Machine Learning-Based Nonlinear Nudging for Chaotic Dynamical Systems

TL;DR

This paper introduces neural network nudging, a data-driven approach to improve data assimilation in nonlinear chaotic systems by learning effective nudging terms, supported by theoretical guarantees and tested on benchmark chaotic models.

Contribution

It proposes a novel neural network-based method for nonlinear nudging in chaotic systems, with theoretical existence results and empirical validation on benchmark models.

Findings

Successful learning of nudging terms for chaotic models

Improved trajectory convergence in Lorenz 96, Kuramoto--Sivashinsky, and Kolmogorov flow

Theoretical foundation based on observer theory

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.