Deep learning of turbulent scalar mixing

Based on recent developments in physics-informed deep learning and deep hidden physics models, we put forth a framework for discovering turbulence models from scattered and potentially noisy spatiotemporal measurements of the probability density function (PDF). The models are for the conditional expected diffusion and the conditional expected dissipation of a Fickian scalar described by its transported single-point PDF equation. The discovered models are appraised against an exact solution derived by the amplitude mapping closure (AMC)–Johnson-Edgeworth translation (JET) model of binary scalar mixing in homogeneous turbulence.

Figure 1

Conditional expected diffusion network: A plain vanilla, densely connected (physics-uninformed) neural network with 10 hidden layers and 50 neurons per hidden layer per output variable (i.e., $2\times 50=100$ neurons per hidden layer) takes the input variables $t$ and $\psi$ while outputting $P$ and $D$. As for the activation functions, we use $\sigma \left(x\right)=x\text{sigmoid}\left(x\right)$ known in the literature as Swish. For illustration purposes only, the network depicted in this figure comprises two hidden layers and five neurons per set of hidden layers. We employ automatic differentiation to obtain the required derivatives to compute the residual (physics-informed) network $R$. The total loss function is composed of the regression loss of the probability density function $P$ and the loss imposed by the partial differential equation $R$. Here, $I$ denotes the identity operator, and the differential operators ${\partial }_t$ and ${\partial }_{\psi }$ are computed using automatic differentiation and can be thought of as “activation operators.” Moreover, the gradients of the loss function are back-propagated through the entire network to train the neural network parameters using the Adam optimizer.

Figure 2

Conditional expected dissipation network: A plain vanilla, densely connected (physics-uninformed) neural network, with 10 hidden layers and 50 neurons per hidden layer per output variable (i.e., $2\times 50=100$ neurons per hidden layer), takes the input variables $t$ and $\psi$ while outputting $P$ and $\mathcal{E}$. As for the activation functions, we use $\sigma \left(x\right)=x\text{sigmoid}\left(x\right)$, known in the literature as Swish. For illustration purposes only, the network depicted in this figure comprises two hidden layers and five neurons per set of hidden layers. We employ automatic differentiation to obtain the required derivatives to compute the residual (physics-informed) network $S$. If a term does not appear in the blue boxes (e.g., $\frac{{\partial }^{2}P}{\partial t^2}$ or $\frac{{\partial }^{2}P}{\partial t\partial \psi }$), its coefficient is assumed to be zero. It is worth emphasizing that unless the coefficient in front of a term is nonzero, that term is not going to appear in the actual “compiled” computational graph and is not going to contribute to the computational cost of a feed-forward evaluation of the resulting network. The total loss function is composed of the regression loss of the probability density function $P$ and the loss imposed by the differential equation $S$. Here, $I$ denotes the identity operator, and the differential operators ${\partial }_t$ and ${\partial }_{\psi }$ are computed using automatic differentiation and can be thought of as “activation operators.” Moreover, the gradients of the loss function are back-propagated through the entire network to train the neural network parameters using the Adam optimizer.

Figure 3

Conditional expected diffusion: The exact probability density function $P(t,\psi )$ alongside the learned one is depicted in the top panels, while the exact and learned conditional expected diffusion $D(t,\psi )$ are plotted in the bottom panels. It is worth highlighting that the algorithm has seen no data whatsoever on the diffusion coefficient.

Figure 4

The relative $L_2$ error as a function of time for probability density function $P(t,\psi )$ (left) and conditional expected diffusion $D(t,\psi )$ (right).

Figure 5

Conditional expected dissipation: The exact probability density function $P(t,\psi )$ alongside the learned one is depicted in the top panels, while the exact and learned conditional expected dissipation $\mathcal{E}(t,\psi )$ are plotted in the bottom panels. It is worth highlighting that the algorithm has seen no data whatsoever on the dissipation coefficient.

Figure 6

The relative $L_2$ error as a function of time for probability density function $P(t,\psi )$ (left) and conditional expected dissipation $\mathcal{E}(t,\psi )$ (right).