Reservoir computing is an efficient implementation of a recurrent neural network that can describe the evolution of a dynamical system by supervised machine learning without solving the underlying mathematical equations. In this work, reservoir computing is applied to model the large-scale evolution and the resulting low-order turbulence statistics of a two-dimensional turbulent Rayleigh-Bénard convection flow at a Rayleigh number $\mathrm{Ra}={10}^7$ and a Prandtl number $\mathrm{Pr}=7$ in an extended spatial domain with an aspect ratio of 6. Our data-driven approach, which is based on a long-term direct numerical simulation of the convection flow, comprises a two-step procedure: (1) reduction of the original simulation data by a proper orthogonal decomposition (POD) snapshot analysis and subsequent truncation to the first 150 POD modes which are associated with the largest total energy amplitudes; (2) setup and optimization of a reservoir computing model to describe the dynamical evolution of these 150 degrees of freedom and thus the large-scale evolution of the convection flow. The quality of the prediction of the reservoir computing model is comprehensively tested by a direct comparison of the results of the original direct numerical simulations and the fields that are reconstructed by means of the POD modes. We find a good agreement of the vertical profiles of mean temperature, mean convective heat flux, and root-mean-square temperature fluctuations. In addition, we discuss temperature variance spectra and joint probability density functions of the turbulent vertical velocity component and temperature fluctuation, the latter of which is essential for the turbulent heat transport across the layer. At the core of the model is the reservoir, a very large sparse random network characterized by the spectral radius of the corresponding adjacency matrix and a few further hyperparameters which are varied to investigate the quality of the prediction. Our work demonstrates that the reservoir computing model is capable of modeling the large-scale structure and low-order statistics of turbulent convection, which can open new avenues for modeling mesoscale convection processes in larger circulation models.

中文翻译：

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二维湍流对流储层计算模型

储层计算是递归神经网络的有效实现，该神经网络可以通过有监督的机器学习来描述动力学系统的演化，而无需求解基础的数学方程式。在这项工作中，油藏计算被用于对二维湍流瑞利-贝纳德对流在瑞利数下的大规模演化和由此产生的低阶湍流统计进行建模。$镭={10}^7$ 和一个Prandtl号码 $镨=7$在长宽比为6的扩展空间域中。我们的数据驱动方法基于对流的长期直接数值模拟，包括两个步骤：（1）减少原始模拟数据通过适当的正交分解（POD）快照分析并随后截断与最大总能量振幅相关的前150个POD模式；（2）建立和优化储层计算模型，以描述这150个自由度的动态演化，从而描述对流的大规模演化。通过直接比较原始直接数值模拟的结果和借助POD模式重建的油田，对储层计算模型的预测质量进行了全面测试。我们发现平均温度，平均对流热通量和均方根温度波动的垂直分布曲线吻合良好。此外，我们讨论了湍流垂直速度分量和温度波动的温度变化谱和联合概率密度函数，后者对于湍流在整个层中的热传递是必不可少的。模型的核心是储层，它是一个非常大的稀疏随机网络，其特征在于相应邻接矩阵的光谱半径以及一些其他的超参数，这些参数通过变化来研究预测的质量。我们的工作表明，油藏计算模型能够对湍流对流的大规模结构和低阶统计量进行建模，