Turbulent Injection assisted by Diffusion Models for Scale Resolving Simulations

Caroline Sainvitu; Gilles Louppe; Joachim Dominique; Lionel Salesses; Margaux Boxho; Michel Rasquin; Tariq Benarama; Thomas Toulorge

REVIEW 4 major objections 6 minor 36 references

Turbulent Injection assisted by Diffusion Models for Scale Resolving Simulations

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A diffusion model conditioned on one Reynolds number can generate realistic turbulent inlet fields for LES/DNS, replacing stored precursor libraries.

desk verdict A promising ML-CFD proof-of-concept whose storage-free inflow generation works reasonably well, but the abstract overstates autocorrelation accuracy and the scalar Re_Lint conditioning does not actually control L_int. read the letter →

arxiv 2508.04318 v1 pith:KV7B6UT3 submitted 2025-08-06 physics.flu-dyn

Margaux Boxho , Joachim Dominique , Tariq Benarama , Michel Rasquin , Lionel Salesses , Caroline Sainvitu , Gilles Louppe , Thomas Toulorge This is my paper

classification physics.flu-dyn

keywords turbulentinflowgenerationdiffusionmodelsdecayinghomogeneousisotropicturbulencelargeeddysimulationclassifier-freeguidanceinletboundaryconditionmomentmatchingposteriorsamplingenergyspectrum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Turbulent inflow remains a bottleneck for scale-resolving simulations: precursor/library methods give realistic fluctuations but require expensive auxiliary simulations and large stored libraries, while synthetic generators are cheap but need a development distance. This paper tries to get the realism of the precursor method without its memory and turnaround cost by training a classifier-free diffusion model on decaying homogeneous isotropic turbulence (DHIT) boxes. The model is conditioned on a single scalar, $Re_{L_{\mathrm{int}}} = \sqrt{2/3}\,k\,L_{\mathrm{int}}/\nu$, which encodes both the turbulent kinetic energy level and the integral length scale. On injection into a free domain, the energy spectrum of the synthetic boxes overlaps that of the true DHIT within about one box length, matching the development distance of the precursor method while storing only the model (about 32 MB) instead of 24 flow fields. The paper is explicit that the hardest statistic to reproduce is the two-point autocorrelation function, with the generated integral length scale overestimated by about 14%.

What carries the argument

The carrying mechanism is a classifier-free guided diffusion model: a UNet score network trained with the variance-preserving SDE objective to denoise $3\times64^3$ velocity boxes, conditioned on the scalar Reynolds number $Re_{L_{\mathrm{int}}} = \sqrt{2/3}\,k\,L_{\mathrm{int}}/\nu$. This single number is the interface between the user-specified turbulence state and the generated field — it is what lets the model interpolate across TKE/scale combinations without retraining. Temporal continuity between successive generated boxes, required for an inlet condition, is imposed by Moment Matching Posterior Sampling, which treats the last $m=16$ slices of the previous box as a noisy measurement co

What would settle it

Generate two DHIT boxes with the same $Re_{L_{\mathrm{int}}}$ but different pairs $(k, L_{\mathrm{int}})$ — for example, double $k$ and halve $L_{\mathrm{int}}$ so the product is unchanged — and condition the model at that Reynolds number. If the model produces the same energy spectrum and integral length scale for both, the conditioning is degenerate and the central premise fails.

Watch

Extended reading notes

Core claim

A classifier-free guided diffusion model trained on 214 DHIT boxes spanning $Re_{L_{\mathrm{int}}} \in [196.58, 1302.16]$ can generate new $64^3$ instantaneous velocity boxes whose a priori statistics (energy spectrum, vorticity distribution, Reynolds-stress anisotropy) closely match ground truth, and after injecting them at the inlet of a free-domain LES the turbulence develops as fast as with the original precursor method — in less than one box length for the energy spectrum at 23.3% turbulence intensity. The conditioning variable $Re_{L_{\mathrm{int}}}$ embeds both the turbulent kinetic energy level and the integral length scale, so a single model covers many target states and interpolate

Load-bearing premise

The load-bearing premise is that a single Reynolds number $Re_{L_{\mathrm{int}}}$ contains enough information to fix the turbulence statistics the model must generate; if two flows with different turbulent kinetic energies and integral length scales share the same Reynolds number, the conditioned samples cannot distinguish them.

Editorial extensions

If this is right

Within the trained Reynolds-number range, the user can query any target TKE/length-scale state without running a new DHIT; 24 boxes are generated in about three minutes on the tested GPU setup.
The inlet library no longer needs to be stored or read from disk: the model itself is the library, and successive boxes can be generated on demand.
The development distance of the generated turbulence is not increased relative to the precursor method (1–2 box lengths, depending on turbulence intensity), so no extra fetch length must be added to the computational domain.
The stochasticity of the diffusion model decorrelates successive inlet boxes: MMPS keeps the time correlation below 10%, unlike simply re-injecting one frozen DHIT box.
The same model can be reused in the trial-and-error loop that matches an inlet condition to measurements at a target station, because regenerating at a new conditioning value costs seconds instead of a full precursor rerun.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not tested here: because conditioning is a single scalar, any two physical states with the same $Re_{L_{\mathrm{int}}}$ but different $(k, L_{\mathrm{int}})$ are indistinguishable to the model; conditioning separately on $k$ and $L_{\mathrm{int}}$ would directly test whether this embedding is lossy.
The observed asymmetry — extrapolating to lower $Re_{L_{\mathrm{int}}}$ works well, extrapolating to higher does not — suggests a learned hierarchy in which lower-Reynolds fields are filtered versions of higher-Reynolds ones. A testable consequence is that a model trained on a longer decay sequence could generate new states by denoising downward rather than by inventing small scales.
The MMPS inlet anisotropy (dominant $u'u'$ component) indicates that conditioning on $y$–$z$ slices constrains the streamwise velocity differently from the cross-stream components; adding a divergence-free projection during sampling, which the paper lists as a perspective, would be a direct test.
Because the training set is built from periodic boxes with translation augmentation, the current proof is tied to homogeneous freestream turbulence; extending to boundary layers or wakes would require a non-periodic architecture and a different training representation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a useful proof-of-concept, but the headline claim is at odds with the paper's own evidence. What is new: the authors adapt classifier-free diffusion to generate 3D DHIT boxes conditioned on a single Reynolds number, use MMPS to make consecutive boxes continuous, and show in a real DG-LES solver that the synthetic inlets develop as fast as precursor-based inlets while avoiding stored precursor libraries. That practical target is real, and the a posteriori spectrum overlap within one box length is credible.

Credit where it's due: they compare against a library-based precursor in a production solver, include two a posteriori cases, report training and inference costs, and cite the relevant diffusion-inflow work. The MMPS time-correlation test is a sensible check. The scope is honest about being DHIT-only, and the paper does not pretend to solve wall-bounded or anisotropic cases.

Soft spots, in increasing order. First, the abstract says samples accurately reproduce two-point autocorrelation functions; Section 4.1 says the opposite: f(r) is underestimated, g(r) is overestimated, and the model predicts nearly the same correlations at each ReLint. The a posteriori integral-scale errors are about 14% and 20%. That is a direct internal contradiction. Second, the conditioning variable is the real issue. ReLint = sqrt(2/3 k) Lint / nu is a single scalar, and the generated samples do not vary Lint in the way the target states do. If a user wants a specific vortex size, this model cannot yet deliver it. The leftover note in the Figure 9 caption reinforces that the authors know this is unresolved. Third, the optimizer hyperparameters and MMPS parameters are chosen by minimizing loss built from the same statistics used for validation, and Barycentric outliers are simply discarded; both inflate the apparent a priori quality. Missing error bars make the quantitative claims hard to interpret. These are real but fixable: use more independent metrics, report variability, and soften the claims.

Who this is for: people working on ML-generated inflow conditions, especially those combining generative models with CFD. It deserves a serious referee rather than a desk reject. If I were editing, I would send it out and ask for claims to be aligned with evidence before acceptance. As it stands, it is a promising engineering demonstration, not a validated method for controlling turbulent length scales via ReLint.

Referee Report

4 major / 6 minor

Summary. The paper proposes a classifier-free diffusion model trained on 214 three-dimensional snapshots of Decaying Homogeneous Isotropic Turbulence to generate instantaneous turbulent velocity boxes for use as inflow boundary conditions in LES/DNS. The model is conditioned on a scalar Reynolds number Re_Lint = sqrt(2/3 k)L_int/nu. Generated boxes are assessed a priori via energy spectra, two-point autocorrelations, barycentric anisotropy triangles, and vorticity PDFs, and a posteriori by injecting them into a free domain with two concatenation methods (blending and moment matching posterior sampling). The authors compare against a library-based precursor method and report similar development distances, lower storage, and faster turnaround for finding target TKE levels.

Significance. The manuscript addresses a real bottleneck in scale-resolving simulations: the high memory and setup cost of precursor-based turbulence inflow. It demonstrates that a diffusion model can generate 3D turbulent boxes with reasonable energy spectra and TKE at multiple Reynolds numbers, and that injected samples develop qualitatively as quickly as the precursor method. The computational cost comparison (Table 2, Fig. 21) and the use of MMPS for temporal continuity are useful contributions. However, the central claim about accurate reproduction of two-point statistics is not supported by the reported results, and the conditioning on a single scalar is not validated for length-scale control. With revisions, the method remains a promising proof-of-concept.

major comments (4)

[Section 4.1 / Fig. 10 / Abstract] The abstract claims samples 'accurately reproduce turbulence statistics, such as the energy spectrum and the two-point autocorrelation functions.' The paper's own a priori results contradict this: §4.1 states there is 'an underestimation of f(r) and an overestimation of g(r)' and that 'The model seems to have difficulty distinguishing between the integral length scales and predicts the same correlations at each ReLint.' A posteriori, §4.2.1 reports L_int overestimated by about 14%; §4.2.2 reports about 20% relative error at x/L=4. Please quantify the errors on f(r), g(r), and L_int and revise the abstract and conclusions accordingly. This is not a presentation issue; it is the paper's headline result.
[Section 4, Eq. (11)] The conditioning strategy assumes a single scalar Re_Lint = sqrt(2/3 k) L_int / nu uniquely determines the target statistics. The paper does not show identifiability: in the 214 training boxes, is L_int a deterministic function of Re_Lint? If two different (k, L_int) states share a Re_Lint value, or if the model cannot vary L_int at fixed Re_Lint, then a user specifying Re_Lint cannot target a desired vortex size. The failure to reproduce L_int (Figs. 10, 14, 18) is consistent with this identifiability problem. Please plot L_int vs Re_Lint for the training data and consider conditioning on TKE and L_int separately (e.g., two-channel conditioning) or justify why Re_Lint is sufficient.
[Section 4.1, Eq. (12) and §3.2] The hyperparameters lambda*, tau*, gamma* and the MMPS parameters m, Sigma_y are selected by minimizing a loss (Eq. 12) that includes the same statistics (k, E(kappa), f/g, barycentric coordinate) used later for validation. Consequently, the reported a priori agreement is not independent evidence of generalization. Please perform a nested validation (e.g., a separate hyperparameter-selection split) or report sensitivity of the metrics to lambda, tau, gamma, m, Sigma_y. Note that the h_{f,g} term is weighted by 100, so the persistent failure of correlation functions is especially telling.
[Sections 4.2.1-4.2.2 / Section 5] The a posteriori comparison rests on only two test conditions and on single trajectories without error bars or statistical significance over the 24 generated boxes. The statement 'The development distance of the two methods is approximately the same' (Section 5) is qualitative. Please define the development length quantitatively (e.g., distance to reach within a specified tolerance of the target TKE and spectrum) and report uncertainties. This is the main practical advantage claimed for the method.

minor comments (6)

[Figure 9 caption] The caption contains a leftover internal note: 'Discuss more the limitation of a pure data-driven model...' Remove it before submission.
[Eq. (5)] The definition of R_ij appears with a comma inside the ensemble average; it should be R_ij(r,t) = <u_i(x+r,t) u_j(x,t)>.
[Section 2.3.1 / Table 1] The text counts 214 boxes and then says 62 boxes and 17 boxes are used for training and validation; the relationship to Table 1's training size 136,102 (after translations) should be stated clearly.
[Figure 15 / Figure 20 captions] Both captions say 'three streamwise locations,' but the figures show eight panels. Correct the captions to match the displayed x/L values.
[Table 2] The symbols ✓, ≈, and ✗ need a legend; the Storage row formatting (32MB GPUs, 72MB CPUs, 3MB CPUs) is confusing and should be clarified.
[Section 2.3.2 / Section 6] 'descent synthetic turbulence' should be 'decent.' The conclusion should be toned down to reflect the L_int limitation rather than stating 'good agreement' without qualification.

Circularity Check

1 steps flagged · score 3.0 of 10

Partial circularity: sampling hyperparameters are tuned on the same statistics later reported as validation; the core generation is otherwise self-contained.

fitted input called prediction [Section 3.2 (MMPS parameters) and Section 4.1, Eq. 12 (grid search for lambda, tau, gamma)]
"In Section 4, m and Σy are set to 16 and 10−4, respectively. These values are obtained by grid search and are those that best preserve the statistics of interest defined in Section 2. ... The optimal λ ∗, τ ∗, and γ ∗ are those how minimize the following loss, L (λ , τ, γ) = αkhk + αEκ hEκ + α f ,gh f ,g + α△h△, (12) where ... hk = |ˆk − k|/k, h△ = ∑ i 1(ξi,ηi) /∈△, hEκ = ... h f ,g = ..."

The sampling hyperparameters of the diffusion model (λ, τ, γ) and the continuation parameters of MMPS (m, Σy) are explicitly chosen by minimizing a loss built from the very statistics used for validation: TKE level, energy spectrum, autocorrelation functions, and barycentric realizability. The reported 'accurate reproduction' of these statistics is therefore a calibrated optimum of the evaluation metrics, not an independent prediction. This is partial rather than total circularity because the diffusion model must still generate the 3D velocity fields and the a posteriori injection comparisons provide a separate physical benchmark, but the headline statistical agreement is in part manufactured by the selection procedure.

full rationale

Most of the derivation is self-contained: the diffusion model is trained on DHIT snapshots and its output is tested against the precursor method in an actual free-domain injection. The cited prior work by co-authors (Rozet & Louppe; Rasquin et al.) supplies open-source sampling code and the baseline precursor method, not an unverified uniqueness theorem or a load-bearing self-citation. The conditioning on the scalar Re_Lint is an identifiability choice rather than a circular definition; indeed, the reported 14–20% errors in L_int show that the conditioning does not force the integral length scale, so the model's imperfect length-scale reproduction is a genuine empirical finding. The main circularity concern is the hyperparameter selection in Section 4.1 and Section 3.2: λ, τ, γ, m, and Σy are tuned to minimize errors on the same TKE, spectrum, correlation, and barycentric statistics that are later presented as validation. This makes the agreement on those metrics a calibrated optimum rather than an independent confirmation. It does not completely collapse the central claim because the generative model must still produce realistic 3D fields and the a posteriori development-length comparison is an independent physical test. The unedited note in the Figure 9 caption ('Discuss more the limitation of a pure data-driven model...') further acknowledges unresolved limitations, which supports a moderate rather than high circularity score.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new physical entities, forces, particles, or dimensions are introduced. The generated boxes are samples from a learned distribution; MMPS is an algorithmic procedure, not an entity. The free parameters are the sampling and MMPS hyperparameters tuned by grid search, plus the loss weights in Eq. (12).

free parameters (6)

lambda (Langevin corrections) = 4
Number of Langevin Monte Carlo corrections in the sampling solver, chosen by grid search over [1,2,4,6,8] minimizing Eq. (12).
tau (reverse SDE step) = 0.1
Discretized reverse SDE step, chosen by grid search over [0.01,0.1,0.25,0.5,0.75,1].
gamma (guidance scale) = 2
Classifier-free guidance scale, chosen by grid search over [1..5].
m (MMPS slices) = 16
Number of y-z slices masked for posterior sampling; obtained by grid search to best preserve statistics (Section 3.2).
Sigma_y (MMPS measurement variance) = 1e-4
Variance of the Gaussian observation model in MMPS; obtained by grid search (Section 3.2).
Loss weights alpha_k, alpha_E, alpha_fg, alpha_delta = 1, 1, 100, 1/ns
Weights in Eq. (12) chosen by hand for the hyperparameter grid search; they encode the relative priority of TKE, spectrum, correlations, and anisotropy.

assumptions (5)

domain assumption Taylor frozen turbulence hypothesis and spatial homogeneity hold at the inlet, so a frozen precursor box can be swept at the mean inlet velocity.
Section 2.2 states these are the two main assumptions of the boundary condition; violation is acknowledged as a possible cause of the TKE drop at high turbulence intensity.
domain assumption Decaying Homogeneous Isotropic Turbulence is a canonical representation of experimental freestream turbulence in front of blades and grids.
Section 2.3 motivates DHIT as the canonical case; the whole method is built on this equivalence.
ad hoc to paper The Reynolds number Re_Lint = sqrt(2/3 k) L_int / nu uniquely determines the turbulent state the model should generate.
Section 4 states both TKE and integral length scale are embedded in this single number; if two states share Re_Lint with different k and L_int combinations, conditioning is ambiguous. This is a modeling choice specific to this paper.
domain assumption Small scales below the DHIT resolution are restored naturally by the flow solver after injection.
Section 2.3 concludes from Figure 3 that a 64^3 box has similar spectrum after one box length as a 128^3 box, so the solver recovers unresolved scales via the turbulent cascade.
standard math The MMPS posterior sampling is a valid way to impose continuity between consecutive generated boxes.
Section 3.2 relies on the moment-matching posterior sampling theory from Rozet and Louppe (2023) and Rozet et al. (2024); the paper cites the original derivation rather than re-proving it.

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Cite this review

Pith. "Pith review of Turbulent Injection assisted by Diffusion Models for Scale Resolving Simulations." pith.science (2026). https://pith.science/paper/KV7B6UT3

@misc{pith2026250804318,
  author       = {Pith},
  title        = {Pith review of: Turbulent Injection assisted by Diffusion Models for Scale Resolving Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KV7B6UT3}},
  note         = {Machine review of arXiv:2508.04318}
}

read the original abstract

The present research proposes a new memory-efficient method using diffusion models to inject turbulent inflow conditions into Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) for various flow problems. A guided diffusion model was trained on Decaying Homogeneous Isotropic Turbulence (DHIT) samples, characterized by different turbulent kinetic energy levels and integral length scales. Samples generated by the diffusion model accurately reproduce turbulence statistics, such as the energy spectrum and the two-point autocorrelation functions, while preserving the ability to generate instantaneous three-dimensional velocity fields with detailed fluctuations. Physical representativeness is also evaluated by injecting the \textit{synthetic} samples into a free domain (i.e., without any wall boundary) through an inlet boundary condition. The method demonstrates promising results regarding energy spectrum, spatial correlation, turbulent kinetic energy level, and integral length scales without increasing the development distance as compared to a library-based method. The following article has been submitted to/accepted by Physics of Fluid (AIP Publishing LLC). After publication, it will be available https://doi.org/10.1063/5.0278541.

Figures

Figures reproduced from arXiv: 2508.04318 by the authors.

**Figure 1.** Sketch of the domain in the x − y plane, comprising the proper boundary conditions, the mesh, the location of the sponge layer, the matching location at which the experimental data needs to be matched, and the trial-and-error loop to define the appropriate conditioning (ReLint) of the diffusion model for the generation of a novel turbulent inlet velocity field u. However, due to computational constraints on the doma… view at source ↗

**Figure 2.** On the left, the evolution of the TKE; on the right, the evolution of the ratio between the TKE and the dissipation. The dashed red lines indicate the theoretical curves. The time at which the DHIT is frozen is usually based on the distance between the inlet plane and the object of interest (in the computational domain) and the pair (TI,Lint) given by the experimentalist, so that the DHIT can reach the desired stati… view at source ↗

**Figure 4.** shows the Barycentric triangle for the four DHITs frozen at different time steps. All points are within the triangle and, as expected, close to the upper vertex, corresponding to 3-component isotropic turbulence. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (16 more)

**Figure 5.** Figure 5: Distributions of the streamwise vorticity ωx at various ReLint for the database DHIT-03. 3 Diffusion Models Diffusion Models (Sohl-Dickstein et al., 2015; Ho et al., 2020) (DM), have been developed to generate realistic data from a statistical distribution of interest …

**Figure 6.** Figure 6: Schematic of the mask used in the MMPS method to impose continuity between successive boxes. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Distribution of the Reynolds number ReLint for the 214 DHITs, the training and validation data in green, blue and orange. The frozen model is used to generate new samples using the predictor/corrector method proposed by Rozet and Louppe (2023). This method contains two…

**Figure 8.** Figure 8: A priori assessment of the CF-DM conditioned at ReLint = {1374,834,571,192}. The energy spectrum at four TKE levels with the ground truth marked by solid circles and the generated samples marked by solid crosses of the same color. At the highest ReLint , several boxes …

**Figure 9.** Figure 9: A priori assessment of the CF-DM conditioned at ReLint = {1374,834,571,192}. Barycentric triangle with the circles indicating the ground truth and the crosses indicating the samples obtained from the CF-DM. Color mapping follows [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: A priori assessment of the CF-DM conditioned at ReLint = {1374,834,571,192}. The longitudinal and transverse correlation functions, on the left and right, respectively, with the ground truth marked in plain circles and the generated samples in plain crosses. The color…

**Figure 11.** Figure 11: A priori assessment of the CF-DM conditioned at ReLint = {1374,834,571,192}. The distribution of the streamwise vorticity ωx. The color mapping is similar to [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

**Figure 12.** Figure 12: Time correlation computed for different blending techniques, the samples are obtained by conditioning the CF-DM at ReLint = 571 using the following hyper-parameters: m = 16, Σy = 10−4 , λ ∗ = 4, τ ∗ = 0.1, and γ ∗ = 2 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

**Figure 13.** Figure 13: shows the streamwise evolution of the TKE. The black circle curve corresponds to the TKE of the DHIT simulation, where time is converted to space evolution using the mean inlet velocity. The blue triangle curve corresponds to the original precursor method, where a si…

**Figure 14.** Figure 14: Freestream turbulence injection at TI= 23.3% to a posteriori assess the CF-DM conditioned with ReLint = 834. Streamwise evolution of the longitudinal length scale. matches the DHIT 2D energy spectrum, because for x/L < 1.5 the TKE retrieved in the free-domain is small…

**Figure 16.** Figure 16: Freestream turbulence injection at TI= 23.3% to a posteriori assess the CF-DM conditioned with ReLint = 834. Streamwise evolution of the Reynolds stress components in the free domain, the different sets of curves are shifted by 10 for readability, the color mapping is…

**Figure 17.** Figure 17: Freestream turbulence injection at TI= 14.7% to a posteriori assess the CF-DM conditioned with ReLint = 571. Streamwise evolution of the TKE. 00 0 0 0 0 0 x/ 000 00 000 0 00 0 000 0 00 / [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: Freestream turbulence injection at TI= 14.7% to a posteriori assess the CF-DM conditioned with ReLint = 571. Streamwise evolution of the longitudinal length scale. vorticity probability distribution of the samples presents similar extreme events as the ground truth. N…

**Figure 19.** Figure 19: Freestream turbulence injection at TI= 14.7% to a posteriori assess the CF-DM conditioned with 571. Streamwise evolution of the Reynolds stress components in the free domain, the different sets of curves are shifted by 10 for readability, the color mapping is similar…

**Figure 21.** Figure 21: Efficiency graph discussed in Section 4.1). Nonetheless, after five trials, the CPU time required to run the DHIT exceeds the initial offset of the ML approach. The power of CF-DM is significant when it comes to generating samples at different levels of TKE. The model…

**Figure 16.** Figure 16: For the velocity fluctuations generated using MMP [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]

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