The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyse a chaotic partial differential equation, the Kuramoto-Sivashinsky (KS), and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data.

翻译：从部分观测中预测并计算混沌系统的稳定性是传统基于方程的方法可能不适用于的任务。在本计算性论文中，我们提出数据驱动方法以实现：(i)推断未观测（隐藏）混沌变量的动力学（全状态重构）；(ii)全状态的时间预报；(iii)推断全状态的稳定性特性。这些任务通过长短期记忆（LSTM）网络完成，该网络仅使用部分状态观测（数据）进行训练：(i)低到高分辨率LSTM（LH-LSTM），它以部分观测为训练输入，并在计算损失时需访问全系统状态；(ii)物理信息增强LSTM（PI-LSTM），其设计将部分观测与动力学系统演化方程的积分形式相结合。首先，我们推导了LSTM的雅可比矩阵。其次，分析了混沌偏微分方程——Kuramoto-Sivashinsky（KS）方程以及Lorenz-96系统。研究表明，所提出的网络能够实现时间精度与统计意义上的隐藏变量预测。表征混沌吸引子稳定性的李亚普诺夫指数及协变李亚普诺夫向量可从部分观测中正确推断。此外，当输入维度小于或接近吸引子的Kaplan-Yorke维度时，PI-LSTM通过成功重构隐藏混沌动力学而优于LH-LSTM。本工作为从部分数据中重构全状态、推断隐藏变量及计算混沌系统稳定性开辟了新途径。