Artificial neural network subgrid models of 2D compressible magnetohydrodynamic turbulence

We explore the suitability of deep learning to capture the physics of subgrid-scale ideal magnetohydrodynamics turbulence of 2D simulations of the magnetized Kelvin-Helmholtz instability. We produce simulations at different resolutions to systematically quantify the performance of neural network models to reproduce the physics of these complex simulations. We compare the performance of our neural networks with gradient models, which are extensively used in the magnetohydrodynamic literature. Our findings indicate that neural networks significantly outperform gradient models in accurately computing the subgrid-scale tensors that encode the effects of magnetohydrodynamics turbulence. To the best of our knowledge, this is the first exploratory study on the use of deep learning to learn and reproduce the physics of magnetohydrodynamics turbulence.

Figure 1

Schematic illustration of a neural network. A multilayer perceptron with two hidden layers is presented. Circles represent neurons, whereas arrows correspond to weights.

Figure 2

Plots of the mass density distribution $\rho$ time slices at $t=1.5$, 5, 10 of our magnetized KHI simulations with a resolution of $N=2048^2$. On the left, we have the $t=1.5$ time slice, in which we can observe vortexes begin to form between the two fluids. The number and size of these vortexes are controlled by an initial sinusoidal perturbation of the fluid velocity in the $y$ direction. The center plot at $t=5$ occurs after many of the aforementioned vortexes have merged together, at which point the flow has become unsteady. This unsteady flow manifests itself in the high-density fluid beginning to break apart. The rightmost plot at $t=10$ depicts the fluids in a turbulent mixing process with two low-density vortexes helping to drive this mixing.

Figure 3

Plots of the density distribution $\rho$ of the test dataset at time slices $t=9.25$ for resolutions $N=512^2$, $1048^2$, and $2048^2$ in the left, middle, and right image, respectively. All three datasets appear to be undergoing a turbulent mixing process at $t=9.25$. We observe that, while the $N=1024^2$ and $N=2048^2$ runs appear to share many of the same general characteristics, the $N=512^2$ run fails to reproduce these same features. This failure implies that $N=512^2$ is not enough to capture the turbulent effects of the magnetized KHI without a SGS model. We note that the testing time slice is of particular importance, because it was used to evaluate the SGS models.

Figure 4

Plots of the energy spectra $\mathcal{E}(k)$ at various time steps throughout the simulation for each resolution. The kinetic energy spectra ${\mathcal{E}}_{\text{kin}}(k)$ is given by the solid lines, while the magnetic energy spectra ${\mathcal{E}}_{\text{mag}}(k)$ is denoted by the dashed lines. The resolutions $N=512^2$, $1024^2$, and $2048^2$ are given by the blue, red, and green lines, respectively. The top left, top right, and bottom left images are taken at approximately $t=1.5$, 5, and 10, respectively, and correspond to the time steps displayed of the density distribution plots in Fig. 2. The bottom right plot provides the spectra of the test dataset used for evaluating the models whose density distribution is featured in Fig. 3.

Figure 5

Time evolution of the total kinetic energy $E_{\text{kin}}$ (left panel) and total magnetic energy $E_{\text{mag}}$ (right panel). The kinetic energy decreases over time as it is being converted into magnetic energy. This energy conversion can be seen in the right panel, where $E_{\text{mag}}$ increases over time. We notice that $E_{\text{kin}}$ decreases more rapidly for the high-resolution runs while $E_{\text{mag}}$ increases more rapidly. This would indicate that this conversion from $E_{\text{kin}}$ to $E_{\text{mag}}$ occurs most efficiently at small scales that high-resolution simulations can best resolve.

Figure 6

Predictions vs target values of SGS tensors for the $N=2048^2$ resolution test dataset with filter size $f=8$ for each of our SGS models. The small transparent green circles represent values of the gradient model and are overlaid on top of the small transparent blue circles, which represent the values of the ANN model. The black dashed line depicts a perfect one-to-one matching between the targets and the predictions. The first row displays the $xx$, $xy$, and $yy$ components of ${\tau }_{\text{kin}}$ from left to right, respectively. The middle row shows the same information for the components of the ${\tau }_{\text{mag}}$ SGS tensor. The bottom row shows the $xy$ component of ${\tau }_{\text{ind}}$ on the left, the $x$ component of ${\tau }_{\text{enth}}$ in the middle, and the $y$ component of ${\tau }_{\text{enth}}$ on the right. We observe that the blue circles of the ANN model appear much closer to the dashed black line for high SGS tensor values than the green circles of the gradient model, especially in the ${\tau }_{\text{mag}}$ and ${\tau }_{\text{ind}}$ SGS tensors.

Figure 7

Histogram of the normalized probability distribution of the values of SGS tensors for the $N=2048^2$ resolution test dataset with filter size $f=8$ for each of our SGS models. The black dot-dashed line represents the distribution of the DNS data, the solid blue line represents the distribution of the ANN model predictions, and the green dashed line represents the distribution of the gradient model predictions. The first row displays the $xx$, $xy$, and $yy$ components of ${\tau }_{\text{kin}}$ from left to right, respectively. The middle row shows the same information for the components of the ${\tau }_{\text{mag}}$ SGS tensor. The bottom row shows the $xy$ component of ${\tau }_{\text{ind}}$ on the left, the $x$ component of ${\tau }_{\text{enth}}$ in the middle, and the $y$ component of ${\tau }_{\text{enth}}$ on the right. We observe that the ANN model predictions more closely resemble the distribution of the DNS values than those of the gradient model.

Figure 8

Plots of the components of the ${\tau }_{\text{mag}}$ SGS tensor of the test dataset for the $N=2048^2$ resolution run with a filter size of $f=8$. The columns depict the SGS tensor values of the exact DNS calculation, the ANN model reconstruction, and the gradient model reconstruction from left to right, respectively. The top, middle, and bottom rows display the $xx$, $xy$, and $yy$ components, respectively. We observe that, while the ANN model appears to reproduce most of the visual features of the DNS calculation, the gradient model appears to struggle in regions with more detailed structure.

Figure 9

Here we plot the correlation coefficient $C$ as a function of filter size $f$ for all resolutions $N$, SGS tensor components $\tau$, and SGS models. The solid lines refer to the ANN model, and the dashed lines refer to the gradient model. The resolutions are given by the color of the line; blue represents the $N=512^2$ simulation, red represents the $N=1024^2$ simulation, and green represents the $N=2048^2$ simulation. We observe that the ANN model has a higher correlation coefficient than the gradient model for all SGS tensor components at all resolutions. We also note that $C$ generally decreases with increasing $f$, but this decay affects the gradient model more significantly.

Figure 10

Here we plot the relative error $E$ as a function of filter size $f$ for all resolutions $N$, SGS tensor components $\tau$, and SGS models. The solid lines refer to the ANN model, and the dashed lines refer to the gradient model. The resolutions are given by the color of the line; blue represents the $N=512^2$ simulation, red represents the $N=1024^2$ simulation, and green represents the $N=2048^2$ simulation. We observe that $E$ is lower for the ANN model than for the gradient model for all SGS tensor components at all resolutions. We also note that $E$ generally increases with $f$ but is more severe for the gradient model than for the ANN model.

Figure 11

Here we plot the rms value as a function of filter size $f$ for all resolutions $N$, SGS tensor components $\tau$, and SGS models including the DNS calculation. The solid lines refer to the ANN model, the dashed lines refer to the gradient model, and the dash-dotted lines refer to the DNS calculation. The resolutions are given by the color of the line; blue represents the $N=512^2$ simulation, red represents the $N=1024^2$ simulation, and green represents the $N=2048^2$ simulation. We observe that the rms values are fairly similar for the ${\tau }_{\text{kin}}$ SGS tensor components (top row) for both SGS models and the DNS result. For the SGS tensors ${\tau }_{\text{mag}}$ (middle row), ${\tau }_{\text{ind}}$ (bottom right), and ${\tau }_{\text{enth}}$ (bottom middle and left), the ANN model has a much closer rms value to the DNS result compared to the gradient model.

Figure 12

The same as Fig. 8 for the SGS tensor components of ${\tau }_{\text{kin}}$.

Figure 13

The same as Fig. 8 for the SGS tensor components of ${\tau }_{\text{ind}}$.

Figure 14

The same as Fig. 8 for the SGS tensor components of ${\tau }_{\text{enth}}$.