Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction

In this paper, we propose a variational approach to estimate eddy viscosity using forward sensitivity method (FSM) for closure modeling in nonlinear reduced order models. FSM is a data assimilation technique that blends model's predictions with noisy observations to correct initial state and/or model parameters. We apply this approach on a projection based reduced order model (ROM) of the one-dimensional viscous Burgers equation with a square wave defining a moving shock, and the two-dimensional vorticity transport equation formulating a decay of Kraichnan turbulence. We investigate the capability of the approach to approximate an optimal value for eddy viscosity with different measurement configurations. Specifically, we show that our approach can sufficiently assimilate information either through full-field or sparse noisy measurements to estimate eddy viscosity closure to cure standard Galerkin reduced order model (GROM) predictions. Therefore, our approach provides a modular framework to correct forecasting error from a sparse observational network on a latent space. We highlight that the proposed GROM-FSM framework is promising for emerging digital twin applications, where real-time sensor measurements can be used to update and optimize surrogate model's parameters.

Figure 1

Evolution of the FOM velocity field, characterized by a moving shock with square wave.

Figure 2

Noisy measurement of velocity fields at $t=0.25$ s and $t=0.50$ s, assuming sensors are located at all grid points.

Figure 3

Temporal evolution of POD coefficients, assuming full-field measurements are available.

Figure 4

Velocity field reconstruction in case of full-field measurements. Top: reconstruction of final velocity field using GROM and GROM with FSM eddy viscosity compared to the FOM and true projection fields. Bottom: RMSE of reconstructed fields at different time instants.

Figure 5

Noisy measurement of velocity fields at $t=0.25$ s and $t=0.50$ s, assuming sensors are located at eight grid points.

Figure 6

Temporal evolution of POD coefficients, where sparse-field measurements are preprocessed to estimate the observed POD coefficients.

Figure 7

Velocity field reconstruction in case of preprocessing sparse-field measurements to compute the observed POD coefficients. Top: reconstruction of final velocity field using GROM and GROM with FSM eddy viscosity compared to the FOM and true projection fields. Bottom: RMSE of reconstructed fields at different time instants.

Figure 8

Temporal evolution of POD coefficients, where sparse-field measurements are compared against POD field reconstruction using the observer operator $\mathbf{C}$.

Figure 9

Velocity field reconstruction where sparse-field measurements are compared against POD field reconstruction using the observer operator $\mathbf{C}$. Top: reconstruction of final velocity field using GROM and GROM with FSM eddy viscosity compared to the FOM and true projection fields. Bottom: RMSE of reconstructed fields at different time instants.

Figure 10

Surface plots for the temporal evolution of velocity fields from (a) FOM, (b) true projection, (c) GROM and FSM with (d) full and (e, f) sparse measurements configurations. Note: (e) shows the results using the method presented in Sec. 5a2a, while (f) refers to the method presented in Sec. 5a2b.

Figure 11

FOM results showing the time evolution of vorticity fields for the 2D turbulence problem: (a) $t=0.0$, (b) $t=2.0$, and (c) $t=4.0$.

Figure 12

Time evolution of the modal coefficients for the 2D turbulence case, assuming full-field measurements are available at $t=1$ and $t=2$.

Figure 13

Reconstructed vorticity fields for the 2D turbulence problem at $t=4$ for (a) true projection, (b) GROM, and (c) GROM-FSM along with the RMSE variation with time (d), assuming full-field measurements are available at $t=1$ and $t=2$.

Figure 14

Time evolution of the modal coefficients for the 2D turbulence case, with full-field measurements and mode-dependent eddy viscosity closure. Note the equal improvements in predictions for all the modes.

Figure 15

Reconstructed vorticity fields for the 2D turbulence problem at $t=4$ for (a) true projection, (b) GROM, and (c) GROM-FSM along with the RMSE variation with time (d), with full-field measurements and mode-dependent eddy viscosity closure. Note the significant decrease in RMSE at earlier times than those in Fig. 13.

Figure 16

Time evolution of the modal coefficients for the 2D turbulence case, with sparse-field measurements and fixed eddy viscosity closure. Note the better improvements in the first few modes compared to the latest ones.

Figure 17

Reconstructed vorticity fields for the 2D turbulence problem at $t=4$ for (a) true projection, (b) GROM, and (c) GROM-FSM along with the RMSE variation with time (d), with sparse-field measurements and fixed eddy viscosity closure. Reduction of RMSE compared to GROM without closure starts to become remarkable around $t=2$.

Figure 18

Time evolution of the modal coefficients for the 2D turbulence case, with sparse-field measurements and mode-dependent eddy viscosity closure.

Figure 19

Reconstructed vorticity fields for the 2D turbulence problem at $t=4$ for (a) true projection, (b) GROM, and (c) GROM-FSM along with the RMSE variation with time (d), with sparse-field measurements and mode-dependent eddy viscosity closure.