Turbulence model reduction by deep learning

A central problem of turbulence theory is to produce a predictive model for turbulent fluxes. These have profound implications for virtually all aspects of the turbulence dynamics. In magnetic confinement devices, drift-wave turbulence produces anomalous fluxes via cross-correlations between fluctuations. In this work, we introduce an alternative, data-driven method for parametrizing these fluxes. The method uses deep supervised learning to infer a reduced mean-field model from a set of numerical simulations. We apply the method to a simple drift-wave turbulence system and find a significant new effect which couples the particle flux to the local gradient of vorticity. Notably, here, this effect is much stronger than the oft-invoked shear suppression effect. We also recover the result via a simple calculation. The vorticity gradient effect tends to modulate the density profile. In addition, our method recovers a model for spontaneous zonal flow generation by negative viscosity, stabilized by nonlinear and hyperviscous terms. We highlight the important role of symmetry to implementation of the alternative method.

Figure 1

Diffusive part of the learned particle flux, i.e., the flux at fixed $U=U^{\prime }=U^{\prime \prime }=0$, as a function of $N^{\prime }$ and $\varepsilon$. The dependence on $N^{\prime }$ may be summarized as linear, plus saturation effects at large $N^{\prime }$.

Figure 2

Nondiffusive part of the learned particle flux, i.e., the flux at fixed $N^{\prime }=U=U^{\prime \prime }=0$, as a function of $U^{\prime }$ and $\varepsilon$. Again, the dependence on $U^{\prime }$ is roughly linear plus saturation effects.

Figure 3

Dependence of particle flux on both gradients: flux at fixed $U=U^{\prime \prime }=0$ and fixed $\varepsilon =20$, as a function of $N^{\prime }$ and $U^{\prime }$.

Figure 4

Plot of learned Reynolds stress against vorticity $U,$ at fixed $N^{\prime }=2$ and $U^{\prime }=U^{\prime \prime }=0$ and several values of the intensity. Near $U=0,$ the behavior is that of a negative viscosity.

Figure 5

Plot of learned Reynolds stress against $N^{\prime }$ at fixed $U=1, \varepsilon =10, U^{\prime \prime }=0$, and several values of $U^{\prime }$. The presence of a gradient in $U^{\prime }$ or $N^{\prime }$ tends to reduce the Reynolds stress.

Figure 6

Plot of learned Reynolds stress against vorticity $U$ at fixed $N^{\prime }=2,U^{\prime }=0, \varepsilon =20$, and several values of $U^{\prime \prime }$. The leading order contribution from $U^{\prime \prime }$ is a stabilizing linear term.

Figure 7

Plot of learned Reynolds stress against $U^{\prime \prime }$ at fixed $N^{\prime }=2,U=U^{\prime }=0$, and several values of the intensity. We should have $\mathrm{\Pi }\to -\mathrm{\Pi }$ under $U^{\prime \prime }\to -U^{\prime \prime }$ here, but the model fails to precisely learn this, which may be attributed to the relatively small contribution to the loss function from the hyperdiffusion term. However, it is clear that this term scales roughly as $\varepsilon U^{\prime \prime }$.