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This research introduces neural network-based nudging as a data-driven solution for controlling nonlinear state space models, specifically targeting chaotic dynamical systems like the Lorenz 96 model and Kuramoto–Sivashinsky equation. By leveraging Kazantzis--Kravaris--Luenberger observer theory to prove theoretical existence, the study demonstrates how neural operators can learn effective control terms where traditional linear assumptions fail. The work advances machine learning applications in scientific computing by bridging rigorous mathematical observer design with deep learning architectures for complex system stabilization.
Neural Operator-Based Nonlinear Nudging for Chaotic Dynamical Systems
arXiv:2508.05778v2 Announce Type: replace Abstract: Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial conditions differ. For linear state space models, such control terms can be derived under mild assumptions. However, designing effective nudging terms becomes significantly more challenging in the nonlinear setting. In this work, we propose neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models. We establish a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory. The proposed approach is evaluated on three benchmark problems that exhibit chaotic behavior: the Lorenz 96 model, the Kuramoto--Sivashinsky equation, and the Kolmogorov flow.
Abstract:Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial conditions differ. For linear state space models, such control terms can be derived under mild assumptions. However, designing effective nudging terms becomes significantly more challenging in the nonlinear setting. In this work, we propose neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models. We establish a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory. The proposed approach is evaluated on three benchmark problems that exhibit chaotic behavior: the Lorenz 96 model, the Kuramoto--Sivashinsky equation, and the Kolmogorov flow.
Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2508.05778 [cs.LG]
(or arXiv:2508.05778v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2508.05778 Focus to learn more arXiv-issued DOI via DataCite
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