Machine Learning-Based Nonlinear Nudging for Chaotic Dynamical Systems
This work addresses the problem of data assimilation in chaotic systems for researchers in dynamical systems and machine learning, representing an incremental advancement by applying neural networks to a known bottleneck in nonlinear nudging.
The paper tackled the challenge of designing effective nudging terms for nonlinear chaotic dynamical systems by proposing a neural network-based data-driven method, achieving results that demonstrate the trajectory of the nudged system approaches the true system trajectory on benchmark problems like the Lorenz 96 model, Kuramoto--Sivashinsky equation, and Kolmogorov flow.
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.