ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees

Andrea Goertzen, Sunbochen Tang, Navid Azizan
Proceedings of The 8th Annual Learning for Dynamics and Control Conference, PMLR 331:209-221, 2026.

Abstract

Chaos is a fundamental feature of many complex dynamical systems, including weather systems and fluid turbulence. These systems are inherently difficult to predict due to their extreme sensitivity to initial conditions. Many chaotic systems are dissipative and ergodic, motivating data-driven models that aim to learn invariant statistical properties over long time horizons. While recent models have made progress in preserving invariant statistics, they are prone to generating unbounded predictions, which prevent meaningful statistics evaluation. To overcome this, we introduce the **Energy-Constrained Operator (ECO)** that simultaneously learns the system dynamics while enforcing boundedness in predictions. We leverage concepts from control theory to develop algebraic conditions based on a learnable energy function, ensuring the learned dynamics is dissipative. ECO enforces these algebraic conditions through an efficient closed-form quadratic projection layer, which provides provable trajectory boundedness. To our knowledge, this is the first work establishing such formal guarantees for data-driven chaotic dynamics models. Additionally, the learned invariant level set provides an outer estimate for the strange attractor, a complex structure that is computationally intractable to characterize. We demonstrate empirical success in ECO’s ability to generate stable long-horizon forecasts, capturing invariant statistics on systems governed by chaotic PDEs, including the Kuramoto-Sivashinsky and the Navier-Stokes equations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v331-goertzen26a, title = {ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees}, author = {Goertzen, Andrea and Tang, Sunbochen and Azizan, Navid}, booktitle = {Proceedings of The 8th Annual Learning for Dynamics and Control Conference}, pages = {209--221}, year = {2026}, editor = {Sukhatme, Gaurav and Lindemann, Lars and Tu, Stephen and Wierman, Adam and Atanasov, Nikolay}, volume = {331}, series = {Proceedings of Machine Learning Research}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v331/main/assets/goertzen26a/goertzen26a.pdf}, url = {https://proceedings.mlr.press/v331/goertzen26a.html}, abstract = {Chaos is a fundamental feature of many complex dynamical systems, including weather systems and fluid turbulence. These systems are inherently difficult to predict due to their extreme sensitivity to initial conditions. Many chaotic systems are dissipative and ergodic, motivating data-driven models that aim to learn invariant statistical properties over long time horizons. While recent models have made progress in preserving invariant statistics, they are prone to generating unbounded predictions, which prevent meaningful statistics evaluation. To overcome this, we introduce the **Energy-Constrained Operator (ECO)** that simultaneously learns the system dynamics while enforcing boundedness in predictions. We leverage concepts from control theory to develop algebraic conditions based on a learnable energy function, ensuring the learned dynamics is dissipative. ECO enforces these algebraic conditions through an efficient closed-form quadratic projection layer, which provides provable trajectory boundedness. To our knowledge, this is the first work establishing such formal guarantees for data-driven chaotic dynamics models. Additionally, the learned invariant level set provides an outer estimate for the strange attractor, a complex structure that is computationally intractable to characterize. We demonstrate empirical success in ECO’s ability to generate stable long-horizon forecasts, capturing invariant statistics on systems governed by chaotic PDEs, including the Kuramoto-Sivashinsky and the Navier-Stokes equations.} }
Endnote
%0 Conference Paper %T ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees %A Andrea Goertzen %A Sunbochen Tang %A Navid Azizan %B Proceedings of The 8th Annual Learning for Dynamics and Control Conference %C Proceedings of Machine Learning Research %D 2026 %E Gaurav Sukhatme %E Lars Lindemann %E Stephen Tu %E Adam Wierman %E Nikolay Atanasov %F pmlr-v331-goertzen26a %I PMLR %P 209--221 %U https://proceedings.mlr.press/v331/goertzen26a.html %V 331 %X Chaos is a fundamental feature of many complex dynamical systems, including weather systems and fluid turbulence. These systems are inherently difficult to predict due to their extreme sensitivity to initial conditions. Many chaotic systems are dissipative and ergodic, motivating data-driven models that aim to learn invariant statistical properties over long time horizons. While recent models have made progress in preserving invariant statistics, they are prone to generating unbounded predictions, which prevent meaningful statistics evaluation. To overcome this, we introduce the **Energy-Constrained Operator (ECO)** that simultaneously learns the system dynamics while enforcing boundedness in predictions. We leverage concepts from control theory to develop algebraic conditions based on a learnable energy function, ensuring the learned dynamics is dissipative. ECO enforces these algebraic conditions through an efficient closed-form quadratic projection layer, which provides provable trajectory boundedness. To our knowledge, this is the first work establishing such formal guarantees for data-driven chaotic dynamics models. Additionally, the learned invariant level set provides an outer estimate for the strange attractor, a complex structure that is computationally intractable to characterize. We demonstrate empirical success in ECO’s ability to generate stable long-horizon forecasts, capturing invariant statistics on systems governed by chaotic PDEs, including the Kuramoto-Sivashinsky and the Navier-Stokes equations.
APA
Goertzen, A., Tang, S. & Azizan, N.. (2026). ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees. Proceedings of The 8th Annual Learning for Dynamics and Control Conference, in Proceedings of Machine Learning Research 331:209-221 Available from https://proceedings.mlr.press/v331/goertzen26a.html.

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