SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 6: Summary of the simulation parameters for all DNS runs [70].
Config. Index	\(U_{\text{in}}/S_L\)	\(u'/S_L\)	Ka_u	Re_t,u	Da_{\text{ign}}	\(t_{\text{end}}/\tau_{\text{ign,0}}\)	\(L_x/L_y\)	\(N_x \times N_y \times N_z\)
\(l_I/l_f=1.5\)
1	2.45	0.5	2.4	53	0.24	13	9	1152 × 128 × 128
2	2.45	1.0	6.8	105	0.12	4.6	9	1152 × 128 × 128
3	2.45	1.5	13.0	158	0.08	3.6	9	1152 × 128 × 128
4	3.67	0.5	2.4	53	0.24	7.4	11	1408 × 128 × 128
5	3.67	1.0	6.8	105	0.12	5.4	11	1408 × 128 × 128
6	3.67	1.5	13.0	158	0.08	3.8	11	1408 × 128 × 128
7	3.67	2.0	19.0	211	0.06	2.9	11	1716 × 156 × 156
8	3.67	3.0	36.0	316	0.04	1.8	11	2816 × 256 × 256
9	4.63	0.5	2.4	53	0.24	7.2	14	1792 × 128 × 128
10	4.63	1.0	6.8	105	0.12	8.2	14	1792 × 128 × 128
11	4.63	1.5	13.0	158	0.08	6.8	14	1792 × 128 × 128
12	5.51	0.5	2.4	53	0.24	6.5	14	1792 × 128 × 128
13	5.51	1.0	6.8	105	0.12	7.4	14	1792 × 128 × 128
14	5.51	2.0	19.0	211	0.06	5.0	14	2184 × 156 × 156
\(l_I/l_f=3.0\)
15	3.67	0.5	1.7	105	0.49	11.0	5.5	1408 × 256 × 256
16	3.67	1.0	4.8	211	0.24	5.3	5.5	1408 × 256 × 256
17	3.67	1.5	8.9	316	0.16	1.6	5.5	1408 × 256 × 256
18	4.63	0.5	1.7	105	0.49	7.8	7.0	1792 × 256 × 256
19	4.63	1.0	4.8	211	0.24	7.4	7.0	1792 × 256 × 256
20	4.63	1.5	8.9	316	0.16	5.1	7.0	1792 × 256 × 256
21	5.51	0.5	1.7	105	0.49	9.3	7.0	1792 × 256 × 256
22	5.51	1.5	8.9	316	0.16	5.8	7.0	1792 × 256 × 256
Figure 14: The top figure shows the DNS configuration of freely propagating flame. The bottom figure shows the cross-sectional views of the  $\mathrm{H}_2\mathrm{O}_2$  contours along with the initial position of the spontaneous ignition front (dashed white line). Reprinted from [70], 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
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 at an equivalence ratio of 0.43 and initial temperature of  $773\mathrm{K}$  is used at the inlet of the domain. This mixture is equivalent to an equivalence ratio of 0.35 and  $T_{u} = 990\mathrm{K}$ , and its ignition delay time  $(\tau_{\mathrm{ign},0})$  and laminar flame speed  $(S_L)$  are identified to be  $0.55\mathrm{ms}$  and  $14.7\mathrm{ms}^{-1}$ , respectively. The reference laminar flame thickness,  $l_f$ , is evaluated to be  $0.66\mathrm{mm}$ . After initialization, the ignition front is superimposed on a turbulent flow-field using a one-to-one correspondence in x-space (Figure 14). Depending on varying  $U_{\mathrm{in}}$  and  $u'$ , the flame may stabilize at a 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  $L_{y} = L_{z} = 5.26l_{I}$ , and the length in the streamwise direction is  $L_{x}$ , which is different for individual configuration. The other associated parameters for all 22 configurations are summarized in Table 6. The turbulent Reynolds number for the unburnt gas is defined as  $\mathrm{Re}_{t,u} = u'l_I / \nu_u$ , where  $\nu_{u}$  is the kinematic viscosity of the unburnt gas. The Karlovitz number,  $\mathrm{Ka}_u$ , is defined as the ratio of the flame characteristic time  $t_f = l_f / S_L$  to the Kolmogorov characteristic time  $t_{\nu u} = (\nu_{u}l / u^{\prime 3})^{1 / 2}$ . The ignition Damkohler number is  $\mathrm{Da}_{\mathrm{ign}} = \tau_I / \tau_{\mathrm{ign},0}$ , with the large-eddy turnover time  $\tau_I = l_I / u'$ . For all configurations, the simulations are run until a 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 it 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 of perfect gases. A detailed chemical mechanism [97] for hydrogen combustion with 9 species and 21 reactions is used for all configurations.
E Additional Momentum128 3D SR Dataset Details
The Momentum128 3D SR dataset is a processed subset of BLASTNet 2.0, and available for download at https://www.kaggle.com/datasets/waitongchung/blastnet-momentum-3d-sr-dataset.
E.1 Data Format and Directory Structure
The Momentum128 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.1.3), and instructions for reading the data in Appendix E.1.4.
Figure 15: Directory structure of the Momentum128 3D SR dataset.
E.1.1 Files on Flow Physics
All 2000 sub-volumes (labels with  $128 \times 128 \times 128$  number of voxels) of density and three velocity components  $[\rho, \mathbf{u}]$  are also presented in little-endian single-precision binary format, in a similar fashion to BLASTNet 2.0 (see Appendix D.1.1), which can be read with np.fromfile or np.memmap. This is shown in Figure 15, which also shows the five data splits described in Section 3.2. In addition, Favre-filtered features for 8, 16, and  $32 \times$  SR are also provided, along with pre-trained 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 are named with <Variable Name and SI Unit>_id<hash value>.dat, where the hash value provides a unique ID based on the spatial coordinates of the sub-volume location and the index of configuration.
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
Figure 16: Statistics of the specific kinetic energy  $\rho e^k$  of each  $128^{3}$  sub-volume in the Momentum128 3D SR dataset. Kaggle links to source (raw) data is available in Table 4.
E.1.4 Reading Data
Similar to BLASTNet 2.0, this data can be read by using np.fromfile/np.memmap to load and reshape the data. Links to tutorials and dataloaders to perform this are also shared in Appendix A. We also attach a 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 a 67 GB dataset that can fit into a 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 16, which is a more detailed version of Figure 2.
E.1.6 Favre-filtered DNS and Coarse-grid Simulations
For the Momentum128 3D SR low-resolution feature samples, we employed Favre-filtering (Equation (3) to generate a canonical surrogate, known as finite-volume optimal LES [74], that has direct
theoretical connections [73, 45] to coarse-grained simulations (also known as implicit LES). While having an implicit LES solution and corresponding DNS data as a 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, 44] and SR [46, 37] have conventionally employed filtered DNS in place of implicit LES flowfields.
Figure 17 demonstrates the quantitative and qualitative relationship between  $8 \times$  low-resolution and super-resolved implicit LES and Favre-filtered 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 a common tool for analyzing the turbulent properties across different lengthscales (up to the DNS wavenumber  $\kappa_{norm}^{DNS}$ ) through Fourier transform operations [26]. Note that the wavenumber  $\kappa$  is inversely proportional to length, i.e., larger wavenumbers correspond to smaller lengthscales. To generate the implicit LES solution, we perform a coarse-grid simulation of the HIT DNS configuration detailed in Appendix D.2.1, with a  $16^3$  spatial grid. It can be seen that TKE spectra is truncated in a similar fashion in both low-resolution implicit LES and Favre-filtered DNS at  $\kappa_{norm}^{LES}$ , demonstrating that turbulence is under-resolved beyond this wavenumber due to the coarse grid. We perform  $8 \times$  SR on both low-resolution flowfields with the 50.2M gradient-loss RRDB model. The super-resolved Favre-filtered 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 an ML model trained on Favre-Filtered DNS can be employed towards super-resolving implicit LES flowfields while maintaining reasonable spectral behavior.
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  $8 \times$  implicit LES (coarse-grid 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 4 and 5.
F.1 Additional Metrics Details
For this work, we employ a conventional definition of normalized root-mean-squared error (NRMSE) for evaluating the ML models:
$$
\operatorname {N R M S E} (\hat {\phi}, \phi) = \frac {\sum_ {i = 1} ^ {N _ {v o x}} \sum_ {j = 1} ^ {N _ {s a m p}} \left(\phi_ {i j k} - \hat {\phi} _ {i j k}\right) ^ {2}}{\sum_ {i = 1} ^ {N _ {v o x}} \sum_ {j = 1} ^ {N _ {s a m p}} \sum_ {k = 1} ^ {N _ {c}} \phi_ {i j} ^ {2}}, \tag {8}
$$
with  $N_{\text{vox}}$  number of voxels and  $N_{\text{samp}}$  number of samples of model prediction  $\hat{\phi}$  and ground truth  $\phi$ . As discussed in Section 4.2, NRMSE is used to evaluate super-resolved density and velocity, along with subgrid-scale stress divergence, volume-averaged kinetic energy, and volume-averaged turbulent dissipation rate. When evaluating the NRMSE of 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 to evaluate super-resolved density and velocity, along with subgrid-scale stress divergence. SSIM is calculated by passing a sliding window of size  $9 \times 9 \times 9$  (similar to the original SSIM paper [82]) across model prediction  $\hat{\phi}$  and ground truth  $\phi$ , and evaluating their statistical quantities. Specifically, SSIM is defined by:
$$
\operatorname {S S I M} (\hat {\phi}, \phi) = \frac {1}{N _ {s a m p} N _ {w} N _ {c}} \sum_ {i = 1} ^ {N _ {s a m p}} \sum_ {j = 1} ^ {N _ {w}} \left(\frac {2 \mu_ {\hat {\phi}} \mu_ {\phi} + c _ {1} ^ {2}}{\mu_ {\hat {\phi}} ^ {2} + \mu_ {\phi} ^ {2} + c _ {1} ^ {2}} \cdot \frac {2 \sigma_ {\hat {\phi} \phi} + c _ {2} ^ {2}}{\sigma_ {\hat {\phi}} ^ {2} + \sigma_ {\phi} ^ {2} + c _ {2} ^ {2}}\right) _ {i j}, \tag {9}
$$
with mean  $\mu_{\{\hat{\phi},\phi\}}$ , variance  $\sigma_{\{\hat{\phi},\phi\}}^2$ , and covariance  $\sigma_{\hat{\phi}\phi}$  for  $N_w$  number of sliding windows. In computer vision applications with RGB images,  $c_1 = 0.01$  and  $c_2 = 0.03$  are typically used to ensure numerical stability [82]. However, we found these values insufficient for numerical stability in this work. Hence, we employed  $c_1 = 0.1$  and  $c_2 = 0.3$ . Note that for  $\mathrm{SSIM}_{sgs}$  (Equation (4b)), the edge voxels of  $\nabla \cdot \tau^{sgs}$  are neglected prior to evaluation, to remove voxels with first-order spatial differencing.
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	1	1	1	1	2	5	8	23
First Channel Size	4	16	32	64	64	64	64	64
Kernel Size	3	3	3	3	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	46	68	86	120
Kernel Size	3	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	1	1	1	10	20	20
Channel Size	26	34	44	60	64	64	64	64
Residual Groups	1	1	1	1	1	2	3	10
Kernel Size	3	3	3	3	3	3	3	3
Residual Scaling	1	1	1	1	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	2	2	2
FNO Channel Size	-	14	20	24	34	46	68	86
Conv-FNO Blocks	-	32	32	32	32	32	32	32
Convolutional 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
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-Residual Dense Block (RRDB) [22], (ii) Enhanced Deep Residual Super-resolution (EDSR) [23], and (iii) Residual Channel Attention Networks (RCAN) [24].
We choose to study these models due to their difference in architecture paradigms. Specifically, RRDB employs a residual layers within residual layers; EDSR features an expanded network width; RCAN utilizes long skip connections and channel attention mechanisms. In addition, we consider a 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 a 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 et al. [78]. Specifically, the architecture settings for RRDB, EDSR, RCAN, and Conv-FNO 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 in 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 5 by increasing the FNO channel size since this approach led to good scaling behavior, especially when compared to increasing the number of Fourier modes.
Table 8: 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	1	2	3	4	5
FNO Channel Size	6	34	34	34	34	34
Conv-FNO Blocks	6	32	32	32	32	32
\( \downarrow \)Val. MSE \( \left\lbrack {\times {10}^{-3}}\right\rbrack \)	278	175	5.3	87.8	182	156
F.3 Additional Loss Details
Similar to other turbulent SR studies [37, 46], all models are trained with mean-squared-error (MSE) loss  $L_{\mathrm{MSE}}$ , unless otherwise stated. Specifically, for a predicted channel quantity  $\hat{\phi}$  and ground truth  $\phi$ :
$$
L _ {\mathrm {M S E}} = \frac {1}{N _ {v o x} N _ {s a m p} N _ {c}} \sum_ {i = 1} ^ {N _ {v o x}} \sum_ {j = 1} ^ {N _ {s a m p}} \sum_ {k = 1} ^ {N _ {c}} \left(\phi_ {i j k} - \hat {\phi} _ {i j k}\right) ^ {2}, \tag {10}
$$
for  $N_{\text{vox}}$  number of voxels,  $N_{\text{samp}}$  number of samples, and  $N_c$  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 at early timesteps, as shown with the 34.6M EDSR  $8 \times$  model in Figure 18. This increased robustness motivated our choice of MSE as a loss function.
In addition, we train variants of the RRDB with a physics-informed gradient-based loss resulting in:
$$
L _ {\text {p h y s}} = (1 - \lambda) L _ {\text {M S E}} + \lambda L _ {\text {g r a d}}, \quad \text {w h e r e} \tag {11a}
$$
$$
L _ {g r a d} = \frac {\Delta^ {2}}{3 N _ {v o x} N _ {s a m p} N _ {c}} \sum_ {i = 1} ^ {N _ {v o x}} \sum_ {j = 1} ^ {N _ {s a m p}} \sum_ {k = 1} ^ {N _ {c}} \sum_ {l = 1} ^ {3} \left[ \left(\nabla \phi_ {i j k}\right) _ {l} - \left(\nabla \hat {\phi} _ {i j k}\right) _ {l} \right] ^ {2}, \tag {11b}
$$
The gradient terms are evaluated using torchgradient, which corresponds to a second-order central differencing scheme that is optimized for GPU calculations. This is done on both super-resolved and ground truth fields before inputting the gradient terms into the MSE function. This gradient term enables the ML models to implicitly learn transport phenomena that arise in flow physics PDEs. For example, advection in the mass conservation equation can be expressed as:
$$
\nabla \cdot (\rho \mathbf {u}) = \mathbf {u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf {u}, \tag {12}
$$
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 be applicable to the advection of momentum, which is the transport term responsible for turbulent phenomena [26].
In Section 5, we employ the weighting factor  $\lambda = 0.99$ , which was determined from a hyperparameter search on RRDB 2.7M models, with the validation MSE for various lambda shown in Table 9.
Table 9: Validation MSE for different weighting factor  $\lambda$  for the gradient-based loss.
Weighting factor λ	0.000	0.100	0.300	0.500	0.700	0.900	0.990	0.999
↓Val. MSE [×10-3]	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  $\nabla \cdot (\rho \mathbf{u})$ , after transformation. The steps to ensure this are summarized in Figure 19, which demonstrates how flip and rotation operations can still result in continuity-consistent transformations on a 2D image. These operations have been extended for 3D random flip and rotation, which we 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 a batch size of 64 across 16 Nvidia V100 GPUs. Learning rate is initialized at  $1 \mathrm{e} - 4$ , and halved every 300 epochs. Both the number of training iterations and learning scheduling are chosen to match other SR studies [22-24] and are found to be sufficient for the SR prediction as 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  $32 \times$  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 the 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://github.com/Lyken17/pytorch-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  $5N_{points}\log N_{points}$  [53].
1. Flip
Continuity-consistent Flip
2. Neg. sign
1. Rotate
Figure 19: Continuity-consistent augmentation on a 2D image that preserves reflective and rotational invariance of the  $\nabla \cdot (\rho \mathbf{u})$ .
Continuity-consistent Rotation
2. Swap axes
Figure 20: Vanilla RRDB 50.2M MSE loss at 8, 16, and  $32 \times$  SR, with initial learning rate  $\mathrm{LR} = 1\mathrm{e} - 4$ .
For tricubic interpolation, an interpolant  $p$  at coordinates  $x, y$ , and  $z$  is defined by [101]:
$$
p (x, y, z) = \sum_ {i = 0} ^ {3} \sum_ {j = 0} ^ {3} \sum_ {k = 0} ^ {3} a _ {i j k} x ^ {i} y ^ {j} z ^ {k}. \tag {13}
$$
For a single point, this totals to 63 additions and 84 multiplications for the matrix multiplication applications, with additional 9 multiplications for the exponents. The elements of tensor  $a_{ijk}$  can be found in a 64-element vector  $\alpha$ , which can in turn be found via. Hence:
$$
\alpha = A _ {1} ^ {- 1} b, \tag {14}
$$
where  $A_{1}$  is a  $64 \times 64$  integer matrix of constants that is predefined in the algorithm, and  $b$  is a 64-element vector which contains derivatives from the input data  $\phi$ . For a uniform spatial grid,  $b$  can be evaluated by evaluating derivatives on a sub-volume with  $4 \times 4 \times 4$  vertices, which can be flattened to a 64-element vector  $\phi_{sub}$ :
$$
\begin{array}{l} 8 \alpha = A _ {1} ^ {- 1} A _ {2} \phi_ {s u b} \\ = B \phi_ {s u b}, \tag {15} \\ \end{array}
$$
where  $A_{2}$  contains integer coefficients for finite-differencing and  $B = A_{1}^{-1}A_{2}$  contains 2765 zero elements. This totals to 4032 additions and 4096 multiplications, when considering dense matrices.
If sparse multiplication is used, this results in 1301 multiplications and approximately 1237 additions, with 64 multiplications for dividing by 8. Putting this together:
$$
\mathrm {F L O P s} _ {\text {t r i}} ^ {\text {d e n s e}} = 8 3 2 8 N _ {\text {v o x}} N _ {c}, \tag {16}
$$
$$
\operatorname {F L O P s} _ {\text {t r i}} ^ {\text {s p a r s e}} = 2 7 3 8 N _ {v o x} N _ {c}, \tag {17}
$$
In this work, our samples contain  $128^{3}$  voxels and 4 channels. Thus, tricubic interpolation costs 23 GFLOPs (reported in Section 5) for sparse matrix multiplication, while employing dense matrix multiplication costs 69 GFLOPs, per inference with batch size of 1.
G.1 Additional Evaluation
In Figure 21, we present normalized absolute error in specific kinetic energy  $|\epsilon_{\rho e^k}| / \rho e_{max}^k$  from model predictions shown in Figure 3.
Figure 21: Normalized absolute error in specific kinetic energy  $\left| \epsilon_{\rho e^{k}} \right| / \rho e_{max}^{k}$  from model predictions shown in Figure 3.
Tables 10 to 12 report additional results from evaluating the models from Tables 2 and 3 with NMRSE metrics. In Table 13, we report the SSIM used for evaluating the scaling behavior of all  $8 \times$  SR models discussed in Section 5.