We introduce a data-driven, linear method for the spatiotemporal prediction of high-dimensional and chaotic dynamical systems. In this method, the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The proper representation of the nonlinear terms is found in a physics-informed way or a purely data-driven fashion, depending on whether any knowledge of governing dynamics is available or not. The unknown matrices appearing in the method are found using the dynamic mode decomposition with control technique. The distinctive features of the method enable it to accurately capture the system's dynamically important unstable modes while suppressing their unbounded growth via the included nonlinearities. The predictive capabilities of the method are demonstrated for well-known prototypes of chaotic dynamics such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the method predictions are accurate for several Lyapunov timescales. Similar performance is shown for two-dimensional lid-driven cavity flows at high Reynolds numbers.

我们介绍了一种数据驱动的线性方法，用于高维和混沌动力学系统的时空预测。在这种方法中，可观测值是矢量值并被延迟嵌入，并且非线性被视为外部强迫。非线性项的正确表示是通过物理通知的方式或纯粹由数据驱动的方式找到的，具体取决于是否掌握控制动力学的任何知识。使用控制技术下的动态模式分解，可以找到方法中出现的未知矩阵。该方法的独特功能使其能够准确地捕获系统的动态重要不稳定模式，同时通过所包含的非线性来抑制其无限增长。证明了该方法的预测能力，适用于著名的混沌动力学原型，例如Kuramoto-Sivashinsky方程和Lorenz-96系统，对于这些方法，该方法的预测在几个Lyapunov时标上都是准确的。对于高雷诺数的二维盖驱动腔流，也显示出类似的性能。