SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 7: Design of the Ahmed-body shapes
Parameters	steps	lower bound	upper bound
Length	20	644	1444
Width	10	239	539
Height	5	208	368
Ground Clearance	2.5	30	90
Slant Angle	2.5	0	40
Fillet Radius	2.5	80	120
Velocity	4	10	70
i.e.  $\hat{i}(x) = (-1,0,0)$ , we predict only the first component of the wall shear stress. Since our results from Table 4 indicate that varying the size of the latent space has a significant effect on predictive performance, we make this the only hyper-parameter of the model and fix all others. We choose latent resolutions varying from 24 to 86. Furthermore, we consider two different loss functions. The first is simply the average of the relative  $L^2$  errors for pressure and wall shear stress. The second includes in this average the relative error in the drag coefficient computed using equation (8). The drag is always computed from our full-field predictions and is never given as part of the output from the model.
To compare the performance of our model against the industry-standard OpenFOAM solver, we perform a full cost-accuracy trade-off analysis. During data generation, we keep track of the drag coefficient predicted by OpenFOAM after every iteration. While the coefficient converges with more iterations, this convergence is not monotone and can often appear quite noisy. This makes computing the error from the raw data not possible. We therefore apply a box filter to the raw signal to compute a filtered version of the drag which acts as smoother. We take as the reference drag, the drag at the last iteration of the filtered signal. To compute the number of iterations it takes for the solver to predict a drag coefficient at a given relative error, we trace back the predictions from the filtered signal and return the first time at which this prediction incurs the given error with respect to the reference drag. An example of this methodology is shown in Figure 3(a). The errors for our GINO model are computed with respect to the true drag coefficient from the last iteration of the solver. This is because we take as ground truth the pressure and wall shear stress from this last iteration and train our model to predict them.
Figure 3(b) shows the cost-accuracy curve, measured in terms of inference time needed for a relative error in the drag coefficient for GINO and OpenFOAM. The cost of GINO is computed as the time, averaged over the test set, needed to predict the drag coefficient by running the model. This time includes both data pre-processing (computing the SDF) as well as the model run-time and the drag calculation given the predicted fields. All models are ran on a single NVIDIA V100 GPU. The cost for OpenFOAM is computed as described in the previous paragraph and is averaged over the test set. The solver is ran on two NVIDIA V100 GPUs in parallel. We observe a four to five order of magnitude speed-up when using GINO. At a  $3\%$  relative error, we find the speed-up from our model which includes drag in the loss to be  $26,000\times$ . As we increase the size of the latent space, the cost of GINO grows, however, we observe a plateau in the drag error. This is common in machine learning models as the error from using finite data starts to dominate the approximation error. Furthermore, we use only the size of the latent space as a hyper-parameter, keeping the number of learnable parameters fixed. It is interesting to explore further how parametrically scaling the model impacts predictive power.
7.3 Data Generation
Industry-standard vehicle aerodynamics simulations are generated in this study, utilizing the Ahmed-body shapes as a foundation [28]. Examples are illustrated in Figure 6. These shapes are characterized by six design parameters: length, width, height, ground clearance, slant angle, and fillet radius, as outlined in Table 7. In addition to these design parameters, we include the inlet velocity to address a wide variation in Reynolds number. The inlet velocity varies from  $10\mathrm{m / s}$  to  $70\mathrm{m / s}$ , consequently resulting in Reynolds numbers ranging from  $4.35\times 10^{5}$  to  $6.82\times 10^{6}$ . This varying input adds complexity to the problem. We identify the design points using the Latin hypercube sampling scheme for space filling design of experiments.
The simulations employ the GPU-accelerated OpenFOAM solver for steady-state analyses, applying the SST  $k - \omega$  turbulence model. Consisting of 7.2 million mesh points in total, including 100k surface mesh points, each simulation is run on 2 NVIDIA V100 GPUs and 16 CPU cores, taking between 7 to 19 hours to complete.
For this study, the focus is solely on the prediction of the pressure field. It is our hope that this dataset can be utilized in future research, potentially aiding in full-field simulation of the velocity field, as well as the inverse design.
Figure 6: Illustrations of the Ahmed-body dataset