Physical invariance in neural networks for subgrid-scale scalar flux modeling

In this paper we present a new strategy to model the subgrid-scale scalar flux in a three-dimensional turbulent incompressible flow using physics-informed neural networks (NNs). When trained from direct numerical simulation (DNS) data, state-of-the-art neural networks, such as convolutional neural networks, may not preserve well-known physical priors, which may in turn question their application to real case-studies. To address this issue, we investigate hard and soft constraints into the model based on classical transformation invariances and symmetries derived from physical laws. From simulation-based experiments, we show that the proposed transformation-invariant NN model outperforms both purely data-driven ones as well as parametric state-of-the-art subgrid-scale models. The considered invariances are regarded as regularizers on physical metrics during the a priori evaluation and constrain the distribution tails of the predicted subgrid-scale term to be closer to the DNS. They also increase the stability and performance of the model when used as a surrogate during a large-eddy simulation. Moreover, the transformation-invariant NN is shown to generalize to regimes that have not been seen during the training phase.

Figure 1

Evolution of the Reynolds number on the Taylor microscale (left) and scalar variance (right). The flow is considered in an established turbulence regime at iteration $\mathcal{S}=20000$. The first 50 samples are used for training, and the remaining 40 samples are used for validation.

Figure 2

Evolution of the Reynolds number on the Taylor microscale (left) and scalar variance (right) in the different regimes (developed turbulence, transitional and scalar decay) for the testing data.

Figure 3

Data slices extracted during the simulation. $\overline{\mathbf{u}}$ are the velocities in each directions, $\overline{\mathrm{\Phi }}$ is the scalar quantity, and ${\mathcal{M}}_{\mathrm{DNS}}$ is the SGS term. The simulation was run with $N=512$ grid points and filtered at $N=64$. The flow reaches a statistically developed turbulence state approximately at iteration $\mathcal{S}=20000$.

Figure 4

Illustration of the subgrid transport neural network architecture (SGTNN). At the top, two nonlearnable steps are applied; the periodic pad replicates boundaries according to the kernel width of convolution kernel and std transforms the velocities to ensures Galilean invariance. Then the two submodels $\mathcal{V}$ (right) for the velocity and $\mathcal{T}$ (left) for the scalar transport are represented in dotted and dashed lines, respectively. The convolution with kernel size $K=1$ corresponding to ${\mathcal{C}}^{+}$ applied to the combined result of the latter is depicted at the bottom. The number of filters within the NN is shown along with the arrows.

Figure 5

Probability distribution function (left), cumulative distribution function (middle), and quantiles-quantiles (right) of the SGS term in developed turbulence regime over the entire testing data. Both distribution function tails are made explicitly visible using log scale on the $y$-axis.

Figure 6

Time evolution of the different metrics in transitional regime. Normalized root-mean-squared error (top left), Pearson's coefficient (top right), Jensen-Shannon distance (bottom left), and integral dissipation error (bottom right).

Figure 7

Data slices of the predicted SGS terms from NN models in transitional regime. It is particularly visible that ${\mathcal{M}}_{\mathrm{MLP}}$ is producing a non-null SGS term during the first iterations, while ${\mathcal{M}}_{\mathrm{CNN}}$ quickly produces large localized values.

Figure 8

Time evolution of the different metrics in scalar decay regime. Normalized root-mean-squared error (top left), Pearson's coefficient (top right), Jensen-Shannon distance (bottom left), and integral dissipation error (bottom right).

Figure 9

Data slices of the predicted SGS terms from NN models in scalar decay regime. It is particularly visible that ${\mathcal{M}}_{\mathrm{MLP}}$ and ${\mathcal{M}}_{\mathrm{CNN}}$ are still predicting a non-null SGS term when the scalar is completely mixed.

Figure 10

Statistics of simulation in developed turbulence regime starting from training data. Scalar variance (left), resolved scalar enstrophy (right).

Figure 11

Energy spectrum of simulation in developed turbulence regime starting from training data at $t=12.5$ (left) and $t=25$ (right).

Figure 12

Statistics of simulation in developed turbulence regime starting from testing data. Scalar variance (left), resolved scalar enstrophy (right).

Figure 13

Energy spectrum of simulation in developed turbulence regime starting from testing data at $t=12.5$ (left) and $t=25$ (right).

Figure 14

Statistics of simulation in scalar transition regime, starting from a newly initialized field. Scalar variance (left), resolved scalar enstrophy (right).

Figure 15

Energy spectrum of simulation in scalar transition regime, starting from a newly initialized field at $t=6$ (left) and $t=25$ (right).

Figure 16

Statistics of simulation in full decay regime starting from testing data. Scalar variance (left), resolved scalar enstrophy (right).

Figure 17

Energy spectrum of simulation in full decay regime starting from testing data at $t=8$ (left) and $t=30$ (right).