The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolves towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function-space approach (approximating the Koopman operator) and a state-space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require $O(10^3)$ and $O(10^5)$ degrees of freedom, respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state-space and function-space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.

中文翻译：

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不变流形上相干态动力学的数据驱动状态空间和库普曼算子模型

流体流动中复杂动力学的精确模拟需要大量的自由度，即高维状态空间。然而，由于粘性扩散而引起的小规模扰动的迅速衰减原则上允许使用显着降低的维度来表示这些流动。随着时间的推移，这种流动的动力学演化为有限维不变流形。在目前的工作中，我们仅使用直接数值模拟的数据来识别流形并确定其动力学的演化方程。我们使用先进的自动编码器框架来自动估计流形的固有维度并提供正交坐标系。然后，我们通过使用函数空间方法（近似库普曼算子）和状态空间方法（近似流形上的矢量场）确定流形上的方程来学习动力学。我们将此方法应用于柯尔莫哥洛夫流和最小流量单位管流的精确相干态。这些情况的完全解析模拟需要 $O(10^3)​​$ 和 $O(10^5)$ 我们分别建立了两个或三个自由度的模型，忠实地捕捉这些流动的动态。对于这些示例，状态空间和函数时空评估都提供了流形坐标中长期动态的高度准确的预测。