Results
This page contains visual results for the paper by McQuarrie et al..
The combustion problem exhibits rich and complex dynamics that are difficult to capture with POD-based reduced-order modeling techniques. Therefore, the reduced-order models produced by regularized Operator Inference are sensitive to the amount of data used in the learning. In this section, we compare ROMs learned through Operator Inference with the first k = 10,000, 20,000, and 30,000 GEMS snapshots as training data. For the dimension r of each ROM (the number of POD modes), we select the smallest integer such that the cumulative energy exceeds 98.5%. Appropriate regularization hyperparameters λ1 and λ2 are chosen via Algorithm 1 of [1].
| k | r | λ1 | λ2 | |
|---|---|---|---|---|
| ROM 1 | 10,000 | 22 | 91 | 32251 |
| ROM 2 | 20,000 | 43 | 316 | 18199 |
| ROM 3 | 30,000 | 66 | 105 | 27906 |
The plots in this section show several learning variable in time at the four monitoring locations indicated below.
While it can be misleading to assess accuracy based on predictions at a single spatial point, the following plots show that the pressure and velocity frequencies are well captured throughout the time domain, though the amplitudes are sometimes less accurate in the prediction regime.
Pressure
x-velocity
y-velocity
The following plots show spatial averages and integrals of the learning variables as functions of time, which give a more global sense of the ROM predictive accuracy and the predicted chemical reaction rate. In each case, the ROMs are able to accurately re-predict the training data and capture much of the overall system behavior in the prediction phase, with slightly more training error as the number of snapshots increases.
Spatial Averages
Species Concentration Integrals
We also compare the three Operator Inference ROMs to the GEMS output over the entire domain by animating each learning variable in time. As with the point traces shown earlier, we see that the ROMs have impressive accuracy over the training region, but lose accuracy as they attempt to predict dynamics beyond the training horizon. However, many of the coherent features are reasonably predicted, especially the recirculation zone dynamics near the dump plane (x = 0). Below we display previews of animations for pressure, temperature x-velocity (horizontal), and methane.
The ROMs produced by regularized Operator Inference hold up well in comparison to more sophisticated methods. In this section, we compare an Operator Inference (OpInf) ROM trained to a state-of-the-art least-squares Petrov-Galerkin POD-DEIM ROM that directly queries the GEMS code for nonlinear residual evaluations. Both ROMs are trained with k = 20,000 snapshots and the minimal number of POD modes r needed achieve 98.5% cumulative energy. Operator Inference represents a significant speedup over POD-DEIM because it is completely independent of the GEMS code.
| GEMS | POD-DEIM ROM | Operator Inference ROM | |
|---|---|---|---|
| Simulation time for 6ms of data | ~1,200 CPU hours | ~30 minutes | ~0.5 seconds |
Pressure
x-velocity
y-velocity
Spatial Averages
Species Concentration Integrals
Below we display previews of animations for pressure, temperature x-velocity (horizontal), and methane.
