用神经网络学习非线性混沌系统的数据同化控制项。
Neural Operator-Based Nonlinear Nudging for Chaotic Dynamical Systems
- 用神经网络自动学习非线性系统的数据同化控制项
- 在洛伦兹96、库朗托-西瓦辛斯基方程等混沌系统上有效
- 理论保证存在性，适合混沌系统状态估计任务
nudging是一种基于观测数据的同化技术，通过在模型动力学中引入观测驱动的控制项，使系统的轨迹随时间逼近真实轨迹，即使初始条件不同。对于线性状态空间模型，在较弱假设下可推导出控制项；但在非线性情况下，设计有效nudging项变得极为困难。本文提出神经网络nudging，一种数据驱动方法，用于学习非线性状态空间模型中的nudging项。基于Kazantzis--Kravaris--Luenberger观测器理论，建立了理论存在性结果。该方法在三个具有混沌行为的基准问题上进行了评估：Lorenz 96模型、Kuramoto--Sivashinsky方程和Kolmogorov流。
原文摘要 · Abstract (English)
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.
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