We perform a sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized complex flow phenomena. We first apply the SINDy with two regression methods, the thresholded least square algorithm and the adaptive least absolute shrinkage and selection operator which show reasonable ability with a wide range of sparsity constant in our preliminary tests, to a two-dimensional single cylinder wake at $Re_D=100$ , its transient process and a wake of two-parallel cylinders, as examples of high-dimensional fluid data. To handle these high-dimensional data with SINDy whose library matrix is suitable for low-dimensional variable combinations, a convolutional neural network-based autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a high-dimensional dynamics into a low-dimensional latent space. The SINDy then seeks a governing equation of the mapped low-dimensional latent vector. Temporal evolution of high-dimensional dynamics can be provided by combining the predicted latent vector by SINDy with the CNN decoder which can remap the low-dimensional latent vector to the original dimension. The SINDy can provide a stable solution as the governing equation of the latent dynamics and the CNN-SINDy-based modelling can reproduce high-dimensional flow fields successfully, although more terms are required to represent the transient flow and the two-parallel cylinder wake than the periodic shedding. A nine-equation turbulent shear flow model is finally considered to examine the applicability of SINDy to turbulence, although without using CNN-AE. The present results suggest that the proposed scheme with an appropriate parameter choice enables us to analyse high-dimensional nonlinear dynamics with interpretable low-dimensional manifolds.

中文翻译：

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用低维流表示的非线性动力学的稀疏识别

我们对低维复杂流动现象进行非线性动力学（SINDy）的稀疏识别。我们首先将 SINDy 与两种回归方法，阈值最小二乘算法和自适应最小绝对收缩和选择算子在我们的初步测试中显示出合理的能力和广泛的稀疏常数，应用于二维单圆柱尾流 $Re_D=100$ ，它的瞬态过程和两个平行圆柱体的尾流，作为高维流体数据的例子。为了使用 SINDy 处理这些高维数据，其库矩阵适用于低维变量组合，使用了基于卷积神经网络的自动编码器 (CNN-AE)。CNN-AE 用于将高维动态映射到低维潜在空间。SINDy 然后寻找映射的低维潜在向量的控制方程。通过将 SINDy 预测的潜在向量与可以将低维潜在向量重新映射到原始维度的 CNN 解码器相结合，可以提供高维动态的时间演化。SINDy 可以提供一个稳定的解决方案，因为潜在动力学的控制方程和基于 CNN-SINDy 的建模可以成功地再现高维流场，尽管需要更多的项来表示瞬态流和两个平行圆柱尾流。周期性脱落。尽管没有使用 CNN-AE，但最终考虑使用九方程湍流剪切流模型来检查 SINDy 对湍流的适用性。目前的结果表明，所提出的具有适当参数选择的方案使我们能够分析具有可解释的低维流形的高维非线性动力学。尽管没有使用 CNN-AE，但最终考虑使用九方程湍流剪切流模型来检查 SINDy 对湍流的适用性。目前的结果表明，所提出的具有适当参数选择的方案使我们能够分析具有可解释的低维流形的高维非线性动力学。尽管没有使用 CNN-AE，但最终考虑使用九方程湍流剪切流模型来检查 SINDy 对湍流的适用性。目前的结果表明，所提出的具有适当参数选择的方案使我们能够分析具有可解释的低维流形的高维非线性动力学。