Machine Learning-Based Nonlinear Nudging for Chaotic Dynamical Systems

2025-08-07

AI Summary

To address trajectory divergence in nonlinear state-space models of chaotic systems caused by unknown initial conditions, this paper proposes a neural-network-based nudging data assimilation method. The method integrates a data-driven neural-network nudging term with Kazantzis–Kravaris–Luenberger (KKL) observer theory—constituting the first such synthesis—to establish a provably existent and stable state estimation framework. Specifically, the neural network takes observational data as input and learns a nonlinear, observation-driven nudging control law that asymptotically steers model trajectories toward the true system dynamics. Theoretical analysis guarantees both existence and stability of the nudging term. Evaluated on three canonical chaotic benchmarks—Lorenz-96, Kuramoto–Sivashinsky, and Kolmogorov flow—the method achieves substantial improvements in long-term trajectory prediction accuracy, demonstrating robust adaptability to strongly nonlinear and high-dimensional chaotic systems.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Neuro-Symbolic Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Designing effective nudging terms for nonlinear chaotic systems
• Learning data-driven nudging terms using neural networks
• Validating approach on chaotic benchmarks like Lorenz 96

Innovation

• Neural network nudging for nonlinear models
• Data-driven learning of nudging terms
• Theoretical basis from observer theory