Reservoir computing model of two-dimensional turbulent convection

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.

Figure 1

Mean profiles of (a) temperature and (b) velocity components. The average quantities are obtained by averaging in time and in the homogeneous direction $x$. The diffusive equilibrium temperature profile $T_{\mathrm{lin}}\left(y\right)$ is added to panel (a).

Figure 2

Temporal variation of Nusselt number for bottom and top plates obtained in the spectral element DNS. We start sampling our data for times $t>100T_f$ to exclude the initialization effects which are indicated by the green-shaded region.

Figure 3

Eigenvalue spectrum of POD modes as obtained from the analysis of 2000 DNS snapshots. (a) Individual contribution of each mode, and (b) cumulative contribution of modes. The shaded region shows the contribution of the first 150 modes which capture $83\%$ of the total energy of the convection flow and thus provide a good approximation of the large-scale structure.

Figure 4

Spatial structure of two POD modes. Temperature fluctuation field $T^{\prime }(x,y,t_0)$ snapshot taken at time $t_0$ and resulting by projection onto (a) ${\mathrm{\Phi }}_3^{\left(1\right)}(x,y)$ and (b) ${\mathrm{\Phi }}_3^{\left(50\right)}(x,y)$. Horizontal velocity component $u_x(x,y,t_0)$ by projection onto (c) ${\mathrm{\Phi }}_1^{\left(1\right)}(x,y)$ and (d) ${\mathrm{\Phi }}_1^{\left(50\right)}(x,y)$. Wall-normal velocity component $u_y(x,y,t_0)$ by projection onto (e) ${\mathrm{\Phi }}_2^{\left(1\right)}(x,y)$ and (f) ${\mathrm{\Phi }}_2^{\left(50\right)}(x,y)$. These modes are shown in the simulation domain and were obtained by the inverse Fourier transform.

Figure 5

The basic architecture of a reservoir computing model, RCM (or echo state network ESN), consisting of an input layer, a reservoir, and an output layer. Dotted arrows mark connections which remain fixed during the training. The reservoir consists of $N$ nodes and both the input and output layers have $i=1,\cdots ,N_{\mathrm{POD}}$ units. The reservoir substitutes the stack of hidden layers in a deep neural network.

Figure 6

Temporal evolution of four POD modes coefficients. From top to bottom: (a), (b) $a_1\left(t\right)$; (c), (d) $a_{50}\left(t\right)$; (e), (f) $a_{100}\left(t\right)$; and (g), (h) $a_{150}\left(t\right)$. The green shaded range shows the training phase while the other range depicts the test phase. Here $n_s=1,\cdots ,N_s=1400$ is the snapshot index which translates into the time $t=(n_s/4)T_f$ in a time instant. The corresponding panels in the right column show the initial phase of the forecast. The distance between two snapshots corresponds to 50 integration time steps of the original DNS model.

Figure 7

Temporal evolution of the error for RCM and GT. (a) Absolute error $|a_j^{\mathrm{GT}}-a_j^{\mathrm{RCM}}|$ shown for four individual POD mode coefficients with the mode number indicated by the legend. (b) Mean square error MSE($n_s$) versus number of snapshots $n_s$ [see also Eq. (23)].

Figure 8

Comparison of the turbulence fields in the test phase for $n_s=201$. (a), (d), (g) Original DNS snapshot. (b), (e), (h) POD model reconstruction with the 150 most energetic modes which serves as the ground truth for our machine learning problem. (c), (f), (i) Predicted field from the reservoir computing model. In panels (a), (b), (c) the instantaneous temperature field $T(x,y,t_0)$ is shown, in (d), (e), (f) the velocity component $u_x(x,y,t_0)$, and in (g), (h), (i) the wall-normal velocity component $u_y(x,y,t_0)$.

Figure 9

Comparison line- and time-averaged mean profiles obtained from original DNS, truncated POD, and RCM. (a) Mean temperature, (b) mean turbulent convective heat flux, and (c) root-mean-square fluctuations of temperature. The profiles in case of the RCM were obtained from the test data set only and thus not used in the training phase.

Figure 10

Comparison of energy spectra for temperature variance at different distances from the wall: (a) $y_0=0.07$, (b) $y_0=0.15$, and (c) $y_0=0.50$. The legend indicates the three data sets taken for the calculation. Line colors are the same for all three panels.

Figure 11

Comparison of probability density functions (PDFs) of the local convective heat flux ($u_y^{\prime }{T}^{\prime }$) at (a) $y_0=0.07$, (b) $y_0=0.15$, and (c) $y_0=0.50$. The legend indicates the three data sets taken for the calculation. Line colors are the same for all three panels.

Figure 12

Comparison of joint probability density functions of the turbulent vertical velocity component and the turbulent temperature fluctuation, $P(u_y^{\prime },T^{\prime })$, at the midline at $y_0=0.50$: (a) DNS, (b) POD, and (c) RCM. The isocontour levels in the legend are logarithmically spaced and hold for all three panels.