SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\operatorname {M e t r i c} _ {\rho , \mathbf {u}} \equiv \frac {1}{4} \left[ \operatorname {M e t r i c} (\hat {\rho}, \rho) + \sum_ {i = 1} ^ {3} \operatorname {M e t r i c} \left(\hat {u} _ {i}, u _ {i}\right) \right], \tag {4a}
$$
$$
\operatorname {M e t r i c} _ {s g s} \equiv \frac {1}{3} \sum_ {k = 1} ^ {3} \operatorname {M e t r i c} \left[ \left(\nabla \cdot \hat {\boldsymbol {\tau}} ^ {\mathrm {s g s}}\right) _ {k}, \left(\nabla \cdot \boldsymbol {\tau} ^ {\mathrm {s g s}}\right) _ {k} \right]. \tag {4b}
$$
with  $\hat{\phi}$  denoting an arbitrary predicted quantity. SSIM is a common image metric, but has also become a popular ML metric for evaluating flow simulations due to its employment of mean, variance, and covariance quantities - suited for evaluating the statistical nature of turbulence [26, 83, 84]. In addition, this metric is intuitive for both readers familiar and unfamiliar with turbulent flows - SSIM of 0 denotes dissimilar fields while SSIM of 1 denotes highly similar fields. Metric $_{\rho,\mathbf{u}}$  evaluates each channel of the predictions via macro-averaging. To measure the suitability of SR for turbulence modeling in coarse-grid simulations, we measure Metric $_{sgs}$ , which evaluates the predicted divergence of the subgrid-scale (SGS) stress  $\nabla \cdot \tau^{\mathrm{sgs}}$ .  $\nabla \cdot \tau^{\mathrm{sgs}}$  represents physics information lost during coarse-graining, and originates from the Favre-filtered/LES momentum equation (Equation (1b)):
$$
\partial_ {t} (\bar {\rho} \widetilde {\mathbf {u}}) + \nabla \cdot (\bar {\rho} \widetilde {\mathbf {u}} \otimes \widetilde {\mathbf {u}}) = - \nabla \bar {p} + \nabla \cdot (\bar {\tau} + \tau^ {\mathrm {s g s}}), \quad \text {w h e r e}
$$
$$
\tau^ {\mathrm {s g s}} = \bar {\rho} \left(\widetilde {\mathbf {u} \otimes \mathbf {u}} - \widetilde {\mathbf {u}} \otimes \widetilde {\mathbf {u}}\right). \tag {5}
$$
We evaluate global physical properties of ML predictions by considering the NRMSE $\{E^{k}, \varepsilon\}$  of turbulent dissipation rate  $\varepsilon$  (rate of conversion of turbulent kinetic energy to heat) and volume-averaged kinetic energy  $E^{k}$  (momentum component in energy conservation of a fixed control volume):
$$
E ^ {k} = \frac {1}{V _ {s}} \int_ {V} \rho e ^ {k} d V, \tag {6a}
$$
$$
\varepsilon = \frac {1}{V} \int_ {V} \frac {\tau}{\rho}: \nabla \mathbf {u} d V, \tag {6b}
$$
with sample volume  $V$  and velocity fluctuation  $\mathbf{u}'$ .
We summarize SSIMs of RRDB, EDSR, RCAN, and Conv-FNO in Table 2, along with model parameters  $N_{p}$  and inferencing cost (in FLOPs for a batch size of 1; see Appendix F.6 for details). The  $8 \times \mathrm{SR}$  models shown here possess the best SSIMs across different sizes for a given model approach, as shown in Appendix G.1. Models with the same network depth and width, are then initialized and trained for 16 and  $32 \times \mathrm{SR}$ . For 8 and  $16 \times \mathrm{SR}$ , RRDB (with gradient loss) performs the best across most of the metrics and evaluation sets, with RCAN demonstrating the highest  $\mathrm{SSIM}_{\rho, \mathbf{u}}$  at  $8 \times \mathrm{SR}$ . At  $32 \times \mathrm{SR}$ , all shown models exhibit lower  $\mathrm{SSIM}_{\rho, \mathbf{u}}$  than tricubic interpolation in the baseline test set, indicating that SR is difficult to learn at high ratios. However, all models exhibit higher  $\mathrm{SSIM}_{sgs}$  than tricubic interpolation for all SR ratios. This indicates that SR models may still be useful for turbulence modeling at high SR ratios. Figure 3, demonstrates that model predictions of specific kinetic energy  $\rho e^{k}$  (Equation (2)) (a physical quantity that combines predictions of all four channels) from all models presented in Table 2 increasingly lose fine turbulent structures as SR ratio increase. Nevertheless, when compared to tricubic interpolation, the SR models can still recover the
magnitudes of the energy at these SR ratios. Further examination of the NRMSE metrics from the  $8 \times$  SR models (see Appendix G.1 for 16,  $32 \times$  SR), in Table 3, also demonstrates that all ML models significantly outperform tricubic interpolation on baseline test and forced HIT sets. Here, gradient loss RRDB performs best in most of the metrics. However, EDSR outperforms with  $\mathrm{NRMSE}_{E^k}$ , as the gradient loss only offers minor improvements to  $E^k$ .
Table 2: Comparison of SSIM of five models at three SR ratios, with tricubic interpolation. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
Models	Baseline Test Set	Param. Variation Set	Forced HIT Set	Size	Cost
↑SSIMρ,u	↑SSIMsgs	↑SSIMρ,u	↑SSIMsgs	↑SSIMρ,u	↑SSIMsgs	Np	↓GFLOPs
Tricubic 8×	0.820	0.431	0.800	0.418	0.951	0.711	-	23
RRDB 8× (+ Grad. Loss)	0.907±0.003	0.715±0.004	0.898±0.003	0.755±0.002	0.997±0.000	0.891±0.003	50.2M	1430
0.936±0.003	0.802±0.003	0.929±0.001	0.825±0.001	0.998±0.000	0.944±0.005
EDSR 8×	0.928±0.004	0.748±0.012	0.916±0.005	0.775±0.010	0.999±0.000	0.937±0.005	34.6M	2122
RCAN 8×	0.928±0.000	0.753±0.002	0.916±0.001	0.778±0.001	0.999±0.000	0.941±0.003	16.4M	671
Conv-FNO 8×	0.846±0.016	0.566±0.019	0.845±0.011	0.614±0.015	0.993±0.001	0.845±0.008	33.0M	1276
Tricubic 16×	0.652	0.175	0.620	0.173	0.876	0.432	-	23
RRDB 16× (+ Grad. Loss)	0.724±0.001	0.506±0.004	0.700±0.001	0.512±0.002	0.971±0.000	0.805±0.003	50.3M	1074
0.739±0.008	0.554±0.001	0.719±0.004	0.556±0.002	0.973±0.000	0.816±0.001
EDSR 16×	0.716±0.005	0.477±0.018	0.693±0.005	0.481±0.019	0.969±0.001	0.783±0.008	37.8M	1944
RCAN 16×	0.672±0.039	0.408±0.066	0.665±0.024	0.415±0.058	0.961±0.009	0.737±0.050	17.3M	573
Conv-FNO 16×	0.629±0.020	0.343±0.027	0.640±0.013	0.355±0.022	0.951±0.006	0.690±0.022	34.6M	1068
Tricubic 32×	0.508	0.060	0.476	0.087	0.758	0.156	-	23
RRDB 32× (+ Grad. Loss)	0.503±0.001	0.194±0.005	0.482±0.000	0.186±0.006	0.845±0.001	0.494±0.011	50.4M	1030
0.505±0.001	0.184±0.009	0.483±0.001	0.188±0.002	0.850±0.000	0.516±0.012
EDSR 32×	0.502±0.004	0.173±0.006	0.481±0.002	0.187±0.004	0.845±0.001	0.463±0.005	40.9M	1921
RCAN 32×	0.473±0.006	0.168±0.007	0.469±0.002	0.185±0.005	0.837±0.003	0.448±0.012	18.2M	561
Conv-FNO 32×	0.476±0.004	0.155±0.012	0.470±0.001	0.178±0.003	0.842±0.002	0.435±0.013	36.2M	1023
Figure 3: Specific kinetic energy  $\rho e^k$  prediction of one sample from the parametric variation set with models from Table 2. Prediction errors are shown in Appendix G.1
Table 3: Comparison of NRMSE for five models at  $8 \times$  SR ratio, with tricubic interpolation. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
Models	Baseline Test Set	Forced HIT Set
↓NRMSEρ,u(×10-2)	↓NRMSEsgs(×10-1)	↓NRMSEEk(×10-4)	↓NRMSEε(×10-1)	↓NRMSEρ,u(×10-3)	↓NRMSEsgs(×10-2)	↓NRMSEEk(×10-6)	↓NRMSEε(×10-4)
Tricubic 8×	5.09	7.51	8.89	4.33	8.82	31.12	734.55	451.68
RRDB 8×	0.92±0.01	2.46±0.04	0.39±0.10	1.16±0.00	0.19±0.01	2.15±0.08	39.83±32.51	0.74±0.29
(+ Grad. Loss)	0.60±0.00	1.41±0.01	0.41±0.17	0.54±0.01	0.13±0.01	1.23±0.17	33.77±21.44	0.55±0.17
EDSR 8×	0.86±0.04	2.30±0.15	0.29±0.06	1.10±0.06	0.10±0.01	1.67±0.25	0.60±0.24	0.21±0.03
RCAN 8×	0.86±0.00	2.31±0.01	0.32±0.01	1.14±0.00	0.09±0.00	1.39±0.11	0.62±0.05	0.23±0.02
ConvFNO 8×	1.46±0.07	4.42±0.23	0.74±0.19	1.64±0.05	0.56±0.11	6.94±0.75	163.50±191.46	3.66±2.22
Scaling behavior of RRDB is shown in Figure 4, which compares ground truth and input values of  $\rho e^{k}$  (shown in the first column) with  $8\times$  SR predictions from tricubic interpolation and variations of RRDB models. For the model predictions, the first row visualizes the specific kinetic energy  $\hat{\rho}\hat{e}^k$ , while the second row shows the error  $|\epsilon_{\rho e^{k}}| = |\hat{\rho}\hat{e}^{k} - \rho e^{k}|$  normalized by  $\rho e_{max}^{k}$ . Our discussion
is focused on the predictions in the cyan box. At  $N_{p} = 0.6\mathrm{M}$ , RRDB is unable to reconstruct  $\rho e^{k}$  accurately. RRDB's prediction is more accurate than tricubic interpolation at  $N_{p} = 4.9\mathrm{M}$ , but spurious structures that originate from the coarse grid can be seen. For  $N_{p} = 50.2\mathrm{M}$ , the model is sufficiently expressive for eliminating the spurious structures from the flow. The addition of the gradient loss term is shown to reduce prediction errors from RRDB 50.2M. This trend in improvement is also visible in the bottom row, which shows the mean divergence of SGS stresses (Equation (5)).
Figure 4: Predictions from various RRDB models, showing gradual improvement in the cyan box.
Figure 5: Scaling behavior of RRDB (with and without gradient-based loss), EDSR, RCAN and Conv-FNO. RRDB, EDSR and Conv-FNO models continue to scale at large model sizes.
Scaling behavior of RRDB (with and without gradient loss), EDSR, RCAN, and Conv-FNO models are examined in Figure 5.  $\mathrm{SSIM}_{sgs}$  scales differently compared to  $\mathrm{SSIM}_{\rho,\mathbf{u}}$ , indicating the importance of evaluating derived physical quantities from model predictions in flow physics applications. For both SSIMs across all evaluation sets, RCAN models demonstrate better performance than EDSR and vanilla RRDB models for  $N_p < 17\mathrm{M}$ , but performance deteriorates after this model size. The gradient loss term improves RRDB predictions for all model parameters explored, resulting in  $\mathrm{SSIM}_{sgs}$  that exceeds RCAN after  $N_p = 1.4\mathrm{M}$  for the baseline test and Parametric Variation sets. Thus, this loss term is shown to benefit moderately sized models ( $N_p = 50.2\mathrm{M}$ ) and data (67 GB), which is in contrast to the notion that physics-based losses are mostly helpful for small models and datasets [50]. Conv-FNO is seen to outperform the baseline tricubic prediction after approximately 20M parameters. FNO layers are memory-intensive due to high number of dimensions found in the spectral convolution weights (six in total: one for batches, two for channels, and three for Fourier modes). This memory-intensive nature has been acknowledged by FNO's original developers, with attempts to address this remaining an active research pursuit [85].
For all models, both SSIMs are found to scale with  $\log_{10}N_p$ . All ResNet-based models share similar slopes in the scaling relationship between  $\mathrm{SSIM}_{sgs}$  and  $\log_{10}N_p$  in the test and Parametric Variation set. However, these slopes can differ when evaluated on another flow configuration. This is seen with the idealized flows in the Forced HIT set, where higher SSIMs from all predictions and baseline are observed.
Figure 6 shows the relationship between  $\mathrm{SSIM}_{sgs}$  and inference cost (in FLOPs) for the five model approaches.  $\mathrm{SSIM}_{sgs}$  for EDSR, RCAN, and RRDB (with gradient loss) models scales with cost in a similar fashion, after approximately 100 GFLOPs. A steeper scaling relationship is observed for both Conv-FNO and vanilla RRDB. Vanilla RRDB models also do not demonstrate a strong linear relationship with  $\log_{10}$  GFLOPs when tested on the Forced HIT set.
In this work, we released BLASTNet 2.0, a public 3D compressible turbulent reacting and non-reacting flow dataset. From this data, we extracted the Momentum128 3D SR dataset, which we employed for benchmarking 3D SR models at 8, 16, and  $32 \times$  SR. SR models are shown to score well in SSIM-based metrics and capture fine turbulent structures at  $8 \times$  SR. For the higher SR ratios, these fine structures cannot be captured, but the SR models can still recover the magnitude of large flow structures. Through our scaling analysis, we demonstrate that benefits from a gradient-based physics
Figure 6: Scaling behavior with cost.
based loss persist with model scale – providing empirical evidence that disagrees with the postulated notion that physics-based methods are useful mostly in small model scenarios [50]. However, we recognize that this observation is applicable only to one type of physics-based ML technique, and is not necessarily extendable to other physics-based ML approaches. We observe that model performance scales with the logarithm of model parameters, and that the scaling relationship between  $\mathrm{SSIM}_{sgs}$  and inference cost are similar for RRDB (with gradient loss), EDSR, and RCAN. We also demonstrate that the choice of model architecture can matter significantly, especially when developing small models for real-time scientific computing applications, and that physics-based losses can improve some metrics of poorly performing architectures. With this work, we demonstrate that BLASTNet 2.0 can provide a rich resource for evaluating models for scientific and engineering turbulent flows.
Limitations and Negative Impact The simple geometry, skeletal finite-rate mechanisms and mixture-averaged transport used in these DNS provide high-fidelity information of fundamental processes, but are not fully representative of real-world systems. However, rectifying this would require complex geometry, detailed mechanisms (introducing an order of magnitude more PDEs) and multi-component transport that can result in intractable calculations [86]. Another limitation is that the DNS data originate from proprietary-licensed numerical solvers (see Appendix D.2), resulting in data that cannot be thoroughly inspected through open-source means. However, expertise, peer review from published research, and solver reputation ensure that these DNS data meet community-accepted standards. In this work, we are limited to using Favre-filtered (Equation (3)) DNS to generate low-resolution inputs in the Momentum128 3D SR dataset, as it is not feasible to obtain pairs of coarse simulation sample that matches a corresponding ground truth DNS sample, as the chaotic nature of turbulence will result in uncorrelated pairs [72]. However, Favre-filtered DNS is a canonical surrogate [74] for coarse simulations with strong theoretical foundations [73, 45]. Further discussion on the validity of employing Favre-filtered DNS is provided in Appendix E.1.6. Data generation incurs up to  $\mathcal{O}(10^{6})$  CPU-hours per case, while this study used 15,000 GPU-hours - resulting in significant carbon emissions. However, we attempted to ameliorate this by curating previously unreleased already-generated DNS from existing publications, and employing mixed-precision training for this study. In addition, this work can improve fundamental knowledge on carbon-free combustion, which can reduce society-wide reliance on hydrocarbons.
The authors acknowledge financial support and computing resources from the U.S. Department of Energy, National Nuclear Security Administration under award No. DE-NA0003968. We also thank the NASA Early Stage Innovation Program with award No. 80NSSC22K0257, the Department of Energy, Office of Science under award No. DE-SC0022222, and Office of Energy Efficiency Renewable Energy (EERE) with award No. DE-EE0008875 for funding this work. Wai Tong Chung is grateful for partial financial support from the Stanford Institute for Human-centered Artificial Intelligence Graduate Fellowship. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory. We are also grateful to members of the Kaggle staff for fruitful discussions that have contributed to this work.
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] We introduce our dataset, describe the benchmark study, and present our conclusions.
(b) Did you describe the limitations of your work? [Yes] This is done in Section 6.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] This is also done in Section 6.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] Our work does not involve human participants. Our data is licensed and consented, while adhering to FAIR [71] principles. Attempts to mitigate negative social impacts have been made.
(a) Did you state the full set of assumptions of all theoretical results? [N/A] This study involves an empirical benchmark approach.
(b) Did you include complete proofs of all theoretical results? [N/A] This study involves an empirical benchmark approach.
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] All data is shared via Kaggle and centralized at https://blastnet.github.io, as mentioned in the abstract and Section 3.1. In addition, links to the code and model weights are shared in Appendix A.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] A summary of training details is provided in Section 4, with data split detailed in Section 3.2. Further details are also provided in Appendix F.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All experiments are performed with three different seeds. Table 2 shows mean and standard deviations from these runs. Figures 5 and 6 include points from all three seeds, with shaded regions demonstrating the standard deviation.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Section 4 mentions that we utilized approximately 13,000 GPU-hours on 16 V100 GPUs on the Lassen supercomputer [80].
(a) If your work uses existing assets, did you cite the creators? [Yes] All previous studies and solvers used for this study have been cited in Section 3.1. Additionally, assets for training and evaluation are cited in Section 4.
(b) Did you mention the license of the assets? [Yes] The previously unreleased data is licensed via CC BY-NC-SA 4.0, as mentioned in Section 3.1.
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] All data is shared via Kaggle and centralized at https://blastnet.github.io, as mentioned in the abstract and Section 3.1. Links to code for this benchmark study is shared in Appendix A.
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes] As mentioned in Section 3.1, all published data originate from the present authors.
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] Our data has no offensive content. For personally identifiable information, it only contains name and institution of data contributors. This is discussed in Section 3.1.
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] No human subjects employed.
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] No human subjects employed.
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] No human subjects employed.
A URL and Links
BLASTNet 2.0 Dataset The landing page for the Bearable Large Accessible Scientific Training Network-of-Datasets (BLASTNet) is hosted at https://blastnet.github.io. Specifically, links for browsing and downloading the dataset is consolidated at https://blastnet.github.io/datasets.
Momentum128 3D SR Dataset This 3D super-resolution dataset for turbulent flows is available at https://www.kaggle.com/datasets/waitongchung/blastnet-momentum-3d-sr-dataset.
Code Code for training and evaluating models in this study are available in https://github.com/blastnet/blastnet2_SR_benchmark, with instructions provided in the README of the repository.
Model Weights All weights of models in this study are also available at https://www.kaggle.com/datasets/waitongchung/blastnet-momentum-3d-sr-dataset.
DOI All BLASTNet datasets [87] share a common DOI at Zenodo: https://doi.org/10.5281/zenodo.7242864.
Tutorials and Browsing Tools Kaggle notebooks for browsing BLASTNet simulation data are attached to each of the Kaggle URLs of specific direct numerical simulation (DNS) data, as listed in Appendix D.2. These notebooks are also consolidated at https://github.com/blastnet/kaggle_tutorials. In addition, a table that summarizes the datasets is also provided in https://blastnet.github.io/datasets. Additional tutorials on reading the data, contributing to BLASTNet, and using the Kaggle command-line API are also provided in https://blastnet.github.io/tutorial.
B Licensing and Author Statement of Responsibility