Symmetries in a dynamical system provide an opportunity to dramatically improve the performance of data-driven models. For fluid flows, such models are needed for tasks related to design, understanding, prediction, and control. In this work we exploit the symmetries of the Navier-Stokes equations (NSE) and use simulation data to find the manifold where the long-time dynamics live, which has many fewer degrees of freedom than the full state representation, and the evolution equation for the dynamics on that manifold. We call this method ''symmetry charting''. The first step is to map to a ''fundamental chart'', which is a region in the state space of the flow to which all other regions can be mapped by a symmetry operation. To map to the fundamental chart we identify a set of indicators from the Fourier transform that uniquely identify the symmetries of the system. We then find a low-dimensional coordinate representation of the data in the fundamental chart with the use of an autoencoder. We use a variation called an implicit rank minimizing autoencoder with weight decay, which in addition to compressing the dimension of the data, also gives estimates of how many dimensions are needed to represent the data: i.e. the dimension of the invariant manifold of the long-time dynamics. Finally, we learn dynamics on this manifold with the use of neural ordinary differential equations. We apply symmetry charting to two-dimensional Kolmogorov flow in a chaotic bursting regime. This system has a continuous translation symmetry, and discrete rotation and shift-reflect symmetries. With this framework we observe that less data is needed to learn accurate data-driven models, more robust estimates of the manifold dimension are obtained, equivariance of the NSE is satisfied, better short-time tracking with respect to the true data is observed, and long-time statistics are correctly captured.

利用流体力学中的对称性，本文运用模拟数据找到了长时间动力学在较低自由度的流形上的演化方程，通过称之为''对称图表''的方法，采用傅里叶变换和自动编码器找到了数据的低维坐标表示，最后利用神经常微分方程在该流形上学习动力学。应用于二维混沌爆破流体中，该方法能够减少所需数据量，获得更稳健的流形维度估计，满足NSE的等变性，能够更准确地进行短时间跟踪并捕捉长时间的统计信息。

基于数据驱动的流体复杂流动模型中引入对称性