Turbulence in Focus: Benchmarking Scaling Behavior of 3D Volumetric Super-Resolution with BLASTNet 2.0 Data
Authors: Wai Tong Chung
Bassem Akoush
Pushan Sharma
Alex Tamkin
Ki Sung Jung
Jacqueline Chen
Jack Guo
Davy Brouzet
Mohsen Talei
Bruno Savard
Alexei Y Poludnenko
Matthias Ihme
Venue: NeurIPS 2023 Datasets and Benchmarks Poster (2023)
Keywords: Super-resolution
3D
Neural Scaling Laws
Physics-informed Loss
Computational Fluid Dynamics
Partial Differential Equations
Turbulent Reacting Flows
Direct Numerical Simulation
Fluid Mechanics
Combustion
Computer Vision
TL;DR: We curate a 2.2 TB compressible turbulent reacting/non-reacting flow database and demonstrate its utility via benchmarking 3D super-resolution, where we also examine neural scaling behavior of 5 model approaches.

Abstract: Analysis of compressible turbulent flows is essential for applications related to propulsion, energy generation, and the environment. 
Here, we present BLASTNet 2.0, a 2.2 TB network-of-datasets containing 744 full-domain samples from 34 high-fidelity direct numerical simulations, which addresses the current limited availability of 3D high-fidelity reacting and non-reacting compressible turbulent flow simulation data.  With this data, we benchmark a total of 49 variations of five deep learning approaches for 3D super-resolution - which can be applied for improving scientific imaging, simulations, turbulence models, as well as in computer vision  applications. We perform neural scaling analysis on these models to examine the performance of different machine learning (ML) approaches, including two scientific ML techniques. We demonstrate that (i) predictive performance can scale with model size and cost, (ii) architecture matters significantly, especially for smaller models, and (iii) the benefits of physics-based losses can persist with increasing model size. The outcomes of this benchmark study are anticipated to offer insights that can aid the design of 3D super-resolution models, especially for turbulence models, while this data is expected to foster ML methods for a broad range of flow physics applications. This data is publicly available with download links and browsing tools consolidated at https://blastnet.github.io.

Turbulence in Focus:
Benchmarking Scaling Behavior of 3D Volumetric Super-Resolution with BLASTNet 2.0 Data
Wai Tong Chung"), Bassem Akoush', Pushan Sharma', Alex Tamkin', Ki Sung Jung", Jacqueline H. Chen", Jack Guo', Davy Brouzet', Mohsen Talei Bruno Savard', Alexei Y. Poludnenko Matthias Ihme Sttanoord University, Sandia National Laboratory, UUniversity of Melbourne, *Polytechnique Montréal, Universiity of Connecticut, SLAC National Accelerator Laboratory
Abstract
Analysis of compressible turbulent flows is essential for applications related propulsion, energy generation, and the environment. Here, we present BLASTNet 2.0, 2.2 TB setwork-ofddataett. containing 744 full-domain samples from 34 high-fidelity direct numerical simulations, which addresses the current limited availability of 3D high-fidelit reacting and non con-reacting compressible turbulent flow simulation data. With this data, we benchmark a total of 49 variations of five deep learning approaches for 3D super-resolution which can be applied for improving scientific imaging, simulations, turbulence models, as well as in com- puter vision applications. We perform neural scaling analysis on these models examine the performance of different machine learning (ML) approaches, including two scientific ML techniques. We demonstrate that (i) predictive performance can scale with model size and cost, (ii) architecture matters significantly, especially for smaller models, and (iii) the benefits physics- based losses can persist with increasing model size. The outcomes of this benchmark study are anticipated to offer insights that can aid the design of 3D super-resolution models, especially for turbulence models, while this data is expected to foster ML methods for broad range of flow physics applications. This data is publicly available with download links and browsing tools consolidated at autttpa:: http://bllsttttt giithu.........................b io
Introduction
In recent years, machine learning (ML) has offered new modeling approaches for natural and engineering sciences. For example, PB ERA5 data [ was employed towards GraphCast (2) an MI. model that outperformed conventional weather modeling techniques. As such, efforts in curating large datasets for scientific ML have been growing across numerous domains including agricultural science [R) geophysics [4], and biology [5]. In many of these examples, the volume of data can significantly exceed the free limit of open-source repositories such as Kaggle (100 GB) - typically used to store and share language and image data due to potentially high dimensions within scientific data. As such, significant resources are required for building and maintaining data storage cloud service providers 7
capabilities, either through institutional collaborations 3 5 6)
In previous work 9 we proposed the Bearable Large Accessible Scientific Training Network- of-datasets (BLASTNet). cost-effective community-driven weakly centralized framework (see Section 3.T) that utilizes Kaggle for increasing access to scientific data, which provided access to 110 full full-domain samples from 10 configurations (225 GB) of 3D high-fidelity flow physics direct
Corresponding authors: (wtchungmin hime estanford.cc..
37th Conference on Neural Information Processing Systems (NeurIPS 2023) Track on Datasets and Benchmarks.

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numerical simulations (DNS). In this work, we present the updated BLASTNet 2.0 dataset, which extended 744 full-domain samples from 34 DNS configurations, as shown in Figure This dataset aims to address limitations in data availability for compressible turbulent non-reacting and reacting flows, which found in automotive 10 11, propulsion [2 13, energy [1 15], and
environmental [66] 17 applications. BLASTNet data has previously been employed for solving MI tasks related to dimensionality reduction, regime classification, and turbuleece-chemiisty closure modeling [9]. Beyond this, BLASTNet is potentially suited for ML problems involving predictions of physical quantities found in turbulent non-reacting and reacting flows, which can also involve inverse problems [8] [9] and physics discovery
20 211.
In this work, we demonstrate the utility of BLASTNet data for 3D super-resolution (SR) of turbulent flows. From BLASTNet 2.0, we pre-process DNS data to form the Momentum122 3D SR dataset for benchmarking this task. To this end, we:
Curate BLASTNet 2.0, diverse public 3D compressible turbulent flow DNS dataset.
Benchmark performance and cost of five 3D ML approaches 19 22 25 for SR with this
publicly accessible dataset.
Show that SR model performance scale with the logarithm of model size and cost.
Demonstrate the persisting benefits of a popular physics- gradient loss term [9 with increasing model size.
We provide overview of related efforts in Section In Section 3 BLASTNet 2.0 and Momentum 128 3D SR datasets. Our benchmark setup is described in Section with results discussed Section 5 before the conclusions in Section
we provide information on the
BLASTNet 2.0
appliad towands
ML Tasks
This Work
Turbulent super-resolution.
Previous Work
Addresses gap availability public 3D compressible turbulent flw physics. 34 configurations, 2.2 TB, 744 full-domain samples, 6o channels (velocity.densityy
Closure madeling.
Regime classification
pressure, temperature, and chemical species), unifomistrctched grids ML-ready format.
Dimensionality
High-fideitty direct numerical simulation data contributed by numerous institutions.
Potential Work
Shared via Kagele. Tutorials and companion code available at httnes//blattncttiithbbb.
Other physical quantity prediction.
sampled to create
Benchmarking 3D Turbulent Reser-Resolution (SR)
Momentum 3D SR Dataset
used benchmork
Models
Labels
Residual in Residual Dense Black
Enhanced Deep Residual Network
128' 3D sub-volumes of velocity and density. configurations, 67 GBB, samples, channels (velocity density), uniform grid.
witlewitoon plusica-bured loss
regulariaation term.
Residual Chameel Atlention Network
Convolational Fourrier Neural Overator
Channal Amarion
Fealures
8X
16K
32X
evaluated via + Metrics and Scaling Behavior
Thooretical FLOPs. model parameter size.
MSE and 3D SSIM of super resolved voxel predictions derived physical quantities related subgrid scale stress, turbulent dissipation, and kinetic energy.
Filter and downsample labels to generate coarse simulation samples.
Figure 1: Summary of this study.
Related Work
Flow Physics Simulation Datasets Numerical simulations accurately describe flow physics, as long as the simulation grid resolves the smallest lengthscales associated with turbulent dissipation [26]. With up (109) voxels, ((10%) core-hours of simulation time, and ((1*)) cores on parallel computing facilities [27-30], high fidelfidelity DNS of many real-world flows cannot be performed

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due to prohibitive costs. Thus, it is common to employ coarser grids with large-eddy simulations quantities with Reynolds- averaged both of which rely on turbulence models that can be discovered from DNS data. DNS data is also useful for applications involving scientific imaging [4 spatio-temporal modeling [55] and solving inverse problems [18]. As shown in Table I many existing flow simulation datasets focus on LES and RANS simulations due to data storage constraints. McConkey et al. [32] released dataset for improving turbulence models in incompressible non- reacting RANS. AirRANN 33 provides both 2D incompressible and compressible non-reacting RANS data, specifically airfoil configurations. For reacting flows, Huang [31] released 2D LES dataset for developing reduced-ord order models. The largest flow physics dataset, the Johns Hopkins Turbulence Database (JHTDB) [36.. provides 3D DNS data from turbulent incompressible non- reacting flow simulations. Since these datasets are either 2D, incompressible, or non-reacting, they not suitable for many applications involving aerodynamies, propulsion, chemical processes. This is one reason why MI studies involving these applications employ self-generated inaccessible datasets (37 38 introducing challenges to open model evaluation. To address these gaps, we curate BLASTNet 2.0, a 3D turbulent compressible reacting and non-reacting flow dataset, which targets a balance of diversity, fidelity, and size.
(LES) T5 3T or by only evolving time-fensemble-avrraa
Navier-Stokes (RANS) simulations [32 3
Table I: Comparison of BLASTNet 2.0 (in bold) with selected flow simulation datasets.
BLASTNet
McConkey al. 32
Datusets
JHTDB 36
AirRANs
33
Huang 31
2.0
1.0 2
Size [TB]
2.2
0.225
490
0.014
0.066
0.209
Configs.
34
10
10
29
1,000
1
Full-domain Samples
744
120
15.791
1,000
30,000
Fidelity
DNS
DNS
RANS
RANS
I.FS
Spatial Dimensions
3D
3D
2D,3D
2D
2D
Primitive Variable
44. u..p.T. p.Yk
u..p.p,b to No
(u<).p)
(p)).(p).((I).p))
p.p.T.P.
Number 2 Compressible Flow?
27 Yes
4 to
7
No
Yo
Yes
Multisphysics?
Reaction
MHL
No
Reactions
Host
Kaggle fp32 binaries
SciServer
o
Kaggle .npy
Sorbonne
UMich
Format
.h5
.vtu.. vtp
asci i
SR for Turbulent Flows Within experimental flow measurements, deep learning- SR ap- proaches have been employed towards improving Schlieren 44), particleimage-velocine [41]. and tomography 42 techniques, which can often be limited by resolution restrictions. many prac- tical engineering analyses, coarse-grained simulations with under-resolved grid sizes are often used to bypass the extensive costs fully-resolved DNS. dominant source of error from this approach involves missing physics that arise from the under-resolved grid, as will be detailed in Section 4.2 While algebraic turbulence models have been traditionally used to represent the under-resolved physics [43] [44, specific algebraic models are effective only in specific flow configurations [45). SR and related upsampling approaches have been proposed as a versatile alternative for correcting under- resolved information from coarse-grid simulations between numerical time-stepping 33 46 47].
which would require considerations of real-time inferencing. Studies on turbulent SR have focused on demonstrating feasibility 377 41 48) mostly by modifying existing image SR models. Due to memory constraints, many of these studies focus on 2D configurations [4] 46 48), with 3D SR investigations only demonstrated recently [37 47). As such, there has not yet been detailed and
reproducible benchmark study comparing SR models for 3D turbulent flows 38 49
Filtered and tine-/ensemble-aveenee quantity is expressed with o and (ø), respectively. In compressible LES, Favre-filtering / 39 is typically used. T refers to temperature; more definitions in Equation
*PDEs for continuity, energy, momentum, chemical species, and magnetic field 6,
Thiis dataset contains blend of compressible and incompressible flow samples.
4 Majority are reacting flows. Some non-reacting flows are present.
'Only one of the ten configurations consider maggeto-hydrodyyaaee (MHD).
3

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Scientific ML Approaches Scientific ML approaches can involve developing custom architectures with implicit biases that suit specific problems and modifying loss functions with constraints related to governing equations [50] Firstly, model architectures such as NUNet [11] MeshGraphNNtt [52),
and Fourier Neural Operators (FNO) [ employ graph and spectral convolution layers to ensure that flow predictions are mesh invariant. While FNOs have been successfully employed towards canonical laminar flow configurations, previous studies have demonstrated that these models are insufficiently expressive for complex configurations, involving turbulent (35] and multi-physics [25] flows. This has been attributed to regularization properties of spectral convolution layers by the FNO's original developers (25). Recently, convolution FNO (Conv-FNO) 25 models have been proposed to ameliorate these underfitting issues by embedding convolutional blocks within FNO blocks. Secondly, modifying the loss functions can involve adding regularization term based on the residuals of the entire governing equations (54] A softer approach involves regularizing with individual operators within the governing equation (such as continuity, advection, diffusion, and source terms) 37 In this work, we benchmark models based on both loss function and architecture approaches, i.e., model with gradient-based loss function 19 (which biases MI. optimization towards turbulence applications; see Appendix F3) and a Conv-FNO model, respectively.
SR Benchmarks SR via deep learning has been subjected to numerous competition and benehmark studies that target various aspects of 2D) image SR, including 2K-images (55), night photography (56), spectral recovery [57]. and satellite images (58). For 3D SR, benchmarks for video [59], medical resonance imaging 660] and 3D microscopy applications [1] have been performed. Another method for studying the behavior of deep learning models involves the construction of empirical scaling relationships [62]. This type of analysis is useful for studying resource requirements of large language models, and was briefly been employed for studying 2D SR with a U-Net model [63]. Given the potential real-time computing applications of turbulent SR, this analysis provides useful information on the relationship between model size, cost, and predictive performance.
3 Dataset
3.1 BLASTNet 2.0
BLASTNet 2.0 consists of turbulent compressible flow DNS data, on Cartesian spatial grids, generated by solving governing equations for mass, momentum, energy, and chemical species, respectively:
3.p + V (pu)
(la)
d((pu) + V (ou @ u) Vp p + V T Ox + V ju(pe" + +p]] - .7.0 a V [(T)-u] u
(lb)
(Ic)
dx(p)r) V (puYr)
j& + d
(Id)
with density p. velocity vector u. pressure p. specific total energy e¹, stress tensor т, and heat flux q. Ys. Jss and We are the mass fraction, diffusion flux and source term for chemical species k = [1, 1]. where N, is the number of species. Molecular fluxes are typically modeled using the mixture-averaged diffusion model.
The BLASTNet 2.0 dataset is developed with these properties in mind:
Fidelity All DNS data is collected from well-established numerical solvers 29 with spatial discretization schemes ranging from 2nd- to 8th- oth-order accuracy, while advncemennne accuracy range from 2nd to 4th-order. Low-order schemes require finer discretizations compared to high-order schemes, to achieve similar accuracy and numerical stability (68). However, all simulations are spatially resolved to the order of the Kolmogorov lengthscale, ranging from 3.9 to 41 jum depending on the configuration, with corresponding temporal discretization that ensures numerical stability.
Size and Diversity BLASTNet 2.0 contains total of 744 full-domain samples (2.2 TB) from a diverse collection of 34 simulation configurations: non-reacting decaying homogeneous isotropic
turbulence (HIT) 20), reacting forced HIT [67], two parametric variations of reacting jet flows 29
six configurations of non-reacting transcritical channel flows [69], a reacting channel flow 28
a prremixy slot burner configuration (27), and 22 parametric variations (with different turbulent and chemical timescales) of a proely-propagating flame configuration [70].

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Hommunntt-involvemeeee BLASTNet 2.0 consists of data contributions from six different insti- tutions. As mentioned in Appendix C our long-term vision and maintenance plan for this dataset involves seeking additional contributions from members of the broader flow community.
Cost-effective Storage, Distribution, and Browsing To circumvent Kaggle storage constraints, we partition the data into a network of < 100 GB subsets, with each subset containing a separate simulation configuration. This partitioned data can then be uploaded separate datasets Kaggle. To consolidate access to this data, all Kaggle download links are presented in
github. ic with the inclusion of bash script for downloading all data through the Kaggle API. In addition, Kaggle notebooks are attached to each subset to enable convenient data browsing on Kaggle's 's cloud computing platform. This approach enables cost-effective distribution of scientific as further detailed in Appendix
data that adheres to FAIR principles
71
Consistent Format Data, generated from different numerical solvers, initially exists in range of formats vtk, vtu, tec, and .dat) that are not readily formatted for training ML models. Thus, all flowfield data are processed into consistent format little-endian single-precision binaries that can be read with np. fromfi Le/np. memmap. The choice of this data format enables high I/O speed in loading arrays. We provide jseon files that store additional information on configurations, chemical mechanisms and transport properties. See Appendix D for more details.
Licensing and Ethics All data is generated by the present authors and licensed via CC BY-NC- SA 4.0. Other than the contributors' names and institutions, no personal-identifanl information published in this data. No offensive content is published with this flow physics dataset. Further discussion on negative impact is provided in Section 6
1200
Standard Deviation pf [Begur 1g-2]
3.2 Momentum 128 3D SR Dataset
1000
BLASTNet 2.0 is further processed for training due to con-
straints in (i) memory and (ii) grid properties. Currently, the single largest sample (92 GB) in BLASTNet 2.0 contains 1.3B voxels and 15 channels, which cannot into typical GPU memory. In addition, the spatial grid is stretched de- pending on the resolution requirements of the flow domain.
800
600
As shown in Figure D circumvent these two issues by
sampling 1283 sub-volumes of density P and velocity u from the uniform-grid regions from all BLASTNet data. This
sults in 12,750 sub-volume samples (427 GB). We choose this sub-volun sub-volume size to enable 32x SR (the resulting feature sub- volume is 4 which is larger than a kernel size of 3), while maintaining low memory footprint. In order to develop a compressible turbulence benchmark dataset that can be easily downloaded, we select 2,000 sub-volumes form 67 GB dataset that can fit into a single Kaggle repository. To ensure that these 2,000 samples are representative of the different flows encountered in each configuration, we:
0-
500
1000
1500
2000
Mean of pe [kgm= 5-1)
Figure Statistics of the specific ki- netic energy pe of each 1283 sub- volume the Momentum 128 3D SR dataset. Marker type represents DNS configuration. See detailed legend in Appendix E.1.5
1. Extract mean, variance, skewness, and kurtosis (statistical moments for characterizing turbulence (26)) from the three velocity components of 12,750 sub-volumes.
2. Apply k-means clustering with the elbow method (using the statistical moments as features) to partition the sub-volumes in 18 clusters.
3. Select 2,000 samples while ensuring that the proportion of clusters are well-balanced.
The resulting sub-volumes form the labels of BLASTNet Momentum 128 3D SR dataset. Figure demonstrate the mean and standard deviation of the specific kinetic energy:
pe p(u13 2 + u2² 2 + uz²)) 2)2
which we use to characterize all channel variables. Each distinct marker represents different simulation configuration. Since flows from the same configuration possess similar statisties, the different configurations from BLASTNet 2.0 can result in dataset with a variety of flow conditions.
5

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Due to stochastic and chaotic nature of turbulence 772). is not possible to obtain matching pairs of coarse (also known as implicit LES (45) and fine DNS data. Thus, we employ a canonical method [7 45 73 for obtaining implicit LES surrogates, known as finite-volume optimal LES [74] (see Appendix F. 1.1.66or on their validity). Specifically, we Favre-filter [39] and downsample the labels by 8, 16, and 32x to obtain representative range of coarse resolution samples (LES is typically an order magnitude coarser than DNS [26 (5)) to generate inputs for turbulent SR:
/ pbdV, PV, V
1
where ō denotes uniform-filtered quantity, is Favre-filtered quantity, and Vj is subvolume with the size of the filter width. In our SR dataset, the channels of each label correspond to (p while the feature channels consist of o, = (p.ū1, ü,g ]. For the purpose of the present benchmark study, we further split the 2,000 sub-volumes as follows:
Train, Validation, and Baseline Test Sets 80: 10:10:10 10 split via random selection with uniform distribution. The training set contains 1,382 samples, and both validation and baseline test sets contain 173 samples each.
Parametric Variation Set A 144-semple subset for model evaluation from an unseen parametric variation configuration with approximately 15% higher mean velocities and velocity fluctuations than the train, validation, and baseline test sets.
Forced HIT Set A 128-sample subset for model evaluation from unseen flow type (forced HIT) with 30-fold higher pressure and 34-fold lower velocity fluctuations.
Benchmark Configuration
4.1 Models and Methods
As shown in Figure 1 three well-studied 2D ResNet-based 76 SR models are modified from their original repositories for 3D SR: (i) -nnettall-i--Reeiiiiee Dense Block (RRDB) (22). (ii) En- hanced Deep Residual Super-resolution (EDSR) (23], and (iii) Residual Channel Attention Networks (RCAN) [24). Convolution networks possess inductive biases that are suitable for problems involving spatial grids such as in flow physics [50 77]. We choose to study these models due to their differences in architecture paradigms. Specifically, RRDB employs residual layers within residual layers; EDSR features an expanded network width; RCAN utilizes long skip connections and channel attention mechanisms. In addition, we consider two additional scientific ML approaches: (i) a Conv-FNO model (25), modified for SR (see details in Appendix [F2). and (ii) an RRDB model regularized with the weighted MSE of the gradient of channel variables AA Ex -= MSE((V)) (Vo)a) 19 to the
loss (1 - A)MS & Φ), where A the distance between each voxel. Details on the gradient-based loss and its weighting factor À are provided in Appendix F. We compare model predictions with a baseline approach, i.e., tricubic interpolation. To investigate the scaling behavior of the model architectures, we vary the number of parameters by changing the network depth and width.
Similar to other turbulent SR studies [37 46), all models are trained with mean--qquardderror (MSE) loss, unless otherwise stated. For evaluation, we select models with the best MSE after training for 1, 1,500 epochs with a batch size of 64 across 16 Nvidia V100 GPUs. Learning rate is initialized at 1e-4, and halved every 300 epochs. Both the number of training iterations and learning scheduling are chosen match other SR studies (22 and are found be sufficient for the SR predictions, as will be shown in Section 5) All other hy pyprraarametess are maintained from their original studies, with He initialization (78) used on all initial model weights. Data augmentation is performed via variants of random rotation and flip modified to ensure that augmented data remains consistent with continuity (Equation (Ta). Training is performed with automatic mixed- preeedsioeiiionn from Li Lightning
1. 6.5 5 79]. Prior to tramnnn, data is normalized with means and standard deviations of density and velocity extracted from the train set. During evaluation, this normalization resulted poor accuracy for the Forced HIT set, due to the significantly different magnitudes of density and velocity. However, Section will show that good performance can be achieved when normalization performed with the mean and standard deviation of each distinct evaluation set. Thus, all evaluation sets are normalized

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with their own mean and standard deviation, prior to testing. All 40 model variations are trained with three different seeds, resulting in a total computational cost of approximately 15,000 GPU-hours on the Lassen Supercomputer [80]. Further information on model hyperparameters, data augmentation. training, normalization, as well as links to code and model weights are found Appendix A F G
4.2 Metrics
We compare the performance of each model by examining local and global quantities of each sample. For the local quantities, we employ Metrics {SSIM, NRMSE), where SSIM is the 3D extension [8]] of the structural similarity image measure [82] and NRMSE the normalized
mottmean-squaree error (see Appendix F.I. For quantities with multiple channels:
1
Metricp.u
Metric p) +
Metric (2; ui)
(4a)
4
3
Metric xgs
Metric 788)) k: (V 880 >0))))
(4b)
with Ô denoting an arbitrary predicted quantity. SSIM is a common image metric, but has also become 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 O denotes dissimilar fields while SSIM of denotes highly similar fields. Metric,, evaluates each channel of the predictions via macro-averaging. To measure the suitability of SR for turbulence modeling in coarse-grid simulations, we measure MetriCags, which evaluates the predicted divergence of the subgrid-scale (SGS) stress 7 V T represents physics information lost during coarse-graining, and originates from the Favre-filtered/LBS. momentum equation (Bquation (bb):
(,(pū) + V pu & ú
Vp (T+
where
A p &
(5)
We evaluate global physical properties of ML predictions by considering the NRMSE (x .c) of turbulent dissipation rate (rate of conversion of turbulent kinetic energy to heat) and volume- averaged kinetic energy E (momentum component in energy conservation of fixed control volume):
E
pe dv,
(6a)
V,
.
1 T
Jv VudV,
(6b)
with sample volume V and velocity fluctuation u'.
5 Experiment Results
We summarize SSIMs of RRDB, EDSR, RCAN, and Conv-FNO in Table 2 along with model parameters N,, and inferencing cost (in FLOPS for a batch size of 1; see Appendix [o66 details). The 8x SR models shown here possess the hest SSIMs across different sizes for a given model approach, as shown in Appendix G.I Models with the same network depth and width, are then initialized and trained for 16 and 32x SR. For 8 and 16x SR. RRDB (with gradient loss) performs the best across most of the metrics and evaluation sets, with RCAN demonstrating the highest SSIM,,, at 8x SR. At 32x SR, all shown models exhibit lower SSIM than tricubic interpolation in the baseline test set, indicating that SR difficult to learn high ratios. However, all models exhibit higher SSIM,gs than tricubic interpolation for all SR ratios. This indicates that SR models may still be useful for turbulence modeling at high SR ratios. Figure demonstrates that model predictions of specific kinetic energy pe (Equation 2 (a physical quantity that combines predictions of all four channels) from all models presented in Table increasing lose fine turbulent structures as SR ratio increase. Nevertheless, when compared to tricubic interpolation, the SR models can still recover the
7

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magnitudes of the energy at these SR ratios. Further examination of the NRMSE metrics from the 8x SR models (see Appendix f.. 16, 32x SR), in T also demonstrates that all ML models significantly outperform tricubic interpolation on baseline test and forced HIT sets. Here, gradient loss RRDB performs best most of the metrics. However, EDSR outperforms with NRMSE g+, as the gradient loss only offers minor improvements to E*
Table 2: Comparison of SSIM of five models three SR ratios, with tricubic interpolation. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
Baseline Test Set
Param.
Variation Set
Forced HIT Set
Size
Cost
Models
PSSIMpa
TSSIMags
1SSIMp,u
1SSIMags
pSSIMpu
1SSIMags
Np
JGFLOPS
Tricubic
0.820
0.418
0.951
0.711
23
RRDB 8x
0.907:+0.003
0.715=10.004
0.898+00.003
0.755=0.002
0.907+000000
0.891=:0.003
50.2M
1430
Grad. Loss)
0.936:10.003
0.802:10.003
0.929:10.011
0.025=0.001
0.998:10.000
0.944:10.005
EDSR 8x
0.928=00.004 0.748=0.012 0.916=0.005
0.775=0.010
0,999+=0.000
0.937=000055
34.6M
2122
RCAN 8x
0.928.10.000
0.753.10.002
0.916110001
0.778_10.001
0.999.10.000
0.941110003
16.4M
671
Conv-FNO 8x
0.846:10.016
0.566:10.019
0.845:0.001
0.614=0.015
0.993:10.001
0.845-10.008
33.0M
1276
Tricubic 16x
0.652
0.175
0.620
0.173
0.876
0.432
23
RRDB 16x
0.724::00001
0.506=00.044
0.700=0.000
0.512=0.002 0.971=000000
0.805=0.0003
50.3M
1074
(+ Grad. Loss)
0.739_10.008
0.55411.000
0.71910.004
0.006_1..02
0.973.10.000
0.81610.001
EDSR 16x
0.716:t0.005
0.477=+0.018
0.693+000005
0.481=0.0119 0.969::0.001 0.783-40.008
37.8M
1944
RCAN 16x
0.672:4:0039
0.408:10.066
0.66540.024
0.415=0.058
0.961:10.009
0.737=0.050 17.3M
573
Com--FNO 16x
0.629:::.020
0.343=0.027
0.640+000013
0.355=0.022
0.951=00.006
0.690=00.02
34.6M
1068
Tricubie 32x
0.508
0,060
0.476
0.087
0.758
0.156
23
RRDB 32x
0.503:+:.001
0.194=0.005
0.482++0.000
08600000
0.845==00001
0.494=0.001
50.4M
1030
(+ Grad. Loss)
0.505:40.001
0.184=2.009
0.483:10.001
0.188=0.002
0.850:10.000
0.516::0.012
EDSR 32x
0.502:100004
0.173:100006 0.481110.002
0.004 0.8451100001 0.001
0.463-10.005
40.9M
1921
RCAN 32x
0.473:+:0006
0. 0.000700007 0.469=000002
0. 0.005 0.837=000003
0.448=0.012
18.2M
561
Conv-FNO x
0.406:+00004
0. 0.155 0.470=0.000 0. 0.000 0.842+000002
0.435=+0.013 36.2M
1023
Ground Truth
Input
Tricubic
RRDB(+ Grad. LosS)
RCAN
Conv-FNO
1750
1500
1250
16x
1000
750
500
32x
250
Figure 3:
Specific kinetic energy pe prediction of one sample from the parametric variation set with models from Table 2] Prediction errors shown in Appendix G.I
Table 3:
Comparison of NRMSE for five models at 8x SR ratio, with tricubic interpolation. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
Baseline Set
Forced IIT
Models
INRMSE INRMSE aga INRMSE gk INRMSE, INRMSE, INRMSE., INRMSE
INRMSE,
(x10
(x10
(x10
(x10-1) (x10
(x10 3)
(x10 -)
(x10
Tricubic 8x
5.09
7.51
8.89
4.33
8.82
31.12
734.55
451.68
RRDB 8x
0.92++0.01
2.46-0.04
0.39++0.10
1.16+0.00
0.19+0.01
2.15=0008
39.83+32.511
0.74+0.29
(+ Grad. Loss)
0.60-10.00
1.41-0.01
0.41100.17
0.54:10.01
0.13-10.01
1.23-0.17
33.77-121.44
0.55-10.17
EDSR
0.86+:0.04
2.30=0015
0.291:0.66
1.10:0.06
0.10=0.01
1.67=0.25
0.60=10.24
0.21-:0.03
RCAN 8x
0.86:+0.00
2.31=0.01
0.32+00.1
1.14
0.09=0.00
1.3900111
0.62+0005
0.23+00.2
ConvENO 8x
1.46.1.0.07
4.42-10.23
0.74
1.64:10.05 0.56:10.11
6.94
163.50
46 3.66-12.22
Scaling behavior RRDB shown in Figure 4 which compares ground truth and input values of pe* (shown in the first column) with 8x SR predictions from tricubic interpolation and variations of RRDB models. For the model predictions, the first row visualizes the specific kinetic energy pē,,
while the second row shows the error |epe lpet pe*| normalized by pe Our discussion
8

---

is focused on the predictions in the cyan box. At = 0.6M, RRDB is unable to reconstruct pe accurately. RRDB's 's prediction is more accurate than tricubic interpolation at N 4.9M, but spurious structures that originate from the coarse grid can be seen. For Np 50.2N 2M, the model is sufficiently expressive for eliminating the spurious structures from the flow. The addition of the gradient loss term shown reduce prediction errors from RRDB 50. 2M. This trend in improvement also visible in the bottom row, which shows the mean divergence of SGS stresses (Equation (5)
Sample
Tricubic
RRDB
Grad.
1e3
Ground Truth
(agm is *1)
Predicted
0.5
Producted Norm. Abs. Ervor le-"llpee war [-]
Ingut por [20 's 51
0.15
0.10
0.05
1 tv 2 Laga 's-- 1
S (T- (kam 15-1
0.00
le5
Gronnn Truit
1.0
0.5
Figure 4: Predictions from various RRDB models, showing gradual improvement in the cyan box.
Baseline
Parametric
1.0
151M, [])
0.8
Tricubic
0.6
x3. 5e-022og N 2.3 Ae 0'2log &
3 Du in N. 1e 12-022o N Dingne 72-02lop
a Se O3looa
3 2e 23n-03
2 4n-d3iin
9.77 9.77-00lo
1.00
0.75
0.50
3e-0
1e-0 1u-01logia
5 40-0022loge
Ze 020g
e92e G2oog
2.3 023
Se-0
odd Oe-0llngge
00-0000 Ingre
0.00
109
107
10a
10s
107
10a
10
10"
10a
Number of Parameters
Sicaling
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 5 SSIMsgs scales differently compared to SSIM 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 NN 17M, but performance deteriorates after this model size. The gradient loss term improves RRDB predictions for all model parameters explored, resulting in and Parametric Variation sets. Thus, this loss term is shown to benefit moderately sized models (N, 50. 50.2M) and data (67 GB), which is in contrast 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 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 active research pursuit [85).
SSIM that exceeds RCAN after N₂ 1.4M for the baseline

---

For all models, both SSIMs are found to scale with 1og10 Np. All ResNet-based models share similar slopes in the scaling relationship between SSIM,gs and lo 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.
Boseline Test set.
Figure shows the relationship between SSIMags and infer- ence cost (in FLOPs) for the five model approaches. SSIMags for EDSR, RCAN, and RRDB (with gradient loss) models scales with cost in a similar fashion, after approximately 100 GFLOPs. steeper scaling relationship is observed for both Conv-FNO and vanilla RRDB. Vanilla RRDB models also do not demonstrate a strong linear relationship with log10 GFLOPs when tested on the Forced HIT set.
1.0
RRDD
XEDSR Conv-FNO RCAN
*+ Grad, Loss)
0.8
0.4
O 33lmg G OPs 17 C.....
0.2
Holg GFLOPs
1SSIM
wOO GPLOPs
0. H4loo GPLOPS
Forced HIT Set
1.0
6 Conclusion
In this work, we released BLASTNet 2.0, public 3D com- pressible turbulent reacting and non-reacting flow dataset. From this data, we extracted the Momentum 128 3D SR dataset, which we employed for benchmarking 3D SR models 8. 16, and 32x SR. SR models are shown to score well in SSIM-based metrics and capture fine turbulent structures at 8x SR. For the higher SR ratios, these fine structures cannot be captured, but the SR models can still recover the magni- tude of large flow structures. Through our scaling analysis, we demonstrate that benefits from a gradient-based physics- 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 rec- ognize 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 SSIM and inference cost 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 rich resource for evaluating models for scientific and engineering turbulent flows.
0.6
Trieuhic
2.2a GPLOPs
G
o GFLOP OPs
O 2 GFLOPS
0.4
00 2 GFLOPs
..n... Scaling Rel.
0.2
Mean
Std. Dev.
10a
10a
10a
104
GFLOPs
Figure 6: Scaling behavior with cost.
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 cepprietar--licensee numerical solvers (see Appendix D.2pp 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-accepteed standards. In this work, we are limited to using Favre-filtered (Equation (3) DNS to generate low-resolution inputs in the Momentum 128 3D SR dataset, as 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 filterfiltere DNS is provided in Appendix E.1.6 Data generation incurs up to (10%) 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 ftready-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.
-reuuuiig
10

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Acknowledgments and Disclosure of Funding
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. SONSSC22K0257, the Department of Energy, Office of Science under award No. DE-SC0022222, and Office of Energy Efficiency Renewable Energy (EERE) with award No. DE-EE0008855 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), 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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Checklist
1. For all authors...
(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
(d) Have you read the ethies 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 FAIR 7 principles. Attempts to mitigate negative social impacts have been made.
2. If you are including theoretical results...
(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.
3. If you ran experiments (e.g. for benchmarks)...
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or as a URL)? [Yes] All data is shared via Kaggle and centralized at https:: /btp///aattttt....m . github. io,, as men- abttpss:/blastnet
tioned in the abstract and Section 3.1 In addition, links to the code and model weights are shared Appendix [A]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] summary of training details provided in Section 4 with data split detailed in Section 3.2 Further details are also provided in Appendix
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] All experiments are performed with three different seeds. Table shows shows
mean and standard deviations from these runs. Figures 3 and includes 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 16 V100 GPUs on the Lassen supercomputer (80).
4. If you using cxisting assets (c.g., code, data, models) or curating/releasing
assets...
(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 licensed via CC BY-NC-SA 4.0, mentioned in Section
3.]
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(c) Did you include any new assets either in the supplemental material or as URL? Yes) All data is shared via Kaggle and centralized at attttps: //lttps:/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'l re using/curating? [Yes] As mentioned in Section 3.1 all published data originate from the present authors.
(c) 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 person- ally identifiable information, it only contains name and institution of data contributors.
This is discussed in Section 3.1
5. If you used crowdsoureing conducted research with human subjects...
(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.
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A URL and Links
The landing page for the Bearable Large Accessible Scientific Training Setwork-of-Dataeett (BLASTNet) is hosted at wtlttps//blasttt...ggn io Specifically, links for browsing and downloading the dataset is consolidated http:///edaaaattttptppp////aa/aa
BLASTNet 2.0 Dataset
thuu
io/datasets
Momentum128 3D SR Dataset This 3D super-resolution dataset for turbulent
flows is available at petps://wwer wawpglee.cm/aaaaatttaaaaaaaooppo Dlastnet-montee -mo-ent um 3d sr -ad-sr-ataset dataset
Code Code for training and evaluating models in this study are available in itttttss://gttttt com/ htmm/b/lastnet/blstneet. pat_rtttt______eeecraaaaaa ber.charrk the repository.
with instructions provided in the README of
Model Weights All weights of models in this study are also wtw. com/datasets, http://aceetts/laaallooccucuaass
moment ---dd-a----t-------
DOI All BLASTNet datasets
87 share common DOI at Zenodo: |attp:://doi.rrg/11
5281 / zenodo. 7242864
Tutorials and Browsing Tools Kaggle notebooks for browsing BLASTNet simulation data 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: //gi thub. com/ blastnet////// Gaggle_tutoree
addition, a table that summarizes the datasets is also provided in httpps //blastnet.githhu.. 1o/ datasets Additional tutorials reading the data, contributing to BLASTNet, and using the Kaggle command-line API are also provided in intttps: //lsss///llttttt Atthub.ic/uuton
B Licensing and Author Statement of Responsibility
All data is generated by the present authors and licensed via CC BY-NC-SA 4.0. bear responsibility in case of violation of rights.
The present authors
C Maintenance Plan and Long Term Preservation
The contributors BLASTNet 2.0 committed maintaining and preserving this dataset. Mainte- nance of this dataset will largely involve tracking and fixing issues that might be discovered after release. To facilitate this, we host issues webpage (https: //t::///itii ggithbb.. com/ blassnet//// blastnet. gig.guh.bb..... /ooosssuess for user feedback. All data is shared via Kaggle, ensuring that the data will be preserved and available in the long-term. In addition, our maintenance plan involves adhering to the FAIR principles [7]] for scientific data management, with the specific details as follows:
Findable All data are indexed and can be easily searched via both Kaggle and BLASTNet platforms. To ensure that the data is findable, phttp:: schema org structured metadata is employed, as detailed in Appendices D and E All BLASTNet datasets share global and persistent DOI at Zenodo: https://dol.ows orc 10. 5281/
7242864
Accessible Both data and descriptive metadata are retrievable via the Kaggle command-ling API. This protocol free and available hhtpps: /tttps://githbb. com/ wwt.////////////. haallll//kaleeeeee authentication and authorization provided through Kaggle account. We provide bash script for users to download all data (shared in multiple repositories) once with this API. Users can also download the data directly from Kaggle repositories.
with
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Interoperable The data and descriptive metadata use accessible formats that can be read by standard python numpy and ison packages. BLASTNet's http:/ /ttp://schema. schema org structured metadata also references the structured metadata of each separate BLASSNNe repository (providing information on specific contributors and Kaggle URIss) We have attempted to use accessible language when generating these metadata.
The descriptive metadata contains information on the flow configuration (initial con- ditions, chemistry, numerics, source publication, efc.). In addition, all Kaggle repositories play a CC BY-SA NC 4.0 license. The structured prttp://schenee rich information that passes the rich results test https://searcis google com/test rich-result s). All data and descriptive metadata are presented in consistent little-endian single-precision binaries and json files, guaranteeing acceptable standards for fast I/O, sufficient floating-pon point precision, and broad accessibility via widely-used python packages.
Reusable
org metadata provides
D Additional BLASTNet 2.0 Details
BLASTNet 2.0 contains pre-processed DNS data shared via network of Kaggle repositories, with links consolidated at the landing page https://blasttett tht.....bi......................
D.1 Data Format and Directory Structure
Data, generated from different numerical solvers (as detailed in Appendix D.2), initially exists in a range of formats (.vtk, vvu,, teec and datt) that are not readily formatted for training ML models. Thus, pre-process all generated data into consistent and convenient format consisting of physical and chemical data (Appendix [DI.I]. descriptive metadata (Appendix[D... and web metadata (Appendix D.1.3). along with instructions for reading the data Appendix D.1.4 This information is summarized in Figure 7
chem, twermo_traatt tran
CTI Chemkin Files
25.CH4.BFER.ct
P_Pa_id000.de
Little-Endian float32 Binaries
RHO
#HO.kgm-3jdd000
Load data via:
T_K_kd00..dat
Nx, Ny,Nz Nz metadata ['giobal'] ["Xxyz"]
D UX
metadata ['local [YYCH4 filename!
UY_ms-1_i8000.e
ché fromfile (ch4 dtype"" "xxx)p"=="x)))
UZ_ms-1,
-d000.ddt
ch4 cb4 reshapa (Nx, Ny, Nz))
YCH4_id000.da.
YCO2_id000.da
YCO_j8000.det
jes:
YH20_id000.dat
Bays variables
YN2_i8000.det
(1UX ni-11, "0Y ns-11, 3p-11, Po', 18H0 1Y02".. Mone
YO2_is000.dst
compreasion.
geid be Ie dot concributors
(1ats "a/araa p.dacl, tyts t "L/esidrt seeiidr....att"" 1ats "./esidra dudat"! with noa-saflecting outdion dn ctgr; "Taylore hypothecis with Re 58.8 and integral lengtheoalo.
grid
Inflovreustlon (with spones)
X_m.dat
hotps: /ttp////////////////// 1017/5tr. 2020.1384 aary scouset and mohsen tolei
Y_m.dat
description
nusbalent Round jet BrEa ché-air Peenixed Plane DES (fcantesa cts": 1./ches themso, tsan/28 CHM (CBR.c11") Spange Lapera tenoved from datt..
Z_m.dat
choa, themmo tean
infojjson
Figure 7: Directory structure and reading instructions for an instance of BLASTNet configuration.
D.1.1 Files Flow Physies and Chemistry
All flowfield data are processed into a consistent format little-endian single-precision binaries that can be read with fromfi wwwofiiee/ppp.mommmmm memmap, shown in Figure 7 The choice of this data format enables high I/O speed loading arrays. As also shown in Figure we also provide jjson files (see Appendix (.1.2) that store additional information on configurations, contributors, solvers. and corresponding source publications. Chemical mechanisms and transport properties are shared through
Cantera [88) mtti/.xm// .yam or Chemkin [89] fortran files. Thus BLASTNet data contains
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all information needed to reconstruct any derived auxiliary quantities (such as vorticity, viscosity. turbulence closure terms, along with heat and chemical transport coefficients) from the conservation equations.
D.1.2 Descriptive Metadata
The binary data described Appendix [....] contains information on physical and chemical data, without much context. Details involving global information such configuration, boundary/initial conditions, solvers, related publications, and spatial grid information, as well as local temporal information (if any) are provided through info. json file in each Kaggle repository. Listings [ and 2 present the python code used to generate global and local information one example of info. json.
('glcbal']
"dataset_id";
"Nxy?"
[129, 129),
anapehote";
96, ["UX_ms-1 "uy_ms- uU
varisbles":
"e_Pa", "I_k*, "SHo_kgm-3",
"Y02",""CH1""
"grid": "compression":
"Nons",
!"x": "./grid/X_m.dar"" "y": "./grid/t_m.dat" "=": "./grid/2_m..dtt"; |"spatial"; *ith order central difeennn
"numerics":
with 2nd arder ENO",
"temporal": "3rd-crder 982-RK3 (non-stifE)
nic": {"0": "HIT Von Karman Pao with Re_t 60 and integral lengthseale of 62.55-Em", semi-implicit RоWP_US (stiff)", "solvert: "Charlesx*),
"bot: "Perlodic in x y and 2-directions,"
"T [K]": 300,
"B [Pa]": 101325,
"Mixture:: "C84-02
cantera counterflos calculations.
"doi": "Ittpa:/ /tte:////oo.o//0000 /ztg//..116///// coccoottflalmmmm alam. 2021 111758", "contsibutors"; "Wai Tong Chung and Matthias lame", "daseriptian':
"Compressible Inert CH4-02 Homogeneous rootropio Turbulence nMs",
cchm_therma_ter
("description": "sRC and Mixture-Averaged Transport
with constant lewis number",
"cantera_xmi""
xm1
Listing 1: Python command for generating global metadata for a BLASTNet Kaggle repository.
metadata [local']
["id": 0,
*Lime [8]": 6.88389e 06,
*UX_ms-1 filename": : ./GGttaUUX_m--Liddh dat""
*ux_ms-1
*U2_ms filename": "/data/ur_mo-__dd00
F_ filename". "/Gata/e_p__id00.dnn
"T_K filename": "/data/t_K_1t000.de
"RRO_kg=-3 f1 felnemme": :/hata/ARo__gmm3_idd00
"Y02 filename": "/daalar0a__Ld00.ddd"""
*yoad filename": "/aaaa/xcH__ee00.ade
["id": 1, ...l,
("id": 97, ...F
Listing 2: Python command for generating local metadata for a BLASTNet Kaggle repository.
21

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D.1.3 Structured Web Metadata
A /http://schema.
schema. org metadata has been added to http:://blasttrt...thub.... io/
datasets and tested with bttps: http:aacc...oooolecccc////e//eea
D.1.4 Reading Data
As shown in Figure BLASTNet data can be read by (i) loading the descriptive metadata with the ison package on python, and (i) using np fromfile/np 1amamaa mennap load and reshape the data. Links to tutorials to perform this are also shared in Appendix A
D.2 DNS Configurations
BLASTNet 2.0 contains data from 34 different DNS configurations. The Kaggle link of all configura- tions are provided in Table 4 The details of each DNS configuration are provided in this section as well.
Table 4: The Kaggle link of all 34 DNS configurations. Ka,, Un and St, denote the Karlovitz number, inlet bulk velocity and laminar burning velocity, respectively.
Index
Sec:
Short Descripton
Kaggle URI
10:0
Isert HET 20
Kagols, dett
Reacting foroed HIT 67
Reacting jet doo 20]
D223
Reacting jet Nlo 20
5-10
Tnert unmseril. cban. Dew 69
X20 KA3914 c0m1
11
Reacting channel now
18
X20 ER3PIC 1. 2009)
wey mir Us
12
Slor burner 22]
N99,2agg1 9 1907 axee 37.0 mny Trren Frame yoe
13
Freely propagating flame
(Kow 2.4, U.n./.... i.
70
2. 4.4
iatarete/vs:: wtt.///tttngggnngg.............. www.errrrooopttiin... wotttti---------a-- www.gie..ooo com/ kww.eeggle
N2-
c00e-
14
[Ka. 6.8, 6.8. 15
2. 2.461
kvew, kequle, tom/ cowe a/va it oun chunn Crew propacel Iig 12 vi
15
(Ka. 13. Unal S
2.45)
1:29 Aa-gur 1 conn datcetre wir 57 wa and emeeeer Fre prop and de vi
16
[Ka
2.4, 2in, /S..//
= 3.677
120 2839 a 20mn mas asa aaaars70 3780 and one mor rea pi Proppaaat Tey gi
17
6.8. 6.8.Uin i
= 3.677
LWV. (a0. a comm assa aassa-rs// 3799 and marchirnn res pr Propanaa Tet by v1
18
(K. 13. U..Un//5 5t.
= =3.67)
120 242911 con 12 wa whi reo- peo--ro PyS
19
(K.. 19. Uun//5 5, = -3.67)
120 K00.91911 L. 20m7) 0113744 s/ve
Thunel rree- wes--pooooooooooo poppar
20
(ka. 36, Unn /S..
+3.67)
1X2 L.22919 100 (o0n//a11
21
(ka. -,., 55 = 44655)
1229 4422915 1...1,... con Kon/ Goteare
22
D.27
(ka. =6.8,Um. 5L 4 4.63)
1220 532015 com) oitorees Wart ehene tree- Proppearie -eree-eit
25
[Ka, 13. U3. Sz
4.63)
24
(ka. 2.4, C.a -,.,
5.511
25
(Ka 6.8. U.S.U..
5.51)
1229 E33011 datt//asaaaa
26
[Ka,
U.n
5.51)
vi
27
1.7. /5L 3.67)
28
(Ka, 4.8. U.S.Un. 5
3.67)
29
/5,.
30
U., 15
31
4.8. 5L
32
8.9, Un, SL
NWM,3agg1e sraggie.com ott.o/tttteeeee shungg rree- propagat
D.2.1 Non-reacting homogeneous isotropic turbulence (HIT)
The HIT DNS simulation 20 is performed on a 3D cubic domain of length L, where a spherical gaseous-oxygen
of radius r 0.25L at 300 K is initialized in gaseous methane environment of 300 K at 1 atm pressure, providing an idealized representation of an inert gaseous fuel-air mixture in rocket engine. The simulation setup is shown in Figure & Periodic boundary conditions used at
all boundaries. A synthetic turbulence generator by Saad et al. [90] based on von Kármán- energy spectrum with zero mean velocity is used to generate the initial velocity profile. Ideal gas law is used as the equation of state (EoS) to relate pressure, temperature and density.
The simulation is performed in an unstructured compressible finite-volume solver 664 The solver uses fourth-order accurate central spatial finite difference scheme. For the time integration, stable third-order Runge-Kan Kutta scheme is employed. As mentioned before in Section 3] aveeued transport properties are used in the DNS.
D.2.2 Reacting forced homogeneous isotropic turbulence
The DNS study 67 involves a statistically steady, isotropic, and homogeneous turbulent flow in an unconfined space. The schematic of the simulation setup presented in Figure 9 The flame is initialized by planar surface separating half of the domain containing methanc/air mixture at
22

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You 1.0
You
Tou 300
TCB 300
Periodic BC
atin
Periodic BC
Figure 8: Schematic of the DNS configuration of inert HIT. & denotes large-eddy timescale, while I denotes integral lengthscale. Adapted from (00), Copyright 2022, with permission from Elsevier.
700 K and 3.04 107 erg cm- pressure, and another half with hot products, and is immersed in high-intensity turbulent flow field with Kolmogorov type spectrum. The idea investigate the process of flame interaction with steady homogeneous isotropic turbulence. However, the flow needs to constantly stirred at the largest scale to ensure a steady energy cascade to smaller scales so that the turbulence-Ilame interaction at the quasi-steady state can be studied. A spectral turbulence- driving method is used the study, the details of which are available in Poludnenko and Oran 667. This driving method produces statistically steady foreed-HIT flows with arbitrarily complex energy spectra. In particular, is possible to achieve Kolmogorov type turbulence with inertial range of energy cascade extending up energy injection scale. The other advantage of this method is that it does not introduce any artificial large-scale anisotropy, compression, or rarefaction. Prior to ignition, all domain boundaries are periodic. At ignition, boundary conditions along the left and right z-boundaries (as shown in Figure ( are switched to zero-order extrapolation to prevent any non-physical pressure build-up in the domain and the formation of artificial large-scale rarefaction waves at the boundaries.
Periodic BC (zero-order extrapolation after ignition)
Periodic BC (zero-order extrapolation after ignition)
Periodic BC
Velocity
CH2O
Initial partition
Reactant side
Flame
Product side
Schematic of the DNS configuration of reacting forced HIT 67 The top and bottom contour plots correspond to the magnitude of velocity and CH2O species mass fractions, respectively.
Figure 9:
The computational domain aspect ratio is 1 x 1 16, with grid size of 257 257 4097, including 16 grid points per unit laminar flame thermal thickness. The cell size is 2.62 x 10-4 cm. The turbulent velocity at energy injection scale (L 0.067 cm) length scale 213.92 cms-! with turbulent root-men mean- squared (RMS) velocity of 245.83 cms resulting in an eddy turnover time of 3. 3.14x 10-4s s. The same velocity quantities corresponding to the integral length scale 0.0196 cm) are 141.93 cm and 132.2 cmns The ignition delay time of the mixture is three times the eddy turn-over time, and the total simulation runtime is times eddy turn-over time. The Damköhler and Karlovitz numbers are 0.66 and 9.97, respectively.
23

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The DNS calculation is performed using the code Athena-RFa RFX [67], which implements higher-order fully conservative Godunov-type methods for integration of fluid equations. The numerics in this work are third-order accurate in space and second-order accurate in time. More details are available in the original paper [67]. The foundational fuel chemistry model (FFCM-1) [91] with 22 species and 107 reactions is used as the chemical mechanism.
D.2.3 Reacting jet flows
The DNS configurations by Brouzet et 29 involve two parametric variations of 3D reacting turbulent premixed methane/air round-jet flames with high-fidelity acoustics to investigate the effect of different chemical mechanisms on flame dynamics. The setup is initialized with methane/air combustion products at adiabatic flame temperature and at atmospheric pressure. The jet Reynolds and Mach numbers are 5300 and 0.36, respectively. A schematic representation of the DNS configuration is shown in Figure [0] The two variations of the reacting jet correspond to two different chemical mechanisms: (i) a semi-global CH,-BFER mechanism with 2 reactions
and (ii) skeletal COFFEE mechanism [93] with 14 species and 38 reactions. In both configurations the domain size 207 167> 167. The grid sizes are 1811 x 721 721 and 1546 x 676 x 676 for the BFER and COFFEE cases, respectively. These meshes correspond to 10 and 12 grid points per unit thermal flame thickness in the streamwise direction, and 12 and 16 points in the transverse and spanwise directions.
Non nDDeing
Figure 10: Schematic of the DNS configuration of reacting jet flows. Reprinted from 29 Copyright 2021, with permission from Cambridge University Press.
The DNS is performed using the code Navier-Stokes equations along with energy and species conservation equations in Cartesian coor- dinates. The solver uses an eight-order explicit central spatial difference scheme and a third-order Runge-Kure Kutta time integration scheme. Ideal gas law and mixture-averaged averaged species-specifts properties are used for the simulations. Further details of the DNS configuration and solver are provided in Brouzet et al. 29
944, which solves fully compressible
D.2.4 Non-reacting transcritical channel flow
The study by Guo et al. [69 involves six different configurations of wall-bounded DNS in the transcritical regime. The schematic of the DNS setup is shown in Figure 11] They used nitrogen N₂ as the working fluid with a critical pressure and temperature of p. = 3.39 MPa and T. = 126.19 19 K. These studies consider the flow of N₂ inside a channel with a hot top and cold bottom wall with temperatures Thot and Teold. respectively. The six variations correspond to different temperature ratio (TR) between the two walls. The channel is periodic in streamwise and spanwise direction, while the wall boundary conditions are enforces at two walls. The domain dimensions are Lg X 2Ly x L., where T.g/ b.y = 2n, T.g/ T. = 4m/ nr33 and the channel height is 2h, = 9.0132 x 10-5 m. Cartesian grid (with mesh size 384 x 256 x 384) is used for all six configurations.
24

---

Configuration
1/
4mt
hau
24,
x.
direction
Tos
2
Contours of Cp
10a
10-1
5/L,
10-2
10-3
2m/33
4TT/3
109
10-2
10
2m/3
4m/3
Figure 11: Schematic of the DNS configuration of non-reacting transcritical channel flows. The contour plots correspond the ep for configurations TR3 and TR1.9, Adapted from 669. Copyright 2022, with permission from Cambridge University Press.
summary of the individual configurations is provided here in Table with more details being available in the original paper [69]. The operating conditions are chosen such that they cover the range of density ratio, Ω Phot/ Poold. between and 20, where p the density the fluid the wall. The name of the configurations are based on different TRs. Configurations TR3, TR1.9, TR1.4 and TRI.3 are transcritical, whereas the other two configurations are sub-critical. The pressure for all configurations is set to be 3.87 MPa, which is higher than P.. The bulk Reynolds number is 10 Table also reports the friction Reynolds number (Re,) for two walls using the channel half-height L, as the length scale.
compressible finite-volume solver (64 is used for these DNS. The governing equations solved using strong tabilitty-peeserving Runge-Kutte scheme with third-order accuracy in time step- ping, and fourth-order accurate central spatial finite difference, which reduces third-order for non-uniform meshes. As the conditions these simulations are in the transcritical regime, the Reng-Robinson EoS is used, which provides better accuracy in predicting thermodynamic variables than ideal gas in the investigated regime. To avoid the pressure oscillations and to obtain physically realizable solutions, an entropy-stable double-flux model 664 used along with second-order accu- rate essentially non-oscillatory (ENO) scheme and Harten Laxt- Lax-Van Leer contact (HLLC) Riemann flux computations.
D.2.5 Reacting channel flow
This DNS configuration by Jiang et al. (28 investigates the flame-wall interaction for methane/air flames diluted by hot combustion products a 3D turbulent V-flame configuration inside a channel with isothermal hot and cold walls. The simulation setup is shown in Figure 12 At the inlet of the channel, the reactant mixture consists of mixture of cold reactants (30%) and hot combustion prod- uets from 1D premixed of Tin 1705 K at alm pressure. The hot and cold wall temperatures are fixed 1200 and 400 K, respectively. The inlet turbulence is generated with non-reacting simulation of the same channel. Then, the results collected sampling plane of x H are fed into the reacting simulation. This turbulence generation allows coupling of the velocity and temperature fluctuations at the inlet.
pelyypprogggttig flame simulation (70%), resulting in an inlet temperature
25

---

Table Summary of the operating conditions of six different DNS configurations of non-reacting transcritical channel flows [69] coolld = Tolld To, = Thot/ T, and Pr.0 Po/Pes where subscript and indicate volume averaged and critical quantity, respectively.
Configs.
Trodold
Tr.
Pr.0
Rér.eold
Ré,,hot
TR3
0.79
2.38
1.16
17.84
433
300
TR1.9
0.79
1.51
1.60
10.05
440
610
TR1.4
0.79
1.11
1.92
5.24
500
1370
TR1.3
0.79
1.03
2.09
2.89
570
1530
TR1.25
0.79
0.99
2.19
1.60
590
1290
TRI
0.79
0.79
2.36
1.00
700
700
(a) Inflow
Outflow
(1200K)
Flame stabiliscr
Cold wall (400K)
(b)
Norallsse HRR
0.5
1..5
2
x/ll
(OPPIIFF)
Flane (FPE/PER)
Figure 12: Schematic of the 3D DNS domain of reacting channel flow, The bottom figure shows snapshot of 2D normalized heat release rate contour along with the black dashed line, which is an iso-line of methane progress variable at particular value. Reprinted from (28). Copyright 2021, with permission from Elsevier.
Velocity fluctuations are first produced using the Passot- Passot-Puqqut spectrum for the turbulent kinetic energy. The inlet turbulence for the non-reacting simulation was then generated by rescaling these fluctuations with the RMS profiles of a fully developed channel flow at a Reynolds number 3200. Next, this is fed into the domain with a convection velocity 25% lower than the mean inlet velocity at the centerline. This accounts for a correction to the Taylor's 's hypothesis due to the high near-wall shear stress. Non-reflecting Navier-Stokes Characteristic Boundary Condition (NSCBC) is used for the outlet boundary, and a periodic boundary condition used the z-direction. For the reacting case to ignite, cylindrical hot patch is imposed y/ H = 0 and x/ H 1 with diameter of 0.03H,, which creates two branches of the V-flame that interact with two walls.
The domain size is 12H x 2H 3H, with grid size of 1000 250 250, which stretches from um at the wall to 30 fam at the centerline in the y-direction, and 30 jm uniform grid in both x- and z-direction, and ensures at least grid point within one wall unit and a mean grid size less than 1.4 times the Kolmogorov length seale. There are around 20 grid points inside the flame thickness as well.
The numerical solver used for the DNS study NTMIX -HIEMKIIIIN 94 Similar to the study of Brouzet et al. [29), this solver features eighth-order central finite difference scheme for spatial derivatives and a third-order Runge-Kure Kutta time integrator. A tenth-order explicit filter also used to eliminate spurious oscillations at high wave numbers. Ideal gas law is used as the EoS. A reduced mechanism for methane/air combustion with species, 12 quasi-steady species and 205 reactions developed for this study.
26

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D.2.6
slot burner
This DNS configuration 27 involves turbulent lifted hydrogen jet flame in heated co-flow air. Figure 13 shows the schematic of the simulation setup. A diluted fuel mixture (65% H2 and 35% N₂ by volume) is issued from the central slot at inlet temperature of 400 K. This central jet is surrounded on either side by co-flowing heated air streams with an inlet temperature of 850 K, at atmospheric pressure. The mean inlet axial velocity Un is given by:
Un = Uc Ujet Ue
tanh y
(U+H/2 H/2
V リ H/H
]
tanh
(7)
2
0.1//
0.1//
where the mean inlet jet (Ujat) and co-flow (U.) velocities are 240 and 2 ms respectively. The jet width the inlet is mm. The other quantities, such temperature and species mass fractions also follow the same profile (Equation (7). The jet Reynolds number is 8000. Velocity fluctuations, u', which is 10% of Ujot. is obtained by generating auxiliary homogeneous isotropic turbulence field. These fluctuations are then fed from the inlet using Taylor's hypothesis. This 2000 x 1600 x 400 computational domain is 15H 20H 3H in the streamwise 2-, transverse yr, and spanwise directions, respectively. resulting in a total of 1.28 billion cells. A uniform grid size of 15 μm is placed in the x- and 2-directions, while the y-directional grid is algebraically stretched outside the flame and shear zones. Improved non-reflecting boundary conditions [95] (6 are adopted in the r- and y-directions, while periodic boundary conditions are applied in the z-direction. The data collected after four jet flow-through times after the flame becomes statistically stationary.
20H
3H
Ourllww
Outlow
Outtlow
1511
Periodic
Fuel
Oxidizer
H
Figure 13: Schematic of the DNS of hut-brrner setup. The contours correspond to OH and HO, species. Reprinted from [27] Copyright 2021, with permission from Elsevier.
The Sandia DNS code, S3D 65 is employed for solving the compressible Navicr-Stokes. species conservation, and total energy equations. Spatial derivatives are approximated with an eighth-order central difference scheme, and tenth-order filter is used to remove any spurious high-freon nigh-frequency fluctuations in the solution. For time integration, fourth-order explicit Runge-Kute Kutta method used. The employed detailed hydrogen-air chemical mechanism composed of 9 species and 21 elementary reaction steps was developed by Li al. 97
D.2.7 Freely-propagating flame
This DNS configuration 770 presents statistically-planar, freely-pro propaaatiipanggg flame. BLASTNet contains 22 parametric variations of this configuration that differ by three essential parameters involving turbulence: (i) turbulence intensity. characterized by the RMS velocity u', (ii) inflow velocity, Un. and (iii) integral length scale, 1,. A schematic diagram of the setup shown in Figure These configurations represent a series of hydrogen-premmss turbulent flames in autoignitive reheat combustion conditions that provide rich information on regimes turbulent spontaneous ignition and turbulent deflagration.
27

---

Table 6: Summary of the simulation parameters for all DNS runs 770.
Config. Index
Unn/St
ul St.
Kau
Rc
Däign
Lend/ Tign.o
La Lay
No No x No
1//1 1.5
2.45
0.5
2.4
53
0.24
13
9
1152 x 128 x 128
2
2.45
1.0
6.8
105
0.12
4.6
9
1132 x 128 x 128
2.45
1.5
13.0
158
0.08
3.6
9
1132 X 128 x 128
3.67
0,24
11
1408 x 128 x 128
3.67
0.12
11
1408 x 128 x 128
3.67
1.5
13.0
1.58
0.08
3.8
11
1408 x 128 x 128
7
3.67
2.0
19.0
211
0.06
2.9
11
1716 X 156 x 156
3.67
3.0
36.0
316
0.04
1.8
11
2816 256 x 256
4.63
0.24
1792 X 128 x 128
10
4.63
1.0
6.8
105
0.12
8.2
14
1792 x 128 x 128
11
4.63
1.5
13.0
158
0.08
6.8
14
1792 x 128 x 128
12
5.51
0.5
2.4
53
0.24
6.5
14
1792 x 128 x 128
13
5.51
6.8
0.12
1792 X 128 x 128
5.51
19.0
211
0.06
14
2184 X 156 x 156
3.0
15
3.67
0.5
1.7
105
0.49
11.0
5,5
1408 x 256 x 256
16
3.67
1.0
4.8
211
0.24
5.3
5,5
1408 x 256 x 256
17
3.67
1.5
8.9
316
0.16
1.6
5.5
1408 x 256 x 256
18
4.63
0.5
1.7
105
0.49
7.8
7.0
1792 256 x 256
19
4.63
4.8
211
0.24
7.0
1792 x 256 x 256
20
4.63
1.5
8.9
316
0.16
5.1
7.0
1792 x 256 x 256
21
5.51
0.5
1.7
105
0.49
9.3
7.0
1792 X 256 X 256
22
5.51
1.5
8.9
316
0.16
5.8
7.0
1792 X 256 x 256
Config.
initial flame position
outflow
constant bulk velocity
X Un
X
Flowfields
(m/s()))
1/₁
36
54
3
68 and 81
Figure 14: The top figure shows the DNS configuration of freely propagating flame. The bottom figure shows the coss-secciooal views of the H,O, contours along with the initial position of the spontaneous ignition front (dashed white line). Reprinted from (7),, Copyright 2019, with permission from Elsevier.
The turbulent flames are initialized with an ignition front. For the initial flat spontaneous ignition front, the thermo-chemical conditions are chosen to be representative of those at the end of the first stage of
28

---

a heavy-duty gas turbine sequential combustor, but at a lower pressure of 1 atm for all configurations. The mixture of fuel and products of first stage hydrogen-air combustion an equivalence ratio of 0.43 and initial temperature of 773 K is used at the inlet of the domain. This mixture is equivalent to an equivalence ratio of 0.35 and Т, 990 K, and its ignition delay time (Tign.0) and laminar flame speed (St) are identified to be 0.55 ms and 14.7 ms respectively. The reference laminar flame thickness, 1, is evaluated to be 0.66 mm. After initialization, the ignition front is superimposed on turbulent flow-field using a one-to-one correspondence x-space (Figure [4]. Depending on varying Un and u', the flame may stabilize at position far away from the inlet (a turbulent spontaneous ignition front) or the introduction of turbulence may trigger the transition to a deflagration, where the flame front propagates towards the inlet.
The width of the domain in the y- and z-directions is Ty = I., 5. 5.261,, and the length in the streamwise direction is Lz. which is different for individual configuration. The other associated parameters for all 22 configurations are summarized in Table [ The turbulent Reynolds number for the unburnt gas is defined as Rer.a = o'lz/Vu. where Vs is the kinematic viscosity of the unburnt gas. The Karlovitz number, Ka,, is defined as the ratio of the flame characteristic time (y = LS,, the Kolmogorov characteristic time Lun (v,l/u y The ignition Damköhler number is Daign = Ti/Tig..0. with the large-eddy turnover time T = 1//n'. For all configurations, the simulations are until statistically steady state is achieved.
The low Mach number form of the governing equations is solved using the energy conservative, finite difference code NGA [66] and high turbulence simulations are enabled by the linear velocity forcing method. NGA is second-order accurate in both space and time, and uses a semi-implicit Crank-Nicolson time integration scheme. A third-order bounded QUICK scheme, BQUICK, is used for scalar transport. Ideal gas law is used as the EoS for a mixture perfect gases. A detailed chemical mechanism 97 for hydrogen combustion with species and 21 reactions is used for all configurations.
E Additional Momentum128 3D SR Dataset Details
The Momentum128 3D SR dataset a processed subset of BLASTNet 2.0, and available for download at httpp:://wwt /www. kaggle com http:///taatttsaattttteeeeoeeeee Mlastnet-momente um sd----dd-rrrrdet
E.1 Data Format and Directory Structure
The Momentum 128 3D SR Dataset contains velocity and density sub-volumes (see Appendix E.1.1 extracted and processed from BLASTNet 2.0, along with descriptive metadata (Appendix E.1.2) web metadata (Appendix E.I.),, and instructions for reading the data in Appendix E.1.4
seed42
40 pre-trained models of different sizes used in this work, along with .csv file with hyperparameters for loading models.
seed43
5eed44
mvodyssuumayy.ccv
dataset
M HR
forcedhit
paramvvr
Labels with the five splits mentioned in the main paper.
test
train
LR_ LR_16x
LR R,3
Features and 32x SR.
LR.8x
jl_datt_summary.csv
Descriptive metadata
Figure 15: Directory structure the Momentum 128 3D SR dataset.
29

---

E.1.1 Files on Flow Physics
All 2000 sub-volumes (labels with 128 X 128 x 128 number of voxels) of density and three velocity components [p.u] are also presented little-endian single-precision binary format, in similar fashion to BLASTNet 2.0 (see Appendix [.I.I). which can be read with np. fromf iomeiee np memmap. This is shown in Figure [5] which also shows the five data splits described in Section 3.2 addition, Favre-filters firte-iiteeed features for 8, 16, and 32x SR also provided, along with pre- t-tirranned weights from all models reported in this study. We include a notebook with the method used for obtaining tricubic interpolation in the code repository described in Appendix A The sub-volume files named with <Variabl Name and SI Un tesitt____h__ unique ID based the spatial coordinates of the sub-volume location and the index of configuration.
value> dat, where the hash value provides a
E.1.2 Descriptive Metadata
In addition to the sub-volumes, we provide csv files that provide information on hash ID, Kaggle ID, short configuration description, k-means cluster index, and spatial grid size for the different dataset splits used in this work.
E.1.3 Structured Web Metadata
/scchma/ org metadata has been added to htt.ps://blasttrtt..tttttt io/ wwweerrrc.ooollee...ooeeoeoo.eee wwm//esst/rcchhreuelllee
da datasets and tested with
Standard Deviation pe [kqm is =]
1200
Preely-propageting (Kau 1.7, UlaSL 3.67)
(Kau 4.8, Uin/SI. 4.63)
1000
(Kau = 1.7, Uia/SL 4.63)
(Kau 6.8, Uin/SI. 2.45)
(Kau = 1.7, Uin/SL 5.51)
(Kau 6.8, Uin/SL 3.67)
(Kau 13, Un /SL 2.45)
(Kau 6.8, Ula/SL 4.63)
800
(Kau 13, Uin /SL 3.67)
(Kau 6.8, Uin/SL 5.51)
(Kau = 13, UlwSL = 4.63)
(Kau 8.9, Uin/SL 3.67)
600
(Kau = 19, Uin /SL 3.67)
(Kau 8.9, Uin/SL 4.63)
(Kau = 19, Uin/SL = 5.51)
(Kau 8.9, Ula/SL 5.51)
(Kau 2.4, Ula/SL 2.45)
Inert HIT
400
(Kau 2.4, UlaSSL 3.67)
Reacting Forced HIT
(Kau 2.4, Uia/SL 4.63)
Reacting Jet Flow (14 Species)
200
(Kau = 2.4, Uin/SL 5.51)
Reacting Jet Flow (6 Species)
(Kau 36, Unn/SL 3.67)
Slot burner
(Kau 4.8, Uia/SL 3.67)
1000 Mean of pe [kgm -1
500
1500
2000
Figure 16: Statistics of the specific kinetic energy pe of each 128 sub-volume in the Momentum 128 3D SR dataset. Kaggle links to source (raw) data is available in Table 4
E.1.4 Reading Data
Similar BLASTNet 2.0, this data be read by using n fromfi weenffleeppp..mm memmap to load and reshape the data. Links to tutorials and dataloaders to perform this are also shared in Appendix A We also attach Kaggle notebook for training and evaluating with this data.
E.1.5 DNS Configurations
We select the 2,000 sub-volumes from BLASTNet 2.0 to form 67 GB dataset that can fit into single Kaggle repository in order to develop labels for a compressible turbulence benchmark dataset that can be easily downloaded. Mean and standard deviation of the specific kinetic energy from the resulting sub-volumes are shown in Figure I6, which a more detailed version of Figure 2
E.1.6 Favre-filtered DNS and Coarse-grid Simulations
For the Momentum122 128 3D SR low-resolution feature samples, we employed Favre-filtering (Equa-
tion 3 to generate canonical surrogate, known finite-volume optimal LES (74), that has direct
30

---

theoretical connections 73 45 to coarse- coarse-grained simulations (also known as implicit LES). While having an implicit LES solution and corresponding DNS data as feature-label pair within BLAST- Net would be ideal, it is not feasible to obtain a matching implicit LES-DNS pair due to the stochastic nature and time-dependency of fluid simulations. Specifically, small changes in the system (such as grid size) can result in widely different flow behavior due to the chaotic nature of turbulence [72]. As such, numerous works involving turbulence modeling, involving analytic [43] [4] and SR [46 37 have conventionally employed filtered DNS in place of implicit LES flowfields.
Figure 17 demonstrates the quantitative and qualitative relationship between 8x low-resolution and super-resolved implicit LES and fivree-iltered DNS through normalized turbulent kinetic energy (TKE) and velocity magnitude flowfields, respectively. Prior to evaluation, we normalized length by the domain length; velocity is normalized by its RMS. The TKE spectra is common tool for analyzing the turbulent properties across different lengthscales (up to the DNS wavenumber k through Fourier transform operations [26]. Note that the wavenumber K. is inversely proportional to length, Le., larger wavenumbers correspond to smaller lengthscales. To generate the implicit LES solution, we perform coarse-grid simulation of the HIT DNS configuration detailed in Appendix D with a 16 spatial grid. It can be seen that TKE spectra is truncated a similar fashion in both low-resolution implicit LES and Favre-filtered DNS at Khog demonstrating that turbulence is under-resolved beyond this wavenumber due to the coarse grid. We perform 8 SR on both low- resolution flowfields with the 50.2M gradient-loss RRDB model. The super- resel-eesolved DNS demonstrates excellent agreement between with ground truth DNS, the super-resolved implicit LES demonstrates maintains reasonable accuracy with the ground truth DNS spectra. The deviation in TKE spectra is within a reasonable margin of error, especially compared with similar analysis in other ML works 35 98 involving turbulence. This result demonstrates that ML model trained on Favre-Filtered DNS can be employed towards super-resolving implicit LES flowfields while maintaining reasonable spectral behavior.
Nom.
ML Input
ML Output
Corresponding DNS
[-]
3,0
DNS lapst)
Implicit LESS
Norn. TKE Spectra [-]
Fore Dies (ML layat)
Coresponding DNS is unfensible to obtain.
2.5
Trgliät LES Dappat
10-1
2.0
Form fibmemee
1.5
10-3
205
Favre filtered DNS
1.0
10.4
005
10'
10a
Norm. Wavenmberrrr Kuco [-]
0.0
Figure 17: Comparison between normalized turbulent kinetic energy (TKE) and velocity magnitudes between ML (RRDB with gradient loss; 50.2M parameters) inputs and outputs from 8x implicit LES (coarse-gr) -rid simulations) and Favre-Filtered DNS of HIT configurations. An ML model trained on Favre-Filtered DNS can be employed towards super-resolving implicit LES flowfields while maintaining reasonable spectral behavior.
F Additional Experiment Details
Here, we provide further information that supplements Sections and
F1
Additional Metrics Details
For this work, we employ conventional definition of normalized soutuee error (NRMSE) for evaluating the ML models:
Nous Σ Ex=1 i=
Neamp (digh Hestt )2
NRMSE (ASE((,))) Ø)=
(8)
ΣN -1=1 E
Naump
Ek= 00
31

---

with Now number of voxels and N samp number of samples of model prediction á and ground truth @. As discussed Section 4.2, NRMSE is used evaluate super-resolves density and velocity, along with subgrid-scale stress divergence, volume-averaged kinetic energy, and volume-averaged turbulent dissipation rate. When evaluating the NRMSE turbulent dissipation rate, the samples are weighted such that the kinematic viscosity is unity, in order to emphasize contributions from the predicted velocity components.
We also evaluate our models with the a 3D version of the structural similarity index measure (SSIM) 82 SSIM is used evaluate super-resolved density and velocity, along with subgrid- scale stress divergence. SSIM is calculated by passing sliding window of size 9x9x9 (similar to the original SSIM paper [82] across model prediction á and ground truth Ø, and evaluating their statistical quantities. Specifically, SSIM is defined by:
Noump No
2paus + oî
20 26
+ 2
SSIM c () N Neamp-NwN No
(9)
/ + m t
+o +
j=1
(o.d) and covariance āe for N number of sliding windows. In computer vision applications with RGB images, numerical stability [82]. However, we found these values insufficient for numerical stability this work. Hence, we employed = 0.1 and c 0.3. Note that for SSIM (Equation (4b)), the edge voxels of V *98 are neglected prior evaluation, to remove voxels with first-order spatial differencing.
with mean (( variance a²
0.01 and Co 0.03 are typically used to ensure
Table 7: Hyperparameters of RRDB, EDSR, and RCAN models investigated in this work.
RRDB Parameters
0.6M
0.9 M
1.4M
2.7M
4.9M
11.4M
17.8M
50.2M
Residual (RRDB) Blocks
5
23
First Channel Size
4
16
32
64
64
64
64
64
Kernel Size
3
3
3
3
RRDB Growth Factor
32
32
32
32
32
32
32
32
Residual Scaling
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
EDSR Parameters
0.5M
1.0M
1.4M
2.8M
5.1M
11.1M
17.8M
34.6M
Residual Blocks
32
32
32
32
32
32
32
32
Channel Size
14
20
24
34
68
120
Kernel Size
3
3
3
3
3
3
3
Residual Scaling
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
RCAN Parameters
0.5M
0.9M
1.5M
2.7M
5.1M
11.8M
16.4M
48.3 M
Residual Blocks
1
1
10
20
20
Channel Size
26
34
44
60
64
64
Residual Groups
1
1
1
1
2
3
10
Kernel Size
3
3
3
3
Residual Scaling
1
1
1
1
Conv-FNO Parameters
0.6M
1.8M
2.6M
5.2M
9.4M
20.6M
32.9M
FNO modes
2
2
2
2
FNO Channel Size
14
20
24
34
46
68
86
Conv-FNO Blocks
32
32
32
32
32
Convolutional Kernel Size
3
3
3
3
3
Residual Scaling
0.1
0.1
0.1
0.1
0.1
0.1
0.1
F.2
Additional Model Details
In this work, three well-studied 2D ResNet-based [76 SR models are modified from their original repositories for 3D SR: (i) Residual-in- Residuall Dense Block (RRDB) 22 (ii) Enhanced Deep
Residual Super-resolution (EDSR) [23], and (iii) Residual Channel Attention Networks (RCAN) 24
32

---

We choose to study these models due to their difference in architecture paradigms. Specifically, RRDB employs residual layers within residual layers; EDSR features an expanded network width; RCAN utilizes long skip connections and channel attention mechanisms. In addition, we consider model that employs Conv-FNO blocks. Specifically, outputs of an FNO layer and a convolutional layer were added to the outputs of each residual block in the EDSR, in similar fashion to both Conv-FNO and U-FNO models [25]. This modification enables us to examine combining FNO layers with convolution blocks that have been demonstrated to perform well in SR applications.
To investigate the scaling behavior of the model architectures, we vary the number of parameters by changing the network depth and width. All other hyperparameters are maintained from their original studies, with all models initialized via He al. [78]. Specifically, the architecture settings for RRDB, EDSR, RCAN, and Conv-FNC -NN are shown in Table 7 In this table, we list the the number of residual blocks, growth factor in RRDB blocks, the channel width, kernel size, number of FNO modes. RCAN residual groups, and residual scaling factors which are arguments for the model objects the code described in Appendix A The hyperparameters within the Conv-FNO model was first determined by comparing two approaches with the same number of parameters (3.0M): (i) one with large number of Fourier modes, and (ii) one with deep and wide Conv-FNO blocks. Since the approach number of modes with deep and wide Conv-FNO blocks demonstrated better validation MSE, another hyperparameter search was performed to determine the optimal number of Fourier modes ranging until the GPU memory was fully consumed at five Fourier modes. We scale the Conv-FNO in Section[]jy increasing the FNO channel size since this approach led to good scaling behavior, especially when compared to increasing the number of Fourier modes.
Table Validation MSE for different hyperparameters of Conv-FNO.
Conv-FNO Parameters
3.0M
3.0M
5.2 M
10.9M
21.7 M
39.8M
FNO Modes
12
2
3
4
5
FNO Channel Size
34
34
34
34
Conv-FNO Blocks
6
32
32
32
32
32
Val. MSE [x10 10 31
278
175
5.3
87.8
182
156
F33
Additional Loss Details
46 all models are trained with mean-squared-error (MSE) loss LMSE, unless otherwise stated. Specifically, for a predicted channel quantity & and ground truth 6:
Similar to other turbulent SR studies 37
Noor Noump No
MSE
Σ (digh - &2gH)2
(10)
Nooz samp
J=1 k=1
for Nvor number of voxels, N Neamp number of samples, and N, number of channel variables. When comparing the use of MSE and mean absolute error (MAE) loss, we found that both models trained with both losses resulted in the similar validation MSE at the end of 1500 epochs. However, MSE loss demonstrated better stability early timesteps, shown with the 34.6M EDSR 8x model in Figure T8 This increased robustness motivated our choice of MSE as loss function.
In addition, we train variants of the RRDB with
informed gradient-based loss resulting in: (1
Lphys (1 --) + >bbgrad, where
Now N
Naamy N.
Ligrad 3N voz
(V (Vos)))d?
(11b)
N Vamp.
Am
The gradient terms are evaluated using torch. gradient, which corresponds to a second-order central differencing scheme that is optimized for GPU caleulations. This is done both super- resolved and ground truth fields before inputting the gradient terms into the MSE function. This gradient term enables the MI models to implicitly learn transport phenomena that arise in flow physics PDEs. For example, advection in the mass conservation equation can expressed as:
(pu) = u Vp + pV u,
(12)
33

---

Trained with MSE loss
10-1
Trained with MAE
Validation MSE
LR x0.5
10-3
300
600
900
1200
1500
Epuch
Figure 18: Validation MSE when comparing EDSR 50.2M models trained with MSE and MAE loss.
which requires the gradients of all channel variables to be predicted correctly. These arguments can also applicable the advection of momentum, which the transport term responsible for turbulent phenomena [26].
In Section 5
employ the weighting factor 0.99, which determined from hyperparameter search on RRDB 2.7M models, with the validation MSE for various lambda shown in Table g
Table 9: Validation MSE for different weighting factor À for the gradient-base loss.
Weighting factor À
0.000
0.100
0.300
0.500
0.700
0.900
0.990
0.999
Val. MSE [x10] [x10-$]
2.18
2.20
2.14
2.12
1.97
1.74
1.42
1.64
F.4
Additional Data Augmentation Details
Data augmentation is performed via variants of random rotation and flip transformations which we modified to ensure augmented data remains consistent with mass conservation. Specifically, this is necessary for maintaining the reflective and rotational invariance of the divergence of momentum (pu)), after transformation. The steps to ensure this are summarized Figure 19 which demonstrates how flip and rotation operations can still result in continutyy-consiseent transformations on image. These operations have been extended for 3D random flip and rotation, which employed during training. Links to code, with implementations of these transformations and corresponding unit tests, are provided in the code described in Appendix A
F.5 Additional Training Details
For evaluation, we select models with the best MSE after training for 1500 epochs with batch size of 64 across 16 Nvidia V100 GPUs. Learning rate is initialized at 1e-4, and halved every 300 epochs. Both the number training iterations and learning scheduling are chosen match other SR studies 24 and are found to be sufficient for the SR prediction shown in Figure 20 where flat
validation loss curves are seen at 1500 epochs for vanilla RRDB 50.2M at all SR ratios shown. We note that at 32x SR, the model begins to show overfitting after 600 epochs. This may be because the SR models cannot learn a generalizable pattern from the insufficient information contained within coarse- coarse-grained features at high SR ratios.
F.6 FLOPs estimation
In this work, theoretical FLOPs for the ML models is estimated via THOPs (https:/) wwt.c//////... httpubb.ccmmnkkeee///ttttrrtt OpCounter) which has been used in other studies [99 100). Einstein summation operations in FNO layers were evaluated through modifying THOPS with numpy.einsum path, while Fourier and inverse Fourier transforms are estimated as 5NN pointss log N ppontts [53].
34

---

Continuity-consistent Flip
Flip
2. Neg. sign
U2
2/2
X2
141
141
X2 X2
41
X22
X1
X1
Continuity-consistent Rotation
1. Rotate
2. Swap axes
Tu U4
222
X2
U1
X1
11
U2
X2 X1
X1
x
X2
x X2
Figure 19: Continutyy-consistent augmentation 2D image that preserves reflective and rotational invariance of the V (pu).
8x SR
16x SR
32x SR
0.10
Train
0.08
Val.
LR x0.5
Loss
0.06
0.04
0.02
0.00
300
1200
300
1500
300
1200
1500
Epech
Figure 20:
Vanilla RRDB 50.2M MSE loss at 8, 16, and 32x SR, with initial learning rate
LR 1e-4.
For tricubic interpolation, an interpolant p at coordinates æ, y, and is defined by [IOI:
3
p(x, y, 2)
aijxz'y 2
(13)
i=0 5=0 k=
For single point, this totals to 63 additions and 84 multiplications for the matrix multiplication applications, with additional multiplications for the exponents. The elements of tensor asjk can be found in 64-element vector which can in turn be found via. Hence:
(14) where A is 64x66 64 integer matrix of constants that is predefined in the algorithm, and b is 64-element vector which contains derivatives from the input data ø. For uniform spatial grid, can be evaluated by evaluating derivatives on a sub-volume with xx4x4 vertices, which can be flattened 64-element vector @sub:
b,
A A2thuu
Bérub
(15)
where Ag contains integer coefficients for finite-differencing and = 1 contains 2765 zero clements. This totals to 4032 additions and 4096 multiplications, when considering dense matrices.
35

---

If sparse multiplication is used, this results in 1301 multiplications and approximately 1237 additions, with 64 multiplications for dividing by 8, Putting this together:
FLOOS PLOPsdesse dense 8328 eow No
Neoz
(16)
FLOPS sparsc 2738Ncorr-Ve. tys
(17)
In this work, our samples contain 128 voxels and 4 channels. Thus, tricubic interpolation costs 23 GFLOPs (reported in Section ()) for sparse matrix multiplication, while employing dense matrix multiplication costs 69 GFLOPs, per inference with batch size of I.
G
Additional Results
G.1
Additional Evaluation
In Figure 21 we present normalized absolute error in specific kinetic energy l k // pe from model predictions shown in Figure
Ground Truth pe [kgm Is 2]
Tricubic
RRDB(+ Grad. Loss) EDSR
Conv-FNo
Irw
[-]
0.5
1750
1500 86
0.4
1250
0.3
1000
0.2
500
0.1
250
0.0
Figure 21: Normalized absolute in specific kinetic energy pe pe from model predictions shown in Figure
Tables 10 to 12 report additional results from evaluating the models from Tables SateesZannne and with NMRSE metrics. In Table 13 we report the SSIM used for evaluating the scaling behavior all 8x SR models discussed in Section
Table 10: Comparison NRMSE five models at and 32x SR ratios, evaluated with the baseline test set. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
JNRMSE p.u (x10-2)
[NRMSE ags (x10-1)
JNRMSE (x10-4)
JNRMSE, (x10-1)
Models
RRDB 16x
5.6+-0.0
4.2+-0.1
5,5++0.1
4.5++0.0
(+ Grad. Loss)
5.3-10.0
3.94-0.1
7.7+1.00
3.7+0..0
EDSR 16x
6.0+00.2
4.5+0..2
6.1.0.1
4.740.11
RCAN 16x
6.740..6
5,340.6
6.740.6
4.940.1
Conv-FNO 16x
7.440.3
5,910.3
9.0-0..8
5.1
Tricubic 32x
28.3
9.5
156.2
9.6
RRDB 32x
20.2++0.1
7.740.11
39.3
8.3+0.0
(+ Grad. Loss)
19.8+0.0
7.5+0.1
48.1+2.2
8.340.0
EDSR 32x
20.1+0.00
7.740.1
45,4+11.
8,420.0
RCAN 32x
20.3+0.1
7.910.1
43.742.8
8.1-0.11
Cony-FNO 32x
20.5+0.1
8.0+H0.1
48.2+3.5
8,410.1
36

---

Table 11: Comparison of NRMSE of five models at 16 and 32x SR ratios, evaluated with the forced HIT set. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
JNRMSE (x10
JNRMSE gg (x10 3)
JNRMSE (x10 ")
JNRMSE (x10-*)
Models
Tricubic 16x
36.2
61.2
7637.2
2517.8
RRDB 16x
3.5
9.7+0.11
40.8+15.1
48.913.6
(+ Grad. Loss)
3.3++.00
9,4+00.2
242.4+89.6
45.6+12.5
EDSR 16x
4.0+t0.3
13.0#1.3
23.3+7.1
83.6#21.9
RCAN 16x
5.1-1.1
18.8+4.9
46.6+22.3
97.9+30.2
Conv-FNO 16x
6.6.10.8
24.8+2.8
124.2
90.0+110.1
Tricubic 32x
118.4
81.0
68869.9
6689.3
RRDB 32x
47.6+-0.7
1.7
1985.71-554.5
1753.9
(+ Grad. Loss)
0..6
4..
2753.6 1260.0
EDSR 32x
48.4+1.00
47.7+118
5709. 1 1100.1
2100.7+
RCAN 32x
4..6+1.9
47.5+2.0
8288.24809.44
1481.6::27..2
Conv-FNO 32x
48.8
49.1
1991,
1906.3#177.4
Table 12: Comparison of NRMSE of five models 8, 16, and 32x SR ratios, evaluated with the parametric variation set. Mean and standard deviation from three seeds are reported here. Bold term represents best mean.
UNRME p.u (x10-
JNRMSE.ge (x10-1)
JNRMSE (x10-1)
INRMSE, (x10-1)
Models
Trieubic 8x
37.5
70.6
301.1
3329.1
RRDB 8x
4.740..1
18.1+0.4
7.0+3.5
551.5+7.0
Grad. Loss)
2.7:10.0
8.7+00.1
15.9
232.7++1.11
EDSR 8x
4.2.10.3
16.2.1..5
1.440..4
487.7139.7
RCAN 8x
4.3
16.3+
1.3
493.345.1
Conv-FNO 8x
8.940088
37.2+2.2
6.3+3..1
830.3438.55
Tricubic 16x
114.7
83.7
1416.9
7139.8
RRDB 16x
39.6:10.3
36.0
83,9422.2
143..
(+ Grad. Loss)
36.0-10.3
30.9
140.9427.9
2923.1
EDSR 16x
42.2+11.7
40.0+1..6
52.615.7
3556.5+108.
RCAN 16x
48.7+5.22
48.3
47.0-16.4
3538.0+108.9
Conv-FNO 16x
54.8+2.9
55.8+2.8
104.5+29.5
3617.7+80.0
Tricubic 32x
242.6
93.0
4743.3
9374.6
RRDB 32x
162.0+1..2
72.2:10.3
615.0+110000
15.
(+ Grad. Loss)
161.5+0..3
70.5
820.2+151.2
7694.2+3.99
EDSR 32x
162.4+0.1
74.0
491.342.5
7623.6-83.00
RCAN 32x
164.6+0.8
73.3+1..
427.5+73..5
6961.8+261.0
Conv-FNO 32x
166.7+11.7
74.2++0.3
590.9491.9
7450.4+278.4
G.2 Efffett of Different Normalization during Evaluation
For the results in Section 5 all evaluation sets normalized with their own mean and standard deviation, as shown in Table 14 prior to testing. Note that we evaluate the same mean and standard deviation for all three velocity components. This is to account for the random rotation applied to the features and labels, as detailed in Appendix F4 which results the velocity channels swapping axes with each other during training.
When comparing the two normalization approaches in Table 15 seen during evaluation the Forced HIT set when normalizing via the train set, with SSIM 0.
we observe that poor performance
37

---

Table 13: Summary of models, of different number of parameters Np. trained 8x SR. Bold term represents best mean for given model approach.
Baseline Test Set
Param. Variation Set
Forced HIT Set
Models
Np
[GFLOPs
1SSIM pm
1SSIM age
1SSIMp.u
TSSIMage
TSSIMpa p.u
TSSIM age
Tricubic
0.820
0.431
0.800
0.418
0.951
0.711
22
0.6M 0.446::0.044
0.022:10.099
0.495:10.050
0.027:00.12
0.10110.078
0.055-100228
10
0.9M 0.754=0.003 0.248=0.026
0.775=00004
0.330=0.004
0.977=0.000
0.523+0.065
76
1.4M 0.841110.003 0.517:10.012 0.83910.000 003 0.574:10.005 0.9911100000 0.791 -0.006
273
2.7M 0.878=:0.001 0.608=10.002 0.867=00.002 0.649:::0002 0.996:1.000 0.845=+0.003
1041
RRDB
4.9M 0.884:10.005 0.635:10.016 0.873:10.000
0.677::0.015 0.99611.000 0.856110.004
1059
11.4M 0,888=0.002 0.657=0.005 0.878=000001 0.701
0.996=00000 0.867
1112
17.8M 0,893:10.001 0.672.10.004 0.884110.000 0.719110.005
1165
50.2M 0,907::0.003 0.715:00.044
0.898:000003
0.755+00002 0.9.000000.099000 0,997:
0.801
1430
0.6M 0.475=0..02 0.069=0.002
0.405=000002
0.083:00.003
0,473=0.0334 0.054=00.004
10
0,9M 0.809 0.011 0.55810.012 0.827
0.988 0.61810..010 0.819
76
1.4M 0.872.10.000
0.66510.003 0.872.10.00 003 0.713110.002 0.9961000000 0.882-10.003
273
2.7M 0.907==0.004 0.733=+0.003 0.899 000 0.768++00002 0.907=000000 0.916=+0.002
1041
(+ Grud. Loss)
4.9M 0.912:+0.00 002 0.745=00003 0.904 002 0.781=0.000 0.997=0.000 0,921=0.003
1059
11.4M 0.920=00.00 000 0.767=00.004 0.911 000 0.001 0.997=0.000 0.931=0.004
1112
17.8M 0.917.10.004 0.7711000022 0.909100002 0.802110.002 0.99710.000 0.923110.003
1165
50.2M 0.936=10.003 0.802:10.003 0.909
0.005 0.000 0.904
1430
0.5M 0.799=10.013 0.480=10.017 0.812+0.009
0.543+000122 0.989=0.002 0.784:10.021
34
1.0M 0.83410.012 0.53110..022 0.83810.000
0.58410.020 0.993110.001 0.82910.011
66
1.4M 0.855=000088 0.567=0.015 0.855
0.617=0.012 0.995=000000 0.857=0.000
94
2.8M 0.883=+00007 0.618=:0.012 0.876:0..004
0.658=00.010 0.907 0.889
181
EDSR
S.IM 0,901=0.004 0.665=#0.011 0.888=0.004
0.600 0.000 0.909=00.00
325
IL.M 0.9151.0.000 0.707110.002 0.902-10.000
0.73510..001 1.9000 0.919
695
17.8M 0.918=00004 0.718=0.011 0.904=000004
0.747=00001
0.998=0.000 0.926=000007
1101
34.6M 0.928=00.004 0.748=00.012 0.916=0.005
0.775=+0.010 0,999=0.0000 0.907
2122
0.5M 0. 0.869:10.006 0.581
0.865-10.005
0.624:10.012
0,996:10.001 0.86911.010
100
0,9M 0.87810.010 0.605100004
0.87110.009
0.645:10.013 0,9971100000 0.886110.011
166
1.5M 0.900=0.002 0.643=0.003 0.890+0.002 0.672+0.000 0.998=0.000 0.911=0.002
272
2.7M 0.905110.001 0.660-10.004 0.893 0.685 0.000 0.914100.0 000
495
RCAN
M. 0.916:10.002 0.712:10.007 0.902:10.003 0.707 0.000 0.9241100004
578
11.9M 0.913:00012 0.713:10.030 1.0013
0.743-10.028 0.008 0.92710.010
633
16.4M 0.928=0.000 0.753=0.002 0.916 0.778=0.001 0.999=0.0000 0.941=0.0003
671
48.3M 0.917100002 0.72411.0007 0.905110.001
0.753 0.000 0.932-10.005
931
0.6M 0.446::0.013 0.038
0.41540.014
0.049
0.491 0.050=1..011
30
1.8M 0.453=+0.00 0.048=0.000
0.473 0.0 0.060=0.000 0.493=0.001 0.058+0.000
77
2.6M 0.457=0.000 0.053=0.000 0.476= 00.001 0.062+0.003 0.493=0.000 0.062+0.001
108
Conv-FNO
5.2M 0 0.401100.002 0.15511..015 0,502100.001 0.17810.002 0.69610.021 0.2051
210
9.4M 0.682+:0.059 0,279 118 0.719+0.001 0.342+0.127 0.96140.014 0.541
376
20.6M 0.802=0.002 0.504=0.002 0.816+0.003 0.571+0.009 0.988=0.001 0.802+0.001
33.0M 0.840=0.0188 0.556=0.022 0.840=0.0013 0.606=0.017 0,992=0.002 0.839=200008
1276
Table 14: Mean and standard deviation of channel quantities. Values rounded to 2 significant figures.
[kgm 33
u; [ms-1]
Split Dataset
Mean
Std. Dev.
Mean
Std.
Dev.
Train Set
0.23
0.068
28
48.0
Baseline Test Set
0.24
0.068
29
48.0
Parametric Variation Set
0.23
0.059
34
55.0
Forced HIT
11.00
4.600
1.4
and low SSIM when compared to tricubic interpolation. This due to the highly different conditions (much higher density and lower velocity) in the Forced HIT set. However, when comparing the two normalization approaches in Table T6] we observe that slightly better performance seen across the two metrics during evaluation on the Parametric Variation set when normalizing via
38

---

Table 15: Evaluating models on the Forced HIT set, with normalization involving means and standard deviations from train and Forced HIT sets.
SSIM
1SII
sgs
Model
N
Normalization with Mean and Std. Dev. from:
Forced HIT Set
Train Set
Forced HIT Set
Train Set
Tricubic 8x
0.951
0.951
0.711
0.711
RRDB 8x
0.997
0.258
0.891=0.0003
0.00010.000
50.2M
(+ Grad. Loss)
0.998110.00
0.255=10000
0.944:100005
0.000:10.000
EDSR 8
34.6M
0.999=00.000
0.259=00000
0.937=0.000
0.000=000000
RCAN 8x
16.4M
0.900=00000
0.261
0.941
0.000=10.000
Conv-FNO 8x
33.0M
0.84910.000
0.61910.014
0.249
-00000=0.000
Tricubic 16x
0.876
0.876
0.432
0,432
RRDB 16x
0.971 +0.000
0.248+0.00
0.805+0.000
0.000-10.000
50.3M
(+ Grad. Loss)
0.973+0.000
0.248
0.816 0.0001
0.000+00.000
EDSR 16x
37.8M
0.909100001
0.249+0.000
0.783 0000
0.000=000000
RCAN 16x
17.3M
0.901
0.244
0.737=10.050
0.000=10.000
Conv-FNO 16x
34.6M
0.644
0.357=0.022
0.245
0.000=000000
Tricubic 32x
0.758
0.758
0.156
0.156
RRDB 32x
0.845=00001
0.242
0.494+00.011
0.000=100000
50.4M
(+ Grad. Loss)
0.000
0.245+0.001
0.516:
0.000=00000
EDSR 32x
40.9M
0.005
0.002
0.463++0.005
0.000=00.000
RCAN 32x
18.2M
0.003
0.239+0.003
0.448
0.000+00.000
Conv-FNO 32x
36.2M
0.472+10.001
0.177+0.002
0.238+40.002
0.000+000000
Table 16: Evaluating models on the Parametric Variation set, with normalization involving means and standard deviations from train and Parametric Variation sets.
SSSI
SSIM ags
Model
Np
Normalization with Mean and Std. Dev. from:
Param. Var. Set
Train Set
Param. Var. Set
Train Set
Tricubic 8x
0,800
0.800
0.418
0.418
RRDB 8x
0.898-40.003
0.901--0.003
0.755+00.002
0.760++0.003
50.2M
(+ Grad. Loss)
0.909
0.932--0.002
0.825 18225+0.001
0.830++0.001
EDSR 8x
34.6M
0.916++0.005
0.917+-0.005
0.775=
0.779++0.010
RCAN 8x
16.4M
0.916110.001
0.9181100000
0.7781100001
0.784410.000
Tricubic 16x
0.620
0.620
0.173
0.173
RRDB 16x
0.700-10.001
0.703-10.001
0.512-10002
0.518
50.3M
(+ Grad. Loss)
0.719-10.004
0.721-1.003
0.556=0.002
0.559=10.001
EDSR 16x
37.8M
0.693
0.695+10.005
0.481
0.48440.019
RCAN 16x
17.3M
0.665+00024
0.668+00.023
0
0.417+
Tricubic 32x
0.476
0.476
0.087
0.087
RRDB 32x
0.482++0.000
0.484+00.000
1186=00006
0.187+ 0.004
50.4M
(+ Grad. Loss)
0.483++0.001
0.485+00.00
0.188+0.009
0.192++0.009
EDSR 32x
40.9M
0.481+00.002
0.482-10.001
0.187=0.004
0.184410.00
RCAN 32x
18.2M
0.469-10.002
0.470-10.002
0.18550.005
0.0005
the train compared to normalizing via the Parametric Variation set (~15% higher mean and standard deviation). Nevertheless, in this work, we choose to normalize via the evaluation sets during inferencing in order to employ a consistent approach that enables sufficiently good performance across all evaluation sets. Note that results on normalization via the test set is not shown, since the mean and standard deviation of the test is similar to those from the train set.
39

---

H
Datasheets for Datasets
The following datasheets for BLASTNet 2.0 and Momentum 128 3D SR Datasets based on Datasheets for Datasets [02
H.1 BLASTNet 2.0
MOTIVATION
For what purpose was the dataset created? Was there specific task mind? Was there specific gap that needed to be filled? Please provide a description.
BLASTNet 2.0 was developed provide researchers in reacting and non-reacting flow physics communities with high-fidelity publicly accessible simulation data for ML applications. With 2.2 TB, 744 full-domain samples, and 34 configurations, BLASTNet can effectively address gaps data availability and aid in fostering open/fair ML development within reacting and non-reacting flow physics communities. This data useful for fluid flows wide range of MI. applications tied to propulsion, energy, and the environment. Specifically, scientific tasks related to these domains may include turbulent closure modeling 37 spatio-temporal modeling [49], and inverse modeling [18]
Who created this dataset (e.g., which team, research group) and behalf of which entity (e.g.. company, institution, organization)?
BLASTNet was initiated by Wai Tong Chung, Ki Sung Jung, Jacqueline H. Chen, and Matthias Ihme 8). The datasets in BLASTNet were generated by the following researchers:
Wai Tong Chung, Jack Guo, Davy Brouzet and Matthias Ihme at Stanford University.
Jacqueline H. Chen and Ki Sung Jung Sandia National Laboratory.
3. Mohsen Talei and Bin Jiang University of Melbourne.
4. Bruno Savard at Polytechnique Montréal.
Alexei Poludnenko at University Connecticut.
Bassem Akoush, Pushan Sharma and Alex Tamkin Stanford University contributed in administer- ing, improving, maintaining, and documenting the dataset curation process.
What support was needed to make this dataset? (e.g.who funded the creation of the dataset? there is an associated grant, provide the name of the grantor and the grant name and number, or if it was supported by a company or government agency, give those details.) This work is funded by:
1. The U.S. Department of Energy National Nuclear Security Administration, under award No. DE-NA0003968.
2. The NASA Early Stage Innovation Program with award No. 8ONSSC22K0257.
3. The Department of Energy Office of Energy Efficiency Renewable Energy (EERE) with award No. DE-EE0000885.
4. Wai Tong Chung received partial financial support from the Stanford Institute for Human-centered Artificial Intelligence Graduate Fellowship.
This research used resources 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, operated under Contract No. DE-AC02-0SCHI123 using NERSC award FRCAP0021046.
Any other comments?
No.
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COMPOSITION
What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)? Are there multiple types of instances (e.g., movies, users, and ratings; people and interactions between them; nodes and edges)? Please provide description.
Each instance consists of 3D domain of velocity, temperature, pressure, density and mass fractions of chemical species. These variables are saved as flat arrays in separate dat binary files, which can be loaded and reshaped to construct the 3D volumes. To enable high I/O speed in loading arrays, the data has consistent little-endian single-precision binaries that can be read with np. fromfi Promfile/pp 1l//p.. memmap.
How many instances there in total (of each type, if appropriate)? BLASTNet contains a total of 744 full-domain samples from a diverse collection of 34 DNS configuration.
Does the dataset contain all possible instances or is it sample (not necessarily random) of instances from a larger set? the dataset is a sample, then what is the larger set? Is the sample representative of the larger set (e.g.. geographic coverage)? If so, please describe how this representativeness was validated/verified. If it is not representative of the larger set, please describe why not (e.g., to cover more diverse range of instances, because instances were withheld or unavailable).
BLASTNet data covers the full physical three-dimensional spatial domain defined during simulation. However, the data contains different number of timesteps, e.g. inert HIT [20 has uniform samples
for 99 timesteps, whereas of the configurations 27 29 67
contain single snapshots.
What data does each instance consist of? "Raw" data (e.g., unprocessed text or images) or features? either case, please provide a description.
Each instance consists of 3D flowfields of velocity, temperature, pressure, density and mass fractions of chemical species. These are raw simulation data that have been pre-processed to a consistent format for ML applications.
Is there label or target associated with each instance? If so, please provide description. Yes. All the thermodynamic and chemical quantities are labeled and can used directly for certain regression problems. On the other hand, the flow-field data can also be used to derive additional labels for particular applications.
Is any information missing from individual instances? If so, please provide description, explaining why this information is missing (e.g., because it was unavailable). This does not include intentionally removed information, but might include, e.g., redacted text. No.
Are relationships between individual instances made explicit (e.g., users' movie ratings, social network links)? If so, please describe how these relationships made explicit. Yes. All instances are related by the same governing equations. Some configurations share multiple time instances. These timesteps are made explicit in a descriptive info. json file.
Are there recommended data splits (e.g., training, development/validattio.. testing)? If so, please provide a description these splits, explaining the rationale behind them.
No.
Are there any errors, sources of noise, or redundancies in the dataset? If so, please provide description.
Yes. Well-established high-order numerical solvers 29 64 67
have been employed, with spatial discretization schemes ranging from 2nd- to 8th-order accuracy and time advancement accuracy ranging from 2nd- to 4th-order. Thus, this data is still subject to small numerical errors. While the
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compressible reacting flow equations (Equation (88), solved to generate this data, are valid for a wide range of conditions, errors can originate from certain assumptions in chemical modeling. Specifically, skeletal finite-rate mechanisms and mixture-averaged transport used in these reacting flow DNS have been validated for their configurations, but are not as fully-representaativ
real-world reactions. However, rectifying this would require detailed mechanisms (introducing an order of magnitude more PDEs) and multi-component transport that can result in intractable calculations [86].
dep- V (pu) =0
(18a)
(pu) + V (pu @ u) = Vp T (pe')) V ju(pe" +p]] = V. + V (()..).
(18b)
(18c)
O.(pYa) (puYr) =- V. je + Wk,
(18d)
Is the dataset self-contained, does it link to otherwise rely external resources (e.g., websites, tweets, other datasets)? If links to or relies on external resources, a) are there guarantees that they will exist, and remain constant, over time; b) are there official archival versions of the complete dataset (i.e., including the external resources as they existed at the time the dataset was created); c) are there any restrictions (e.g., licenses, fees) associated with any of the external resources that might apply to a future user? Please provide descriptions of all external resources and any restrictions associated with them, as well as links or other access points, as appropriate. BLASTNet is self-contained.
Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege by doctor-patient confidentiality, data that includes the content of individuals' non-public communications)? so, please provide a description. No.
Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, or might otherwise cause anxiety? If so, please describe why. No.
Does the dataset relate to people? If not, you may skip the remaining questions in this section. No.
Does the dataset identify any subpopulations (e.g., by age, gender)? If so, please describe how these subpopulations are identified and provide a description of their respective distributions within the dataset. No.
Is it possible to identify individuals (i.e., one or more natural persons), either directly or indirectly (i.e., in combination with other data) from the dataset? If so, please describe how. No.
Does the dataset contain data that might be considered sensitive in any way (e.g., data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic data; forms of government identification, such as social security numbers; criminal history)? If so, please provide a description. No.
Any other comments? No.
COLLECTION
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How was the data associated with each instance acquired? Was the data directly observable (e.g.. raw text, movie ratings), reported by subjects (c.g., survey responses), or indirectly inferred/derived from other data (e.g., part-of-speech tags, model-based guesses for age or language)? If data was reported by subjects or indirectly inferred/derived from other data, was the data validated/verified? so, please describe how.
BLASTNet represents a collection of multiple DNS datasets, with analysis previously published in 69 70) However, the data was not publicly available until the release of this work.
27 67
20
Over what timeframe was the data collected? Does this timeframe match the creation timeframe of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please describe the timeframe in which the data associated with the instances was created. Finally, list when the dataset was first published.
This dataset was collected from published work between years 2019 and 2022.
What mechanisms or procedures were used to collect the data (e.g., hardware apparatus or sensor, manual human curation, software program, software API)? How were these mechanisms or procedures validated?
The DNS cases are performed using well-established numerical solvers 29 in this research community, with analysis of the flowfields typically validated via peer-review.
What was the resource cost of collecting the data? (e.g. what were the required computational resources, and the associated financial costs, and energy consumption estimate the carbon footprint. See Strubell et al. [103] for approaches in this area.)
Compute cost of all DNS is summarized in Table 17 However, we note that BLASTNet curates
previously-uneleasse aiready-generated data. Thus, the additional compute cost in developing this dataset is insignificant.
Table 17: Computing cost for dataset collection.
Dataset
Cost [CPU-hr]
Non-reacting HIT 20
0.029 10
Reacting forced HIT
67
106
Reacting jet flows (BFER case)
29
0.54 x 106
Reacting jet flows (COFFEE case
29
0.64 106
Non-reacting transcritical Channel Flow
69
0.384 x 106
Reacting channel flow 28
0.5 x 106
Partially premixed slot burner
27
2.5 x 106
Freely Propagating Flame 70
12 x 106
If the dataset is a sample from a larger set, what was the sampling strategy (e.g., deterministic, probabilistic with specific sampling probabilities)?
There is no sampling involved in BLASTNet 2.0, as it contains all potential instances within three-dimensional spatial domains.
Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)? Neither crowdworkers nor contractors were used in data collection. The BLASTNet datasets were created and processed by the authors of this work for no additional payment outside of typical salary and stipend.
Were any ethical review processes conducted (e.g., by an institutional review board)? If so, please provide a description of these review processes, including the outcomes, as well link or other access point any supporting documentation.
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No.
Does the dataset relate to people? If not, you may skip the remainder of the questions in this section.
No.
Did you collect the data from the individuals in question directly, or obtain via third parties or other sources (e.g,, websites)? No.
Were the individuals in question notified about the data collection? If so, please describe (or show with screenshots or other information) how notice was provided, and provide link or other access point to, or otherwise reproduce, the exact language of the notification itself. N/A.
Did the individuals in question consent to the collection and use of their data? If so, please describe (or show with screenshots or other information) how consent was requested and provided, and provide link or other access point to, or otherwise reproduce, the exact language to which the individuals consented. N/A.
If consent was obtained, were the consenting individuals provided with a mechanism to revoke their consent in the future for certain uses? If so, please provide description, well a link or other access point to the mechanism (if appropriate) N/A.
Has an analysis of the potential impact of the dataset and its use on data subjects (e.g., a data protection impact anelysis)been conducted? If so, please provide description of this analysis, including the outcomes, as well as a link or other access point to any supporting documentation. N/A.
Any other comments? No.
PREPROCESSING: / CLEANING / LABELING
Was any prepoocesingddeenig/aabiiii of the data doneee-g.diseretizatioo tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? If so, please provide description. If not, you may skip the remainder of the questions in this section.
bucketing,
Yes. The data was generated from different numerical solvers, initially exists range of formats (. (tvk,, vtu, ttec, and .dat) that are not readily formatted for training ML models. Thus, all data are processed into consistent format little-endian single-precision binaries that can be read with mp. fromf: 1e/np. memmap. The choice of this data format enables high I/O speed in loading arrays. We provide json files that store additional information on configurations, chemical mechanisms and transport properties.
Was the "raw"" data saved in addition to the preproessed/deennedlabeeed data (e.g., to support unanticipated future uses)? If so, please provide link or other access point the "raw" data. No, raw data are not available for public. This is because flow physics datasets are typically stored in a variety of double-precision formats that cannot fit into open repositories, introducing challenges, along with auxiliary derived quantities. However, the shared data are single-precision ver- sions of the original data, with additional files shared in metadata for obtaining the auxiliary quantities.
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Is the software used to preprocess/cdenn/lbeel the instances available? If so, please provide link other access point.
Python is used to preprocess the raw data in convenient format for MI. applicaaas...s..
Any other comments? No.
USES
Has the dataset been used for any tasks already? If so, please provide a description. Yes, part of the dataset was introduced previously BLASTNET 1.0 in [ and was used for semantic segmentation and reaction closure modeling.
there a repository that links to any or all papers or systems that use the dataset? so, please provide link other access point.
No.
What (other) tasks could the dataset be used for? BLASTNet can be used for multiple tasks such as subgrid-scale modeling 37,. spatial and temporal prediction [49] Specifically, this dataset is useful for developing closure models 38 in computational fluid dynamics (in both reacting and non-reacting flows) to capture the interactions between resolved and unresolved turbulence scales and generate closure terms or correction factors to add to the existing turbulence models. Lastly, BLASTNet can also be employed for solving inverse problems [18] via ML methods.
Is there anything about the composition of the dataset or the way was collected and preprocesedd/deened//ate
that might impact future uses? For example, is there anything that future user might need know to avoid uses that could result in unfair treatment of individuals or groups (e.g., stereotyping, quality of service issues) or other undesirable harms (e.g.. financial harms, legal risks) If so, please provide a description. Is there anything future user could do mitigate these undesirable harms? No.
Are there tasks for which the dataset should not be used? If so, please provide description. BLASTNet 2.0 data contains data from reacting non-reacting DNS configurations, which covers specific range of flow physics conditions. Consequently, this dataset cannot be for unrepresented conditions such as rarefied, multi-phase, micro- fluid, Non- Non-Newton,an, or augneto-hydrodynaan flows.
Any other comments?
No.
DISTRIBUTION
Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) on behalf which the dataset was created? If so, please provide a description. Yes, the dataset is publicly available. To circumvent Kaggle storage constraints, we partition our data into a network of <100 GB subsets, with each subset containing a separate simulation configuration. This partitioned data then be uploaded separate datasets Kaggle, with links consolidated in https::/blastner git.ub.............
How will the dataset will be distributed (c.g., tarball on website, API, GitHub)? Does the dataset have digital object identifier (DOI)?
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The data is available through multiple Kaggle repositories, with links consolidated at Each dataset shares common DOI at When will the dataset be distributed?
http://///...aat....ttttaaaaaaaa 7242864
7242864
The BLASTNet 1.0 dataset is available online and can be accessed and downloaded via Kaggle. We are planning to release fully-tested version of BLASTNet 2.0 prior to NeurIPS 2023, depending on the outcome the review process. However, a publicly-availabe version that can be peer-reviewed is released on June 14 2023.
Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)? If so, please describe this license and/or ToU, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, well any fees associated with these restrictions. Licensed via CC BY-SA NC 4.0.
Have any third parties imposed IP-based or other restrictions on the data associated with the instances? If so, please describe these restrictions, and provide link other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions. No.
Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any supporting documentation. No.
Any other comments? No.
MAINTENANCE
Who is supporiinghottin/naantiiin the dataset?
BLASTNet is hosted on Kaggle. Stanford Laboratory of Fluids in Complex Systems
(https: /ettps://web: //web//wb......// wttp:///w....t.ordd..... wtw.:/oooo.e.eeeerroooo......pe.
http://g--biiii http:////w.iiio//oa/aaaaaaaaaaaa Thhme)
will maintain this dataset, along with the support of contributors from Sandia National Laboratory. University of Melbourne, Polytechnique Montr/ cal, and University of Connecticut.
How can the ownerrcurraorrmaaager of the dataset be contacted (e.g., email address)? The BLASTNet team can be reached at blatt...... net www.:eeettdaaaaaaaammeeemmm wwt.ttddaaaaaallccmoommmmmm. com Users are encouraged www.lll.oooobbaaaaaaa
to submit any issues/inquiries our GitHub page allasneet.github.... wtiiihhab.i/isssseee Lastly, Wai Tong Chung (wt (wtcchun@sstanford edu) and Matthias Ihme (mi (mihme@stanfordd edu) can be reached for further inquiries and collaborations.
Is there erratum? If so, please provide link other access point. Yes. A detailed issue tracker is htt.taas/gttuuu..cm///attttt/tea github. / //sssues
in
Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)? so, please describe how often, by whom, and how updates will communicated to users (e.g., mailing list, GitHub)?
Yes, the team will release minor updates the dataset in case of any identified errors. These updates will be communicated via the landing page httpp::/blasttne github. io and GitHub
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the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g,, individuals in question told that their data would be retained for fixed period of time and then deleted)? If so, please describe these limits and explain how they will be enforced. N/A.
Will older versions of the dataset continue to be supporteennsteemmiitannddd?? If so, please describe how. If not, please describe how its obsolescence will be communicated to users. Yes. BLASTNet 2.0 builds on data from BLASTNet 1.0. Thus, all previous data will be preserved with each version update. In addition, version histories and logs from individual Kaggle repositories will be used to inform users of changes. Changes will also be communicated to users via the landing page https:: githla..tttett..tu....... .ithub.. io
If others want to extend/augment/build on/contribute to the dataset, is there mechanism for them to do so? If so, please provide description. Will these contributions validated/verified? so, please describe how. If not, why not? Is there process for commuiicaing/ddssiittingg these contributions to other users? If so, please provide description.
Yes. We plan accept contributions via https: //bl astnet. cotubb.oo/oottriiii Any contributions will be checked for accuracy, robustness, reliability, and formats prior to new release. New contributions will be tracked via version history.
Any other comments?
No.
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H.2
Momentum 128 3D SR
MOTIVATION
For what purpose was the dataset created? Was there specific task mind? Was there specific gap that needed to be filled? Please provide a description.
Momentum 128 3D SR consists of BLASTNet 2.0 data processed for 3D super-resolution (SR) of density and velocity at 8, 16 and 32x SR, while mitigating constraints in memory and grid properties. The single largest sample in BLASTNet is 92 GB (with 1.3B voxels and 15 channels), which is too large to fit in typical GPU memory. In Momentum125 128 3D SR. the BLASTNet 2.0 data is downsampled to 128 sub-volumes of density and velocity fields to maintain low memory footprint. This dataset is useful for training/evaluating 3D SR models, as well as developing closure models (38] in computational fluid dynamics (in both reacting and non-reacting flows) to capture the interactions between resolved and unresolved turbulence scales.
Who created this dataset (e.g,, which team, research group) and on behalf of which entity (e.g., company, institution, organization)?
Momentuml 28 3D SR is created by Wai Tong Chung. The raw datasets used to develop Momen- tum128 3D SR are contributed by the following researchers:
1. Wai Tong Chung, Davy Brouzet and Matthias Ihme Stanford University.
2. Jacqueline H. Chen and Ki Sung Jung at Sandia National Laboratory.
3. Mohsen Talei at University of Melbourne.
Bruno Savard at Polytechnique Montréal.
Alexei Poludnenko at University Connecticut.
Bassem Akoush, Pushan Sharma and Alex Tamkin Stanford University contributed in administer-
ing, improving, maintaining, and documenting the dataset curation process.
What support was needed make this dataset? (e.g.who funded the creation of the dataset? there is an associated grant, provide the name of the grantor and the grant name and number, or if it was supported by a company or government agency, give those details.) This work is funded by:
1. The U.S. Department of Energy National Nuclear Security Administration, under award No. DE-NA0003968.
2. The NASA Early Stage Innovation Program with award No. 8ONSSC22K0257.
3. The Department of Energy Office of Energy Efficiency Renewable Energy (EERE) with award No. DE-EE0000885..
4. Wai Tong Chung received partial financial support from the Stanford Institute for Human-centered Artificial Intelligence Graduate Fellowship.
This research used resources the National Energy Research Scientific Computing Center (NERSC), U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-0SCHI 1231 using NERSC award ERCAP0021046.
Any other comments? No.
COMPOSITION
What do the instances that comprise the dataset represent (e.g,, documents, photos, people, countries)? Are there multiple types of instances (e.g., movies, users, and ratings; people and interactions between them; nodes and edges)? Please provide description.
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Each instance consists of 3D domain of velocity, temperature, pressure, density and mass fractions of chemical species. These variables are saved as flat arrays in separate dat binary files, which can be loaded and reshaped to construct the 3D volumes. To enable high I/O speed in loading arrays, the data has consistent little-endian angle-precision binaries that can be read with np. f fromfi e/np. memmap.
How many instances are there in total (of each type, if appropriate)? Momentum 128 3D SR contains a total of 2000 sub-volumes of density and velocity samples from diverse collection of 27 DNS configuration.
Does the dataset contain all possible instances is it sample (not necessarily random) of instances from a larger set? If the dataset is a sample, then what is the larger set? Is the sample representative of the larger set (e.g., geographic coverage)? If so, please describe how this representativeness was validated/verified. If it is not representative of the larger set, please describe why not (e.g., to cover more diverse range of instances, because instances were withheld or unavailable).
Momentum 128 3D SR subset of BLASTNet 2.0. (https://blasttt geeeit.ggtuuubb. io), downsampled to preserve statistical characteristics of the larger dataset. This is done via maintaining good proportion of similar clusters (obtained via k-means with four statistical moments of three velocity components used as inputs).
What data does each instance consist of? "Raw" data (e.g., unprocessed text or images) or features? In either case, please provide a description.
Labels consists of 3D flowfields of DNS-fidelity velocity and density (of size 128*). Features consist of corresponding 8,16, and 32x Favre-filtered flowfields from the labels.
Is there label or target associated with each instance? If so, please provide description. Yes. The data is fully labeled for 8,16, and 32x SR turbulent flow applications.
Is any information missing from individual instances? If so, please provide description, explaining why this information is missing (e.g., because it was unavailable). This does not include intentionally removed information, but might include, e.g., redacted text.
No.
Are relationships between individual instances made explicit (e.g., users' movie ratings, social network links)? If so, please describe how these relationships are made explicit. Yes. Features and labels from the same DNS configuration and spatial location share unique hash identifier.
Are there recommended data splits (e.g., training, developmenz/vlidattion, testing)? If so, please provide description of these splits, explaining the rationale behind them.
Yes. We provide train, validation, and test splits in 80: 10:10 ratio. Two additional split sets from unseen DNS configurations are recommended for evaluating out-of-distribution behavior and data normalization.
Are there any errors, sources of noise, redundancies in the dataset? If so, please provide description.
Yes. Well-established high-order numerical solvers 29 64 67
have been employed, with spatial discretization schemes ranging from 2nd- to 8th-order accuracy and time advancement accuracy ranging from 2nd- to 4th-order. Thus, this data is still subject to small numerical errors.
Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)? If links to or relies on external resources, a) are there guarantees that they will exist, and remain constant, over time; b) are there official archival versions of the complete dataset (i.c., including the external resources as they existed at the time the dataset
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was created); c) are there any restrictions (e.g., licenses, fees) associated with any of the external resources that might apply a future user? Please provide descriptions of all external resources and any restrictions associated with them, as well as links or other access points, as appropriate. Momentum122 3D SR self-contained.
Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor-patient confidentiality, data that includes the content of individuals' non-public communications)? so, please provide a description. No.
Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, might otherwise cause anxiety? If so, please describe why. No.
Does the dataset relate people? not, you may skip the remaining questions in this section. No.
Does the dataset identify any subpopulations (e.g,, by age, gender)? If so, please describe how these subpopulations are identified and provide a description of their respective distributions within the dataset. No.
Is it possible to identify individuals (i.e., one or more natural persons), either directly or indirectly (i.e., in combination with other data) from the dataset? If so, please describe how. No.
Does the dataset contain data that might be considered sensitive in any way (e.g., data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, locations; financial or health data; biometric genetic data; forms of government identification, such as social security numbers; criminal history)? so, please provide description. No.
Any other comments? No.
COLLECTION
How was the data associated with each instance acquired? Was the data directly observable (e.g.. raw text, movie ratings), reported by subjects (e.g., survey responses), indirectly inferred/derived from other data (e.g., part-of-speech tags, model-based guesses for age or language)? If data was reported by subjects or indirectly inferred/derived from other data, was the data validated/verifid?? If so, please describe how.
Momentum 128 3D SR is sampled from BLASTNet 2.0, which was in turn validated through analysis in previous publications [27
Over what timeframe was the data collected? Does this timeframe match the creation timeframe of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please describe the timeframe which the data associated with the instances was created. Finally, list when the dataset was first published.
This dataset was collected from work done between years 2019 and 2022.
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What mechanisms or procedures were used to collect the data (e.g,, hardware apparatus or sensor, manual human curation, software program, software API)? How were these mechanisms or procedures validated?
The DNS cases are performed using well-established numerical solvers community.
in this research
What was the resource cost of collecting the data? (e.g. what were the required computational resources, and the associated financial costs, and energy consumption estimate the carbon footprint. See Strubell et al. [103] for approaches in this area.)
Compute cost of all employed DNS is summarized Table [7] However, we note that this data curates -reerrdgeeeeated data. Thus, the additional compute cost in developing this dataset is insignificant.
Table 18: Computing cost for dataset collection.
Dataset
Cost [CPU-hr]
Non-reacting HIT 20
0.029 10
Reacting forced HIT 67
106
Reacting jet flows (BFER case
29
0.54 x 106
Reacting jet flows (COFFEE case)
29
0.64 x 106
Partially premixed slot burner 27
2.5 x 106
Freely Propagating Flame 70
12 x 106
If the dataset is a sample from a larger set, what was the sampling strategy (e.g., deterministic, probabilistic with specific sampling probabilities)?
Momentum 28 3D SR is created by sampling 128 sub-volumes of density and velocity from the uniform regions of all BLASTNet 2.0 datasets. This results in 12750 sub-volume samples, out of which 2000 sub-volumes are sampled to create Momentum125 3D SR. To ensure statistical representativeness of the original dataset, these 2000 samples are collected these steps:
1. Extracting four statistical moments of the three celocity of each of the 12750 sub- volumes.
2. Applying k-means clustering with the elbow method to partition the sub-volumes in 18 clusters. Sampling 2000 sub-volumes with balanced proportions from each cluster to generate the labels.
4. Favre-filtering the labels to generate the features.
Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)? Neither crowdworkers nor contractors were used in data collection. The BLASTNet datasets were created and processed by the authors of this work for no additional payment outside of typical salary and stipend.
Were any ethical review processes conducted (e.g., by institutional review board)? If so, please provide a description of these review processes, including the outcomes, as well as a link or other access point any supporting documentation. No.
Does the dataset relate to people? not, you may skip the remainder of the questions in this section.
No.
Did you collect the data from the individuals in question directly, or obtain via third parties or other sources (e.g., websites)?
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No.
Were the individuals in question notified about the data collection? If so, please describe (or show with screenshots other information) how notice was provided, and provide link other access point to, or otherwise reproduce, the exact language of the notification itself. N/A.
Did the individuals in question consent to the collection and of their data? If so, please describe (or show with screenshots or other information) how consent was requested and provided, and provide link or other access point to, otherwise reproduce, the exact language to which the individuals consented. N/A.
If consent was obtained, were the consenting individuals provided with mechanism to revoke their consent in the future or for certain uses? so, please provide a description, as well as a link or other access point to the mechanism (if appropriate) N/A.
Has an analysis of the potential impact of the dataset and its use on data subjects (e.g., data protection impact anee conducted? If so, please provide description of this analysis, including the outcomes, as well as a link or other access point to any supporting documentation. N/A.
Any other comments? No.
PREPROCESSING / CLEANING CAAOHING/LLBBBLIIE LABELING
Was any prepoccesingdllenigg/abiiii of the data disereiiiiinrttiitaiinn bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? If so, please provide description. If not, you may skip the remainder of the questions in this section.
Momentum 128 3D SR is created by sampling 128 sub-volumes of density and velocity from the uniform regions of all BLASTNet 2.0 datasets. This results in 12750 sub-volume samples, out of which 2000 sub-volumes are sampled create Momentum 3D SR. To ensure statistical representativeness of the original dataset, these 2000 samples are collected in these steps:
1. Extracting four statistical moments of the three celoeety of each of the 12750 sub- volumes.
2. Applying k-means clustering with the elbow method to partition the sub-volumes in 18 clusters.
3. Sampling 2000 sub-volumes with balanced proportions from each cluster to generate the labels. Favre-filtering the labels to generate the features.
Was the "raw" data saved in addition to the preproessed/cceancd//bbeeed data (e.g., to support unanticipated future uses)? If so, please provide link or other access point to the "raw" data. Yes. BLASTNet 2.0 is publicly available via/nztps: piettps:/bblastneee www.inneettttttut...eeeeee github.
Is the software used to preprocees/cdean/laeel the instances available? If so, please provide link or other access point.
Python is used to sample from BLASTNet 2.0. https: / twwb. http:///nnmmmm/ooaaattt/////laaa
Code for Favre-filtering is attached to sr benchmark
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Any other comments?
No.
USES
Has the dataset been used for any tasks already? If so, please provide description. No.
Is there repository that links to any all papers systems that use the dataset? If so, please provide a link or other access point. No.
What (other) tasks could the dataset be used for?
The Momentum 3D SR dataset was specifically sub-sampled for the purpose super-resolution and turbulence closure modeling. This dataset can be used for closure modeling via direct regression of the subgrid-scale terms. This high-fidelity data can also be employed for generative and deterministic reconstruction.
Is there anything about the composition of the dataset the way it was collected and preprocesed/dccnne//lbbet that might impact future uses? For example, is there anything that future user might need to know to avoid uses that could result in unfair treatment of individuals or groups (c.g.. stereotyping, quality of service issues) or other undesirable harms (c.g., financial harms, legal risks) If so, please provide description. Is there anything future user could do mitigate these undesirable harms? No.
Are there tasks for which the dataset should not be used? If so, please provide description, This dataset is specifically generated for SR turbulent closure modeling purpose from BLASTNet 2.0. Thus, it should not be used for other tasks.
Any other comments? No.
DISTRIBUTION
Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) behalf of which the dataset was created? If please provide description. Yes, the dataset is publicly available at https::/www kaggla com/ datasets/ -a-
How will the dataset will be distributed (e.g., tarball website, API, GitHub)? Does the dataset have a digital object identifier (DOI)?
The data is available through Kaggle at https://www kaggla le com datm///ttses///////m wa waitongcchngg/be
This dataset shares a com-
mon DOI with BLASTNet 2.0, at /lttttps://oi.ooo//110/ ailcppes//ooloorr//0555///e/0 724.388
7242864
When will the dataset be distributed?
We are planning to release a fully-tested version of Momentum122 3D SR prior to NeurIPS 2023, depending the outcome of the review process. However, publicly-availe version that can be peer-reviewed released on June 14 2023.
Will the dataset be distributed under a copyright or other intellectual property (TP) license, and/or under applicable terms of use (ToU)? If please describe this license and/or ToU, and
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provide link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, as well any fees associated with these restrictions. Licensed via CC BY-SA NC 4.0.
Have any third parties imposed IP-based or other restrictions on the data associated with the instances? If so, please describe these restrictions, and provide link or other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions. No.
Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? If please describe these restrictions, and provide a link or other access point to, otherwise reproduce, any supporting documentation. No.
Any other comments? No.
MAINTENANCE
Who is hyapoortiighsstinnmmmaaaaiinggnt the dataset?
Momentum128 3D SR is hosted on Kaggle. Stanford Laboratory of Fluids in Complex Systems (https://wet) http:://ww..........t.tt.t.et..t stanford..d...om heab:/gropp/mmmerrrr///gggg//// http:///gg-bbi//attaaaaaaaaaaa Ihme) will maintain this dataset, along with the support of contributors from Sandia National Laboratory, University of Melbourne, Polytechnique Montreal, and University of Connecticut.
How can the owner/cuurator/manager the dataset be contacted (e.g., email address)? The BLASTNet team can be reached at [blast net data@@aaail.cccommm com The users are encouraged submit any inquiries GitHub page httpss:/github. wwm http:///ttuu..cmo/mmaassttttee blastnet github..tiioo issues Lastly, Wai Tong Chung wrtchungesttanforrd edu) and Matthias Ihme (mihme@stanford) edu) can reached for further inquiries and collaborations.
Is there erratum? If so, please provide link other access point. Yes. A detailed issue tracker is provided in github. sssee
Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)? so, please describe how often, by whom, and how updates will communicated to users (e.g.. mailing list, GitHub)?
Yes, the team will release minor updates to the dataset case of any identified errors. Theses updates will be announced in the landing page https: https:/ /ttp::://iithb.... com/b aataaatttlassteeettttttt wwwplasttee..iihutte www.....tuug...ggg wta.chub.ooooo issues
and GitHub
If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g., were individuals in question told that their data would be retained for fixed period of time and then deleted)? so, please describe these limits and explain how they will be enforced. N/A.
Will older versions of the dataset continue be supporte/hsscclnaattiidddd If so, please describe how. If not, please describe how its obsolescence will be communicated to users. Yes. Version histories and logs from the Kaggle repository will be used to inform users of changes.
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If others want to extend/augmeent/uiid on/contribute to the dataset, is there mechanism for them do so? If so, please provide description. Will these contributions be validated/verified? If so, please describe how. If not, why not? Is there process for communicaiingdisstrttiing these contributions other users? If so, please provide description.
Yes. Kaggle hosts comments section that enables user feedback and suggestions. In addition, contact information the data curators are also provided in m/lttps://blast.e github. io
Any other comments?
No.
55


Reviews:
[{"review_id": "P6TlGQfrSR", "reviewer": "Reviewer_AcWj", "summary": "", "questions": "", "limitations": "See above for limitations. Additionally, the lack of a mechanism to generate more simulations and expand the dataset is a significant limitation as the solvers are proprietary.", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "1. The dataset is for 3D simulations which is more complex than 2D simulations and can capture more realistic flow behavior. \n2. DNS of compressible reacting flows involves solving the full set of governing equations for fluid dynamics and chemical reactions without making any simplifying assumptions or turbulence models. These simulations are computationally intensive and complex, requiring significant computational resources and time to carry out accurately. The availability of such a high-fidelity dataset can significantly foster ML research in this challenging scientific domain.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "4ax5EBefgd", "reviewer": "Reviewer_yabe", "summary": "", "questions": "", "limitations": "The authors have provided a paragraph discussing the limitations of this paper. Please also see my comments in **Opportunities For Improvement** and **Correctness**.", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "- This 3D turbulence dataset is large-scale and high-fidelity and has good promise for future research in scientific machine learning. \n\n- This paper includes a discussion of neural scaling analysis. It is crucial for analyzing model scalability, especially for the super-resolution of high-dimensional scientific data. \n\n- This paper considers the physics-informed loss and compares the performance between the data-driven method and the physics-informed strategy.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "iTItWJnskw", "reviewer": "Reviewer_5NmK", "summary": "", "questions": "", "limitations": "Limitation is well discussed.", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "- This paper curates a diverse public 3D compressible turbulent flow dataset\n- This paper benchmarks the performance and cost of 3D versions of three popular SR models with the proposed dataset.\n- This paper shows that SR model performance can scale with the logarithm of model size and cost.\n- This paper demonstrates the persisting benefits of a popular gradient-based physics-informed loss with increasing model size.\n- The proposed dataset comes with a well-established website and tutorial, which is convenient for researchers to use.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "czaPEKP1Vf", "reviewer": "Reviewer_o9yf", "summary": "", "questions": "", "limitations": "The authors adequately addressed the limitations and potential negative societal impact of their work.", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "The previous studies on flow physics simulation datasets have predominantly focused on Reynolds averaged Navier Stokes simulations (RANS) and large eddy simulations (LES) using two-dimensional model. However, it is important to note that two-dimensional flow simulations have limitations in capturing turbulence due to the inability to resolve vortex and eddy structures inherent in three-dimensional models.\nIn contrast, BLASTNet 2.0 addresses this limitation by employing a three-dimensional model for simulating compressible turbulent flows. This choice enables BLASTNet 2.0 to effectively capture and represent the complex structures associated with turbulence. By utilizing a three-dimensional approach, BLASTNet 2.0 enhances the fidelity of the simulations and provides a more accurate representation of turbulent flow phenomena. \n", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}]

Author rebuttal:

Meta-review: Paper delivers comprehensive data set of generic compressible turbulent flows.    
Proof of concept for super-resolution reconstructions are provided.
Paper is clear and and reproducible, includes website and tutorial.
An extension to time series and more advanced super-resolution strategies is welcome future work,.
Decision: Accept (Poster)