Extreme learning machine for reduced order modeling of turbulent geophysical flows

We investigate the application of artificial neural networks to stabilize proper orthogonal decomposition-based reduced order models for quasistationary geophysical turbulent flows. An extreme learning machine concept is introduced for computing an eddy-viscosity closure dynamically to incorporate the effects of the truncated modes. We consider a four-gyre wind-driven ocean circulation problem as our prototype setting to assess the performance of the proposed data-driven approach. Our framework provides a significant reduction in computational time and effectively retains the dynamics of the full-order model during the forward simulation period beyond the training data set. Furthermore, we show that the method is robust for larger choices of time steps and can be used as an efficient and reliable tool for long time integration of general circulation models.

Figure 1

A schematic of the SLFN utilized for the stabilization framework in this study. Our inputs are resolved ROM variables ($P=3$), and our prediction is a mode-dependent eddy viscosity ($J=1$).

Figure 2

POD analysis by the snapshot data for $\text{Re}=450$ and $\text{Ro}=3.6\times {10}^{-3}$. (a) The distribution of eigenvalues; (b) their energy levels with $k$ denoting the modal index.

Figure 3

Mean stream function and vorticity fields retrieved from the standard Galerkin projection method. (a) $\psi$ with $M=10$ modes; (b) $\psi$ with $M=20$ modes; (c) $\psi$ with $M=30$ modes; (d) $\omega$ with $M=10$ modes; (e) $\omega$ with $M=20$ modes; (f) $\omega$ with $M=30$ modes.

Figure 4

A comparison of the standard Galerkin approach (ROM) and the proposed ANN-based stabilized approach (ROM-ANN) for $M=10$ modes. (a) $\psi$ by FOM; (b) $\psi$ by ROM; (c) $\psi$ by ROM-ANN; (d) $\omega$ by FOM; (e) $\omega$ by ROM; (f) $\omega$ by ROM-ANN.

Figure 5

A sensitivity analysis with respect to the number of neurons in ELM using $M=10$ modes. (a) $\psi$ with $Q=20$ nodes; (b) $\psi$ with $Q=40$ nodes; (c) $\psi$ with $Q=60$ nodes; (d) $\omega$ with $Q=20$ nodes; (e) $\omega$ with $Q=40$ nodes; (f) $\omega$ with $Q=60$ nodes.

Figure 6

Time series for the first temporal coefficient ${\alpha }_1\left(t\right)$.

Figure 7

Time series for the CFL criterion assuming a fixed time step of $\mathrm{\Delta }t=2.5\times {10}^{-5}$ for the FOM simulation.

Figure 8

Mean stream function and vorticity contours obtained by the proposed ROM-ANN with various time steps. (a) $\psi$ with $\mathrm{\Delta }t=5\times {10}^{-4}$; (b) $\psi$ with $\mathrm{\Delta }t=2.5\times {10}^{-3}$; (c) $\psi$ with $\mathrm{\Delta }t={10}^{-2}$; (d) $\omega$ with $\mathrm{\Delta }t=5\times {10}^{-4}$; (e) $\omega$ with $\mathrm{\Delta }t=2.5\times {10}^{-3}$; (f) $\omega$ with $\mathrm{\Delta }t={10}^{-2}$. Note that our ROM-ANN implementation uses $M=10$ and $Q=40$.

Figure 9

Temporal mode evolution of ${\alpha }_1$ for various time steps.

Figure 10

A comparison of the standard Galerkin approach (ROM) and the proposed ANN-based stabilized approach (ROM-ANN) for $M=10$ modes at $\text{Re}=200$ and $\text{Ro}=1.6\times {10}^{-3}$ where the training data have been extracted from the snapshots at $\text{Re}=450$ and $\text{Ro}=3.6\times {10}^{-3}$. (a) $\psi$ by FOM; (b) $\psi$ by ROM; (c) $\psi$ by ROM-ANN; (d) $\omega$ by FOM; (e) $\omega$ by ROM; (f) $\omega$ by ROM-ANN. Note that the ROM-ANN uses $Q=40$ hidden nodes.

Figure 11

Time series for of first temporal coefficient ${\alpha }_1\left(t\right)$ for the out-of-sample forecast. Predictive performance is shown for $\text{Re}=200$ and $\text{Ro}=1.6\times {10}^{-3}$, while the training has been performed using the data generated at $\text{Re}=450$ and $\text{Ro}=3.6\times {10}^{-3}$.