Long-Term Forecasting of Ergodic Dynamical Systems. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:25370-25395 Available from https://proceedings.mlr.press/v235/kostic24a.html.

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