Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures

Jian-Xun Wang; Meng Wang; Xiantao Fan; Yi Liu

REVIEW 3 major objections 5 minor 259 references

Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Wall-modelled LES of boundary layers is accurate only when the subgrid-scale and wall closures are trained together inside the solver, and a model trained that way extrapolates well beyond its training Reynolds numbers.

desk verdict Solid demonstration of joint end-to-end training of SGS and wall closures in a differentiable WMLES solver; the extrapolation claim leans on the prescribed skin-friction law, but the coupling evidence holds. read the letter →

arxiv 2607.17357 v1 pith:DRTRBBPE submitted 2026-07-19 physics.flu-dyn

Xiantao Fan , Yi Liu , Meng Wang , Jian-Xun Wang This is my paper

classification physics.flu-dyn PACS 47.27.Eq

keywords wall-modelledlarge-eddysimulationsubgrid-scaleclosurewall-stressmodeldifferentiableCFDneuralend-to-endtrainingturbulentboundarylayerturbulencestatistics

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

The paper argues that in coarse wall-modelled large-eddy simulation the subgrid-scale (SGS) closure, the wall-stress model, and the numerical discretization are not separable: through the resolved velocity field each one shapes what the others see, so calibrating any of them in isolation cannot fix the coupled errors. To make the coupling learnable, the authors build a differentiable finite-volume solver in which both closures are neural operators constrained to the physical structure of their conventional counterparts, a U-Net feeding a fixed eddy-viscosity form for the SGS stress and a U-Net adding fluctuations to an empirical mean wall-stress law. Both are optimized end-to-end by back-propagating a loss on low-order statistics (mean velocity, r.m.s. fluctuations, friction velocity) through a rollout of the coupled solver. Across nine a posteriori tests of a zero-pressure-gradient turbulent boundary layer, the jointly trained model beats two standard WMLES baselines, reproduces the logarithmic mean-velocity region and resolved energy spectra even though neither is in the training objective, and extrapolates to more than four times the highest training Reynolds number. Ablations in which only one closure is learned fail in complementary ways, which the authors take as direct evidence that joint optimization is necessary.

What carries the argument

The carrying mechanism is the composed neural operator: for each closure, a trainable convolutional U-Net (3-D for SGS, 2-D for wall) supplies only the degrees of freedom that the conventional form leaves open, followed by a fixed differentiable physical layer that guarantees structure. For the SGS stress the fixed layer is the Vreman eddy-viscosity functional, which auto-switches off in laminar and pure-shear regions; for the wall stress it is an empirical skin-friction law for the mean plus the non-negative wall-eddy-viscosity imposition of the total stress. Because every operation in the solver is differentiable, the statistics-based loss is back-propagated through a 100-step rollout of t

What would settle it

Run the trained closures on a zero-pressure-gradient boundary layer at Re_theta around 10,000, well beyond the tested range, and check whether the mean-velocity log region and the Cf–Re_theta trend still hold; alternatively, retrain with the empirical mean wall-stress law replaced by a constant or deliberately biased value and see whether the emergent log region collapses — if it does, the empirical anchor, not the joint learning, is carrying the extrapolation.

Watch

Extended reading notes

Core claim

The central claim is that the SGS closure, the wall closure, and the discretization are coupled through the resolved field, and that only their joint, in-the-loop optimization recovers the full set of turbulence statistics. The authors demonstrate this with Hybrid-Joint: a differentiable solver in which a structure-constrained neural SGS closure (a trained coefficient field inserted into the Vreman eddy-viscosity form) and a structure-constrained neural wall closure (a learned correction and fluctuation added to an empirically scaled mean wall stress) are trained together from statistics alone. The resulting model outperforms conventional WMLES, transfers to unseen grids and domains, and — w

Load-bearing premise

The load-bearing premise is that the mean wall shear stress is reliably anchored by an empirical skin-friction correlation evaluated from an online running-averaged velocity profile, with the training loss additionally pinning the friction velocity; if that empirical input drifts at extrapolated conditions, the claimed generalization rests entirely on the learned correction, which is not independently verified.

Editorial extensions

If this is right

If the coupling claim is right, a posteriori WMLES accuracy cannot be guaranteed by improving either the SGS model or the wall model in isolation; both must be co-optimized in the discretized solver.
Low-order statistics, not pointwise SGS-stress labels, are sufficient training signals for the coupled closures, making the method applicable where only statistical reference data exist.
The trained closures are frozen and reused across Reynolds numbers, domains, and grids with no retraining, so the one-time training cost is amortized over many deployments.
Recovering the logarithmic velocity region and resolved spectra as emergent, un-forced behavior implies the closure pair is not overfitting the single-point statistics it was trained on.
Because the neural SGS closure inherits the eddy-viscosity form, it cannot represent backscatter; representing that would require relaxing the structure constraint, a limitation the paper itself states.

Reading between the lines

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

The coupling argument generalizes: if SGS and wall closures are coupled through the resolved field, then the numerical discretization is a third partner in the same loop, and one could co-learn discretization parameters such as stencil weights, time-step weighting, or numerical viscosity alongside the closures.
A sharper falsifier of the claimed emergent log law would be to train with the empirical mean wall-stress law replaced by a deliberately wrong scaling; if the log region persists, the structure constraint is doing the work, whereas if it collapses, the learned correction is merely compensating around the empirical anchor.
The framework's reliance on a running-averaged momentum-thickness Reynolds number suggests that an online estimator of the same integral from wall quantities could be learned, removing the empirical correlation from the loop and testing how much of the extrapolation was due to the scaling law.
A natural stress test beyond ZPG TBL is to apply the same joint-training recipe to an adverse-pressure-gradient or separated boundary layer, where the equilibrium wall-stress assumption is known to fail, and ask whether joint optimization of the same two structured closures still recovers the statistics.

Share X Bluesky LinkedIn Reddit HN

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

This paper earns its central claim: joint end-to-end training of the SGS and wall closures inside a differentiable solver is new, and the authors make a credible case for it. The composed-operator architecture—trainable U-Net plus fixed Vreman form, trainable U-Net plus scaling-law mean stress—is a sensible way to keep the learned parts well-posed, and stable training over 100 gradient-tracking steps with no pretraining is a real result for this chaotic-flow setting. The ablations are the strongest part: Hybrid-WallOnly and Hybrid-SGSOnly fail in complementary ways, which genuinely supports the conclusion that the two closures and the discretization are coupled and need to be optimized together. The transfer tests across Reynolds numbers, domains, and grids are also impressive, and the paper is honest about its scope (ZPG TBL only) and about the eddy-viscosity form's inability to produce backscatter.

The soft spots are real but not fatal. The stress-test note is right: Eq. (2.11) prescribes the mean wall stress via the Smits et al. skin-friction law, Eq. (2.12) evaluates Re_theta online, and Eq. (2.16) anchors u_tau in the loss. So the Cf–Re_theta agreement and the inner-scaled log region are largely by construction. The high-Re cases show that the learned correction remains small and stable, not that the closures have learned the Reynolds-number scaling themselves. The mean-only ablation in Fig. 16 makes this explicit: remove the fluctuating network term and the mean profile still looks fine because the imposed mean stress supplies the correct inner scaling. I'd also note the lack of error bars on single-rollout statistics, the fact that the Re_theta=5000 ground truth is wall-resolved WMLES rather than DNS, and the absence of released code/data for the in-house Diff-FlowFSI solver. Each is addressable.

None of this undermines the joint-training conclusion. The paper is a well-executed, significant step for differentiable WMLES, and it deserves serious refereeing. For revision, I'd ask for error bars or multiple independent rollouts, a clearer separation of what is imposed by the scaling law versus what is learned, and either released code or a detailed reproducibility appendix.

Referee Report

3 major / 5 minor

Summary. The paper presents a differentiable hybrid neural–CFD framework for wall-modelled LES of zero-pressure-gradient turbulent boundary layers. The SGS and wall closures are composed operators: a 3-D U-Net feeding a fixed Vreman-type eddy-viscosity layer, and a 2-D U-Net supplying a fluctuating/corrective wall stress on top of a Smits et al. skin-friction scaling law. Both closures are trained jointly and end-to-end through a differentiable finite-volume solver against a low-order statistics loss (mean velocity, r.m.s. velocity and pressure fluctuations, and friction velocity). Training is performed at Re_theta = 600 and 1200; deployment covers nine a posteriori cases up to Re_theta ≈ 6500, including changed domains, grids, and a spatially developing boundary layer. The authors report that Hybrid-Joint outperforms Smagorinsky and Vreman baselines, recovers the log region and resolved spectra, and that ablations learning only one closure fail in complementary ways, supporting the central claim that SGS closure, wall closure, and discretization must be trained jointly.

Significance. If the results hold, this is a substantive advance: it demonstrates stable end-to-end training of two coupled closures inside a differentiable solver from low-order statistics only, without pretraining, and with genuine deployment from a random Blasius inflow. The complementary failure pattern of the single-closure ablations is credible evidence for the coupling claim, and the paper is refreshingly explicit about limitations (ZPG-only scope, no backscatter, empirical wall-stress anchor). However, the headline extrapolation and log-region claims are weaker than the abstract suggests, because the mean wall stress is prescribed by Eq. (2.11) and u_tau is anchored in the loss, Eq. (2.16). Figure 5(b) is therefore not an independent test of learned Reynolds-number scaling. The central joint-training conclusion survives this concern, but the manuscript needs revision to separate constrained generalization from fully learned generalization, and to support single-rollout quantitative claims with uncertainty estimates.

major comments (3)

[§2.2.3 and §2.3, Eqs. (2.11)–(2.12), (2.15)–(2.16)] The extrapolation/log-region claim is substantially predetermined by the empirical Smits et al. skin-friction law. Equation (2.11) sets the mean wall stress at every Reynolds number, Eq. (2.12) evaluates Re_theta online, and Eq. (2.16) explicitly anchors u_tau to the ground truth. Thus the Cf–Re_theta agreement in Fig. 5(b) and the inner scaling that produces the log region are to first order consequences of a correlation calibrated for ZPG TBLs, not of the learned closures. I recommend relabelling this as “constrained extrapolation around an empirical anchor” and quantifying the learned correction’s contribution, e.g. by comparing against the mean-only wall model at Re_theta = 5000 or by showing sensitivity to replacing Eq. (2.11) with a different correlation. This does not undermine the joint-training conclusion, but it changes what Fig. 5 demonstrates.
[§3, Figs. 2, 5, 8, 10] No uncertainty estimates are reported. All statistics are based on single rollouts (final 20 flow-through times), and no ensemble of initial conditions or training runs is shown. Given the chaotic dynamics and the stochastic training procedure, statements such as “agrees closely” and “extrapolates without appreciable degradation” need run-to-run variability or at least a sensitivity study to the averaging window. Without error bars, it is difficult to judge whether Hybrid-Joint’s advantage over Baseline: Vreman in Figs. 2, 8, and 20 is significant relative to natural fluctuations.
[§3, Case 8 and Appendix C, Fig. 20] The Re_theta = 5000 reference is described in Fig. 20 as “wall-resolved-quality fine-mesh WMLES,” not DNS. This is a model-dependent target, so the claim that Hybrid-Joint “agrees closely” and the quantitative extrapolation assessment are conditional on the fidelity of that reference. Please state explicitly how this reference is validated and, where possible, compare against DNS at overlapping lower Reynolds numbers. The Case 4 reference-domain mismatch is acknowledged in §3.3, but the same caution should be stated for Case 8.

minor comments (5)

[§2.2.3, Eq. (2.12)] The averaging window T is defined as 10 delta_99 / U_infinity, but delta_99 is not defined before this equation, and the integral upper limit is written as infinity rather than delta_99. Please clarify notation.
[§3.2, first paragraph] The sentence “This agreement is a genuine test of the model” should be softened or qualified in view of Eq. (2.11), which fixes the mean wall stress and hence the inner scaling. See major comment 1.
[§3.2, Fig. 5(b)] Because Cf is prescribed by Eq. (2.11), the comparison with Fernholz & Finley data and Smits/Nagib correlations is not independent. Consider plotting the learned correction (Cf_model − Cf_Smits)/Cf_Smits to show what the network actually adds.
[§4.4, Table 3] The CPU-to-GPU cost comparison is labeled indicative, which is appropriate. However, the claim of a “540× reduction” combines solver efficiency, mesh coarsening, and amortization; this should be broken down more clearly to avoid overstatement.
[General] No code or data availability statement is included. Given the complexity of the framework, releasing the differentiable solver wrapper and trained weights would materially help reproducibility.

Circularity Check

3 steps flagged · score 6.0 of 10

Extrapolation and skin-friction evidence are largely predetermined by the empirical Smits law and the u_tau anchor; the joint-training conclusion itself is independently supported.

self definitional [§2.2.3, Eqs. (2.11)–(2.12); presented as extrapolation evidence in §3.2, Fig. 5(b)]
"Cf = ⟨τ_w,1⟩/(1/2 ρ U∞^2) = 0.024 Re_theta^{-1/4}, ⟨τ_w,3⟩=0 ... The resulting Re_theta is substituted into equation (2.11) to give the reference mean stress."

The mean wall shear stress used by the neural wall closure is prescribed at every time step by the Smits et al. empirical skin-friction law, with Re_theta evaluated online from the running-averaged profile. Therefore the simulated Cf is not an independent output of the learned closures: it is an input of the closure. The paper's claim that the predicted Cf follows the expected Re_theta scaling across 600–6500 is thus largely a restatement of Eq. (2.11), not a test of learned Reynolds-number generalization. The paper itself states this: the scaling law 'provides the correct Re-scaling, which the network need not relearn.'
fitted input called prediction [§2.3, Eq. (2.16); invoked in §3.2 as evidence of accurate inner scaling and extrapolation]
"L(θ) = w1‖⟨u+1⟩−⟨u+1,GT⟩‖^2 + ... + w4‖uτ − uτ,GT‖ ... the final term anchors the absolute magnitude of uτ, which inner scaling alone leaves undetermined."

The friction velocity u_tau is directly included in the training loss, and u_tau is derived from the wall stress produced by the wall closure. Hence the accurate inner-scaled mean profile and the u_tau values at the training Reynolds numbers are fitted quantities, not predictions. At extrapolated Reynolds numbers, the same u_tau is dominated by the prescribed Smits-law mean (Eq. 2.11), so the Reynolds-number trend of the inner scaling is inherited from the empirical law rather than learned. Reporting this agreement as evidence of extrapolation is therefore circular with respect to the training objective.

1 more flagged steps

other [§4.1.2, Fig. 16]
"The mean-only model gives a reasonable, slightly overpredicted mean velocity profile, since the imposed mean wall stress still provides the correct inner scaling."

This ablation explicitly shows that the reasonable mean velocity profile — the basis for the log-region and inner-scaling claims — is obtained even when the learned wall-stress fluctuations are removed, because the mean wall stress is imposed by the scaling law. It confirms that the mean-profile agreement is not produced by the learned correction but by the empirical input. The learned fluctuations are needed for the Reynolds shear stress, which is genuine, but the mean-profile and Cf-based extrapolation evidence is tied to the prescribed input.

full rationale

The paper's central methodological claim — that the SGS closure, wall closure, and discretization are coupled and must be trained jointly — is independently supported by the ablation experiments (Hybrid-WallOnly and Hybrid-SGSOnly fail in complementary ways), and several reported results are genuinely emergent: the SGS coefficient near c≈0.07, the resolved spectra, the wall-stress fluctuation PDFs, and the need for fluctuations to recover the Reynolds shear stress. However, the Reynolds-number extrapolation claim is substantially prebuilt. The mean wall stress is set by the Smits et al. skin-friction law (Eq. 2.11) with online Re_theta (Eq. 2.12), and the loss contains an explicit u_tau anchor (Eq. 2.16). Consequently, the Cf–Re_theta agreement, the inner-scaled mean profile, and the log-region behavior at extrapolated Reynolds numbers are not independent tests of a learned Reynolds-number-scaling law; they largely reflect the empirical input plus a small trained correction. The paper openly acknowledges that the scaling law 'provides the correct Re-scaling, which the network need not relearn', and the mean-only ablation corroborates that the mean profile is reasonable even without the learned fluctuation. This is partial circularity in the generalization evidence, not in the joint-optimization conclusion, so a score of 6 is appropriate rather than higher.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central new content is the joint training procedure and the network corrections; the physical structure (Vreman form, Cf scaling law) is imported from prior literature and is part of why the 'predicted' friction scaling is not fully emergent. No new physical entities are postulated.

free parameters (6)

Neural SGS U-Net weights (θ_sgs) = trained (architecture in Table 4)
3-D U-Net maps resolved fields to Vreman coefficient; all parameters optimized against the statistics loss; this is the learned part of the SGS closure.
Neural wall U-Net weights (θ_wall) = trained (architecture in Table 4)
2-D U-Net produces wall-stress correction and fluctuations; joint optimization of this network with the SGS network is the central claim.
Smits et al. skin-friction law constants (0.024, -1/4) = 0.024 Re_theta^-1/4
Imposed mean wall stress via Eq. (2.11); empirical constants adopted from prior literature make the reported Cf-Re_theta agreement partly by construction.
Training Reynolds number set = {600, 1200}
Cases chosen for training; all extrapolation and interpolation claims are defined relative to this set.
Momentum-thickness averaging window T = 10 δ99/U∞
Eq. (2.12) uses this ad hoc running-average window to estimate Re_theta online, which in turn sets the imposed mean wall stress.
Loss weights w_k = 1/|L_k| = recomputed per epoch, detached
Inverse-magnitude weighting balances the four loss terms; procedural but affects the training trajectory and final closure.

assumptions (7)

domain assumption Filtered incompressible Navier-Stokes equations (2.1) govern the resolved field with implicit filtering by the mesh
Basis for both closures; no explicit filter is applied, so commutation errors are neglected.
domain assumption Eddy-viscosity form for SGS stress, τ_ij = -2ν_t S_ij
Eq. (2.5); inherited from conventional closures and fixes the sign of SGS energy transfer to non-negative; paper notes it excludes backscatter.
domain assumption Vreman functional (2.7)-(2.8) as fixed output layer remains a suitable tensor structure for the learned coefficient
Chosen for realizability and near-wall vanishing; an architectural assumption, not proved for all learned coefficients.
domain assumption Empirical skin-friction law of Smits et al. (2.11) gives accurate mean wall shear stress
Mean wall closure relies on this correlation; the paper's Cf predictions inherit its accuracy.
domain assumption Local resolved fields at the third off-wall cell contain enough information for the wall network to predict stress fluctuations
Eq. (2.13); a modeling assumption on the receptive field of the wall closure.
domain assumption Statistics accumulated over the final 20 flow-through times are converged without error bars
All comparisons use single-rollout statistics; no confidence intervals or ensemble averaging are reported.
domain assumption Recycling-rescaling inflow (Lund et al. 1998) provides sufficient turbulence for autonomous transition and development
Inference starts from Blasius plus random perturbations and relies on inlet recycling; no sensitivity analysis is given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures." pith.science (2026). https://pith.science/paper/DRTRBBPE

@misc{pith2026260717357,
  author       = {Pith},
  title        = {Pith review of: Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRTRBBPE}},
  note         = {Machine review of arXiv:2607.17357}
}

read the original abstract

Wall-modelled large-eddy simulation (WMLES) treats the subgrid-scale (SGS) closure, wall closure and numerical discretization as independent components, although their effects are coupled through the same resolved field. We present a differentiable hybrid neural--CFD framework in which the SGS and wall closures are learned jointly, end-to-end, within a differentiable flow solver, using only low-order statistics as training targets. Each closure is a composed neural operator: a trainable neural network followed by a fixed differentiable layer that preserves the structure of its conventional counterpart, so that the network learns only the functions left undetermined by the conventional form. Because every operation is differentiable, gradients of the training loss are back-propagated through the coupled solver, allowing both neural closures to be optimized consistently against the flow field, rather than fitted offline or in isolation. We demonstrate the framework, denoted Hybrid-Joint, on a zero-pressure-gradient turbulent boundary layer across a posteriori tests spanning Re_\theta = 600--6500, computational domains and mesh resolutions. The model outperforms WMLES baselines, extrapolates to more than four times the highest training Reynolds number, and transfers to grids and domains absent from training. It recovers a logarithmic mean-velocity region, not imposed by the wall closure, and reproduces the resolved energy spectra accurately, although spectral information is excluded from the training objective. Ablation studies show that learning either closure alone is insufficient and that only joint optimization recovers the full set of statistics, confirming that SGS closure, wall closure and discretization are coupled and must be trained jointly. Once trained, the closures are reused without retraining across all cases, so that training cost is amortized over repeated deployment.

Figures

Figures reproduced from arXiv: 2607.17357 by the authors.

**Figure 2.** Turbulence statistics for Case 1 (𝑅𝑒 𝜃 = 600), compared with the two baselines and the ground truth (DNS). (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square (RMS) velocity fluctuations 𝑢 + 𝑖,rms. (e) Reynolds shear stress −⟨𝑢 ′ 1 𝑢 ′ 2 ⟩ + . All statistics are normalized by the friction velocity 𝑢𝜏 . lead to an underprediction of the friction velocity 𝑢𝜏 by approximately 12% and hence to the observed lo… view at source ↗

**Figure 3.** One-dimensional wavenumber energy spectra of the velocity components for Case 1 (𝑅𝑒 𝜃 = 600) at 𝑦 + = 138. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off wavenumber associated with the same mesh. Baseline: Smagorinsky (c) Ground truth (a) (b) Hybrid-Joint (d) Baseline: Vreman [PITH_FULL_IMAGE:figures/ful… view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: Comparison of the instantaneous velocity field predicted by different models for Case 1 (𝑅𝑒 𝜃 = 600). (a) Baseline: Smagorinsky (b) Baseline: Vreman (c) Hybrid-Joint (d) Ground truth, obtained by filtering the DNS solution and downsampling it onto the same coarse mesh.…

**Figure 5.** Figure 5: (a) shows the mean velocity profiles over the range. A clear logarithmic region develops at every Reynolds number, in agreement with the law of the wall, with a slight departure only at the first off-wall point. This agreement is a genuine test of the model: unlike con…

**Figure 6.** Figure 6: One-dimensional wavenumber energy spectra of the velocity components for Case 2 (𝑅𝑒 𝜃 = 800) at 𝑦 + = 170. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off wa…

**Figure 7.** Figure 7: One-dimensional wavenumber energy spectra of the velocity components for Case 8 (𝑅𝑒 𝜃 = 5000) at 𝑦 + = 600. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off w…

**Figure 8.** Figure 8: Turbulence statistics at 𝑅𝑒 𝜃 = 1200 for Case 3 (uniform grid, 100𝜃 domain) and Case 4 (wallnormal stretched grid, 400𝜃 domain), compared with the two baselines and the ground truth (DNS). (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square velocity fluctua…

**Figure 9.** Figure 9: Streamwise velocity for Case 9 (𝑅𝑒 𝜃,in = 5000). (a) Mean streamwise velocity contour, boundarylayer thickness, and velocity profiles at selected streamwise locations. For clarity, the velocity profiles are scaled by a factor of 50. (b) Instantaneous streamwise veloci…

**Figure 10.** Figure 10: Streamwise development of wall-friction quantities for Case 9 (𝑅𝑒 𝜃,in = 5000). (a) Skin-friction coefficient along the streamwise direction. (b) Predicted friction Reynolds number 𝑅𝑒𝜏 as a function of the Reynolds number based on momentum thickness, 𝑅𝑒 𝜃 . only the j…

**Figure 11.** Figure 11: Prediction of the neural SGS and wall models. (a) Mean subgrid viscosity 𝜈𝑡 over wall-normal distance, combining the interior SGS contribution and the wall contribution ⟨𝜈 wall 𝑡 ⟩ (first point). (b) Time history of the wall-parallel-averaged Vreman coefficient 𝑐 [PI…

**Figure 12.** Figure 12: Instantaneous subgrid viscosity 𝜈𝑡 predicted by the neural SGS model. (a) Case 1, 𝑅𝑒 𝜃 = 600. (b) Case 2, 𝑅𝑒 𝜃 = 800. (c) Case 3, 𝑅𝑒 𝜃 = 1200. (d) Case 5, 𝑅𝑒 𝜃 = 2400. Bae 2025) and used as the fixed coefficient in Baseline: Vreman. This is an emergent outcome of trai…

**Figure 13.** Figure 13 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

**Figure 14.** Figure 14: Instantaneous streamwise wall-shear stress 𝜏𝑤,1 predicted by different models for Case 1 (𝑅𝑒 𝜃 = 600). (a) Hybrid-Joint model. (b) Ground truth obtained by filtering the DNS solution and downsampling it to the same coarse mesh. (c) Baseline: Smagorinsky. (d) Baseline:…

**Figure 15.** Figure 15: PDF of the instantaneous streamwise wall-shear stress 𝜏𝑤,1 across all Reynolds numbers tested. (a) Hybrid-Joint. (b) Two WMLES baselines, Smagorinsky (solid) and Vreman (dashed), which are nearly indistinguishable at every 𝑅𝑒 𝜃 . To isolate the contribution of the flu…

**Figure 16.** Figure 16: Effect of the wall-stress fluctuation, evaluated at 𝑅𝑒 𝜃 = 5000 (Case 8). Hybrid-Joint (Mean only) retains the scaling-law mean wall stress but removes the network-predicted stress fluctuation. (a) Mean velocity profile. (b) Reynolds shear stress 𝜏12. the mean wall st…

**Figure 17.** Figure 17: Turbulence statistics for case 8 (𝑅𝑒 𝜃 = 5000) under different ablation settings. (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square velocity fluctuations 𝑢 + 𝑖,rms. (panel a), and the outer-layer turbulent kinetic energy is strongly underestimated (panels…

**Figure 18.** Figure 18: shows the training histories for several choices of 𝑁1 and 𝑁2. Increasing the tracking window to 𝑁2 = 150 or 200 leads to unstable training in both cases: the gradient norm eventually grows by several orders of magnitude and the loss plateaus at an elevated level inst…

**Figure 19.** Figure 19 [PITH_FULL_IMAGE:figures/full_fig_p034_19.png]

**Figure 20.** Figure 20 [PITH_FULL_IMAGE:figures/full_fig_p035_20.png]

**Figure 21.** Figure 21: Instantaneous streamwise velocity contour for Case 9 from two baselines (𝑅𝑒 𝜃,in = 5000): (a) Baseline: Smagorinsky. (b) Baseline: Vreman. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p036_21.png]

**Figure 22.** Figure 22: Energy transfer between the resolved and subgrid scales for a priori test at 𝑅𝑒 𝜃 = 600. (a) Mean SGS energy-transfer rate ⟨𝛱⟩ in the wall-normal direction. (b) Probability density function (PDF) of the instantaneous energy-transfer rate 𝛱 at 𝑥2/𝛿99 = 0.2. 0 X0-36 [P…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

259 extracted references · 1 canonical work pages

[1]

Physics of Fluids 27 (3)

Abkar, Mahdi & Port \'e -Agel, Fernando 2015 Influence of atmospheric stability on wind-turbine wakes: A large-eddy simulation study . Physics of Fluids 27 (3)

2015
[2]

Theoretical and Applied Mechanics Letters p

Akhare, Deepak , Luo, Tengfei & Wang, Jian-Xun 2025 HybridNDiff-UQ : Uncertainty quantification for hybrid neural differentiable modeling . Theoretical and Applied Mechanics Letters p. 100609

2025
[3]

Fluids 6 (3), 112

Bae, H Jane & Lozano-Dur \'a n, Adri \'a n 2021 Effect of wall boundary conditions on a wall-modeled large-eddy simulation in a finite-difference framework . Fluids 6 (3), 112

2021
[4]

In 13th Fluid and Plasmadynamics Conference\/ , p

Bardina, Jorge , Ferziger, J & Reynolds, WC 1980 Improved subgrid-scale models for large-eddy simulation. In 13th Fluid and Plasmadynamics Conference\/ , p. 1357

1980
[5]

Computer Physics Communications 282 , 108527

Bezgin, Deniz A , Buhendwa, Aaron B & Adams, Nikolaus A 2023 Jax-fluids: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows . Computer Physics Communications 282 , 108527

2023
[6]

Annual Review of Fluid Mechanics 50 , 535--561

Bose, Sanjeeb T & Park, George Ilhwan 2018 Wall-modeled large-eddy simulation for complex turbulent flows . Annual Review of Fluid Mechanics 50 , 535--561

2018
[7]

2021 Integral measures of the zero pressure gradient boundary layer over the R eynolds number range 0 R_ <

Cantwell, Brian J. 2021 Integral measures of the zero pressure gradient boundary layer over the R eynolds number range 0 R_ < . Physics of Fluids 33 (8), 085108

2021
[8]

AIAA Journal 17 (12), 1293--1313

Chapman, Dean R 1979 Computational aerodynamics development and outlook . AIAA Journal 17 (12), 1293--1313

1979

Show all 259 references

[9]

Journal of Fluid Mechanics 1024 , A28

Chedevergne, Fran c ois 2025 Characterisation and modelling of interscale energy transfers in high R eynolds number boundary layers . Journal of Fluid Mechanics 1024 , A28

2025
[10]

Journal of Fluid Mechanics 91 (1), 1--16

Clark, Robert A , Ferziger, Joel H & Reynolds, William Craig 1979 Evaluation of subgrid-scale models using an accurately simulated turbulent flow . Journal of Fluid Mechanics 91 (1), 1--16

1979
[11]

arXiv preprint arXiv:2603.15920

Du, Pan , Li, Yongqi , Xu, Mingqi & Wang, Jian-Xun 2026 DiFVM : A vectorized graph-based finite volume solver for differentiable cfd on unstructured meshes . arXiv preprint arXiv:2603.15920

2026
[12]

Annual Review of Fluid Mechanics 51 (1), 357--377

Duraisamy, Karthik , Iaccarino, Gianluca & Xiao, Heng 2019 Turbulence modeling in the age of data . Annual Review of Fluid Mechanics 51 (1), 357--377

2019
[13]

Physics of Fluids 37 (9)

Fan, Guoqing , Zhao, Weiwen & Wan, Decheng 2025 a\/ Near-wall turbulent fluctuations and coherent structures in wall-modeled large-eddy simulation . Physics of Fluids 37 (9)

2025
[14]

Computer Methods in Applied Mechanics and Engineering 433 , 117478

Fan, Xiantao , Akhare, Deepak & Wang, Jian-Xun 2025 b\/ Neural differentiable modeling with diffusion-based super-resolution for two-dimensional spatiotemporal turbulence . Computer Methods in Applied Mechanics and Engineering 433 , 117478

2025
[15]

Computer Methods in Applied Mechanics and Engineering 448 , 118455

Fan, Xiantao , Liu, Xin-Yang , Wang, Meng & Wang, Jian-Xun 2026 Diff-FlowFSI : A GPU -optimized differentiable CFD platform for high-fidelity turbulence and FSI simulations . Computer Methods in Applied Mechanics and Engineering 448 , 118455

2026
[16]

Journal of Computational Physics 496 , 112584

Fan, Xiantao & Wang, Jian-Xun 2024 Differentiable hybrid neural modeling for fluid-structure interaction . Journal of Computational Physics 496 , 112584

2024
[17]

Progress in Aerospace Sciences 32 (4), 245--311

Fernholz, Hans-Hermann & Finley, PJ 1996 The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data . Progress in Aerospace Sciences 32 (4), 245--311

1996
[18]

Journal of Fluid Mechanics 934 , A44

Fowler, Mitchell , Zaki, Tamer A & Meneveau, Charles 2022 A lagrangian relaxation towards equilibrium wall model for large eddy simulation . Journal of Fluid Mechanics 934 , A44

2022
[19]

Journal of Computational Physics p

Franz, Aleksandra , Wei, Hao , Guastoni, Luca & Thuerey, Nils 2025 PICT --a differentiable, GPU -accelerated multi-block PISO solver for simulation-coupled learning tasks in fluid dynamics . Journal of Computational Physics p. 114433

2025
[20]

Physical Review Fluids 4 (6), 064603

Fukami, Kai , Nabae, Yusuke , Kawai, Ken & Fukagata, Koji 2019 Synthetic turbulent inflow generator using machine learning . Physical Review Fluids 4 (6), 064603

2019
[21]

Physics of Fluids A: Fluid dynamics 3 (7), 1760--1765

Germano, Massimo , Piomelli, Ugo , Moin, Parviz & Cabot, William H 1991 A dynamic subgrid-scale eddy viscosity model . Physics of Fluids A: Fluid dynamics 3 (7), 1760--1765

1991
[22]

Journal of Computational Physics 458 , 111090

Guan, Yifei , Chattopadhyay, Ashesh , Subel, Adam & Hassanzadeh, Pedram 2022 Stable a posteriori LES of 2D turbulence using convolutional neural networks: Backscattering analysis and generalization to higher R e via transfer learning . Journal of Computational Physics 458 , 111090

2022
[23]

Journal of Fluid Mechanics 928 , A27

Guastoni, Luca , G \"u emes, Alejandro , Ianiro, Andrea , Discetti, Stefano , Schlatter, Philipp , Azizpour, Hossein & Vinuesa, Ricardo 2021 Convolutional-network models to predict wall-bounded turbulence from wall quantities . Journal of Fluid Mechanics 928 , A27

2021
[24]

Physical Review Fluids 11 (1), 014602

Huang, Xinyi , Leung, Sze Chai & Bae, H Jane 2026 Consistency requirement of data-driven subgrid-scale modeling in large-eddy simulation . Physical Review Fluids 11 (1), 014602

2026
[25]

Physics of Fluids 24 (1)

Kawai, Soshi & Larsson, Johan 2012 Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy . Physics of Fluids 24 (1)

2012
[26]

Kim, Hojin , Shankar, Varun , Viswanathan, Venkatasubramanian & Maulik, Romit 2025 Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics, arXiv:arXiv: 2307.13533

2025
[27]

Proceedings of the National Academy of Sciences 118 (21), e2101784118

Kochkov, Dmitrii , Smith, Jamie A , Alieva, Ayya , Wang, Qing , Brenner, Michael P & Hoyer, Stephan 2021 Machine learning--accelerated computational fluid dynamics . Proceedings of the National Academy of Sciences 118 (21), e2101784118

2021
[28]

Neural Networks p

Kohl, Georg , Chen, Li-Wei & Thuerey, Nils 2026 Benchmarking autoregressive conditional diffusion models for turbulent flow simulation . Neural Networks p. 108641

2026
[29]

Mechanical Engineering Reviews 3 (1), 15--00418

Larsson, Johan , Kawai, Soshi , Bodart, Julien & Bermejo-Moreno, Ivan 2016 Large eddy simulation with modeled wall-stress: recent progress and future directions . Mechanical Engineering Reviews 3 (1), 15--00418

2016
[30]

Physics of Fluids 35 (7)

Li, Zhijie , Peng, Wenhui , Yuan, Zelong & Wang, Jianchun 2023 Long-term predictions of turbulence by implicit U - N et enhanced f o urier neural operator . Physics of Fluids 35 (7)

2023
[31]

NCAR Report

Lilly, D 1966 On the application of the eddy viscosity concept in the inertial sub-range of turbulence . NCAR Report

1966
[32]

In AIAA SCITECH 2025 Forum\/ , p

Ling, Yuenong & Lozano-Duran, Adrian 2025 Numerically consistent data-driven subgrid-scale model via data assimilation and machine learning. In AIAA SCITECH 2025 Forum\/ , p. 1280

2025
[33]

Computer Methods in Applied Mechanics and Engineering 433 , 117441

List, Bjoern , Chen, Li-Wei , Bali, Kartik & Thuerey, Nils 2025 Differentiability in unrolled training of neural physics simulators on transient dynamics . Computer Methods in Applied Mechanics and Engineering 433 , 117441

2025
[34]

Journal of Fluid Mechanics 949 , A25

List, Bj \"o rn , Chen, Li-Wei & Thuerey, Nils 2022 Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons . Journal of Fluid Mechanics 949 , A25

2022
[35]

Journal of computational physics 140 (2), 233--258

Lund, Thomas S , Wu, Xiaohua & Squires, Kyle D 1998 Generation of turbulent inflow data for spatially-developing boundary layer simulations . Journal of computational physics 140 (2), 233--258

1998
[36]

Physical Review Fluids 6 (5), 050502

MacArt, Jonathan F , Sirignano, Justin & Freund, Jonathan B 2021 Embedded training of neural-network subgrid-scale turbulence models . Physical Review Fluids 6 (5), 050502

2021
[37]

Journal of Fluid Mechanics 858 , 122--144

Maulik, Romit , San, Omer , Rasheed, Adil & Vedula, Prakash 2019 Subgrid modelling for two-dimensional turbulence using neural networks . Journal of Fluid Mechanics 858 , 122--144

2019
[38]

Scientific Data 8 (1), 255

McConkey, Ryley , Yee, Eugene & Lien, Fue-Sang 2021 A curated dataset for data-driven turbulence modelling . Scientific Data 8 (1), 255

2021
[39]

Annual Review of Fluid Mechanics 32 (1), 1--32

Meneveau, Charles & Katz, Joseph 2000 Scale-invariance and turbulence models for large-eddy simulation . Annual Review of Fluid Mechanics 32 (1), 1--32

2000
[40]

Physical Review Fluids 6 (10), 104607

Mons, Vincent , Du, Yifan & Zaki, Tamer A 2021 Ensemble-variational assimilation of statistical data in large-eddy simulation . Physical Review Fluids 6 (10), 104607

2021
[41]

Journal of Fluid Mechanics 1003 , A2

Nabae, Yusuke , Inagaki, Kazuhiro , Kobayashi, Hiromichi , Gotoda, Hiroshi & Fukagata, Koji 2025 Large-eddy simulation of high- R eynolds-number turbulent channel flow controlled using streamwise travelling wave-like wall deformation for drag reduction . Journal of Fluid Mecha...

2025
[43]

PhD thesis, Stanford University

Piomelli, Ugo 1988 Models for large eddy simulations of turbulent channel flows including transpiration . PhD thesis, Stanford University

1988
[44]

Progress in Aerospace Sciences 44 (6), 437--446

Piomelli, Ugo 2008 Wall-layer models for large-eddy simulations . Progress in Aerospace Sciences 44 (6), 437--446

2008
[45]

Annual Review of Fluid Mechanics 34 (1), 349--374

Piomelli, Ugo & Balaras, Elias 2002 Wall-layer models for large-eddy simulations . Annual Review of Fluid Mechanics 34 (1), 349--374

2002
[46]

International Journal of Heat and Fluid Flow 24 (4), 538--550

Piomelli, Ugo , Balaras, Elias , Pasinato, Hugo , Squires, Kyle D & Spalart, Philippe R 2003 The inner--outer layer interface in large-eddy simulations with wall-layer models . International Journal of Heat and Fluid Flow 24 (4), 538--550

2003
[47]

Physics of Fluids A: Fluid Dynamics 3 (7), 1766--1771

Piomelli, Ugo , Cabot, William H , Moin, Parviz & Lee, Sangsan 1991 Subgrid-scale backscatter in turbulent and transitional flows . Physics of Fluids A: Fluid Dynamics 3 (7), 1766--1771

1991
[48]

Physics of Fluids 8 (1), 215--224

Piomelli, Ugo , Yu, Yunfang & Adrian, Ronald J 1996 Subgrid-scale energy transfer and near-wall turbulence structure . Physics of Fluids 8 (1), 215--224

1996
[49]

Journal of Fluid Mechanics 415 , 261--284

Port \'e -Agel, Fernando , Meneveau, Charles & Parlange, Marc B 2000 A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer . Journal of Fluid Mechanics 415 , 261--284

2000
[50]

Computers & Fluids 185 , 34--58

Rezaeiravesh, Saleh , Mukha, Timofey & Liefvendahl, Mattias 2019 Systematic study of accuracy of wall-modeled large eddy simulation using uncertainty quantification techniques . Computers & Fluids 185 , 34--58

2019
[51]

Journal of Fluid Mechanics 659 , 116--126

Schlatter, Philipp & \"O rl \"u , Ramis 2010 Assessment of direct numerical simulation data of turbulent boundary layers . Journal of Fluid Mechanics 659 , 116--126

2010
[52]

Physical Review Fluids 10 (2), 024605

Shankar, Varun , Chakraborty, Dibyajyoti , Viswanathan, Venkatasubramanian & Maulik, Romit 2025 Differentiable turbulence: Closure as a partial differential equation constrained optimization . Physical Review Fluids 10 (2), 024605

2025
[53]

Journal of Fluid Mechanics 966 , A26

Sirignano, Justin & MacArt, Jonathan F 2023 Deep learning closure models for large-eddy simulation of flows around bluff bodies . Journal of Fluid Mechanics 966 , A26

2023
[54]

the basic experiment

Smagorinsky, Joseph 1963 General circulation experiments with the primitive equations: I. the basic experiment . Monthly Weather Review 91 (3), 99--164

1963
[55]

Journal of Ship Research 27 (03), 147--157

Smits, AJ , Matheson, N & Joubert, PN 1983 Low- R eynolds-number turbulent boundary layers in zero and favorable pressure gradients . Journal of Ship Research 27 (03), 147--157

1983
[56]

International Journal of Heat and Fluid Flow 26 (2), 173--190

Temmerman, L , Had z iabdi \'c , M , Leschziner, MA & Hanjali \'c , K 2005 A hybrid two-layer URANS -- LES approach for large eddy simulation at high R eynolds numbers . International Journal of Heat and Fluid Flow 26 (2), 173--190

2005
[57]

International Journal of Heat and Fluid Flow 27 (5), 789--799

Tessicini, F , Temmerman, L & Leschziner, MA 2006 Approximate near-wall treatments based on zonal and hybrid RANS -- LES methods for LES at high R eynolds numbers . International Journal of Heat and Fluid Flow 27 (5), 789--799

2006
[58]

Journal of the Aeronautical Sciences 23 (11), 1007--1011

Van Driest, Edward R 1956 On turbulent flow near a wall . Journal of the Aeronautical Sciences 23 (11), 1007--1011

1956
[59]

Physics of Fluids 16 (10), 3670--3681

Vreman, AW 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications . Physics of Fluids 16 (10), 3670--3681

2004
[60]

Physical Review Fluids 2 (3), 034603

Wang, Jian-Xun , Wu, Jin-Long & Xiao, Heng 2017 Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on DNS data . Physical Review Fluids 2 (3), 034603

2017
[61]

Journal of Computational Physics 267 , 210--224

Wang, Qiqi , Hu, Rui & Blonigan, Patrick 2014 Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations . Journal of Computational Physics 267 , 210--224

2014
[62]

Wang, Qi , Wang, Mengze & Zaki, Tamer A 2022 What is observable from wall data in turbulent channel flow? Journal of Fluid Mechanics 941 , A48

2022
[63]

Journal of Fluid Mechanics 906 , A14

Wang, Wenkang , Pan, Chong & Wang, Jinjun 2021 Energy transfer structures associated with large-scale motions in a turbulent boundary layer . Journal of Fluid Mechanics 906 , A14

2021
[64]

& Font, B

Weymouth, G.D. & Font, B. 2025 Water L ily.jl: A differentiable and backend-agnostic J ulia solver for incompressible viscous flow around dynamic bodies . Computer Physics Communications 315 , 109748

2025
[66]

Journal of Fluid Mechanics 869 , 553--586

Wu, Jinlong , Xiao, Heng , Sun, Rui & Wang, Qiqi 2019 Reynolds-averaged N avier-- S tokes equations with explicit data-driven reynolds stress closure can be ill-conditioned . Journal of Fluid Mechanics 869 , 553--586

2019
[67]

Physical Review Fluids 4 (3), 034602

Yang, XIA , Zafar, S , Wang, J-X & Xiao, H 2019 Predictive large-eddy-simulation wall modeling via physics-informed neural networks . Physical Review Fluids 4 (3), 034602

2019
[68]

Physics of Fluids 33 (1)

Yang, Xiang IA & Griffin, Kevin P 2021 Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation . Physics of Fluids 33 (1)

2021
[69]

Physical Review Fluids 2 (10), 104601

Yang, Xiang IA , Park, George Ilhwan & Moin, Parviz 2017 Log-layer mismatch and modeling of the fluctuating wall stress in wall-modeled large-eddy simulations . Physical Review Fluids 2 (10), 104601

2017
[70]

Journal of Computational Physics p

Zhang, Fengshun , Yang, Xiaolei & He, Guowei 2026 A differentiable wall-modeled large-eddy simulation method for high- R eynolds-number wall-bounded turbulent flows . Journal of Computational Physics p. 114835

2026
[71]

Journal of Fluid Mechanics 949 , A26

Zhang, Xin-Lei , Xiao, Heng , Luo, Xiaodong & He, Guowei 2022 Ensemble K alman method for learning turbulence models from indirect observation data . Journal of Fluid Mechanics 949 , A26

2022
[72]

Journal of Computational Physics 506 , 112948

Zhou, Di & Bae, H Jane 2024 Sensitivity analysis of wall-modeled large-eddy simulation for separated turbulent flow . Journal of Computational Physics 506 , 112948

2024
[74]

Physical Review Fluids , volume=

Differentiable turbulence: Closure as a partial differential equation constrained optimization , author=. Physical Review Fluids , volume=. 2025 , publisher=

2025
[75]

Physical Review Fluids , volume=

Synthetic turbulent inflow generator using machine learning , author=. Physical Review Fluids , volume=. 2019 , publisher=

2019
[76]

Neural Networks , pages=

Benchmarking autoregressive conditional diffusion models for turbulent flow simulation , author=. Neural Networks , pages=. 2026 , publisher=

2026
[77]

Physics of Fluids , volume=

An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications , author=. Physics of Fluids , volume=. 2004 , publisher=

2004
[78]

Physics of Fluids , volume=

A dynamic subgrid-scale eddy viscosity model with a global model coefficient , author=. Physics of Fluids , volume=. 2006 , publisher=

2006
[79]

Physics of Fluids , volume=

A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries , author=. Physics of Fluids , volume=. 2007 , publisher=

2007
[80]

The basic experiment , author=

General circulation experiments with the primitive equations: I. The basic experiment , author=. Monthly Weather Review , volume=
[81]

Physics of Fluids A: Fluid dynamics , volume=

A dynamic subgrid-scale eddy viscosity model , author=. Physics of Fluids A: Fluid dynamics , volume=. 1991 , publisher=

1991
[82]

Meneveau, Charles and Lund, Thomas S and Cabot, William H , journal=. A. 1996 , publisher=

1996
[83]

Fluids , volume=

Effect of wall boundary conditions on a wall-modeled large-eddy simulation in a finite-difference framework , author=. Fluids , volume=. 2021 , publisher=

2021
[84]

Annual Review of Fluid Mechanics , volume=

Wall-modeled large-eddy simulation for complex turbulent flows , author=. Annual Review of Fluid Mechanics , volume=. 2018 , publisher=

2018
[85]

Physical Review Fluids , volume=

Log-layer mismatch and modeling of the fluctuating wall stress in wall-modeled large-eddy simulations , author=. Physical Review Fluids , volume=. 2017 , publisher=

2017
[86]

Smits, AJ and Matheson, N and Joubert, PN , journal=. Low-. 1983 , publisher=

1983
[87]

1950 , publisher=

The boundary layers in fluids with little friction , author=. 1950 , publisher=

1950
[88]

Journal of Fluid Mechanics , volume=

The law of the wake in the turbulent boundary layer , author=. Journal of Fluid Mechanics , volume=. 1956 , publisher=

1956
[89]

Progress in Aerospace Sciences , volume=

The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data , author=. Progress in Aerospace Sciences , volume=. 1996 , publisher=

1996
[90]

Large eddy simulations of turbulent channel and boundary layer flows at high

Lee, Jungil and Cho, Minjeong and Choi, Haecheon , journal=. Large eddy simulations of turbulent channel and boundary layer flows at high. 2013 , publisher=

2013
[91]

Journal of Fluid Mechanics , volume=

Universal law of skin-friction coefficient in a fully developed zero pressure gradient axisymmetric turbulent boundary layer flow , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

2023
[92]

arXiv preprint arXiv:2604.09947 , year=

Unified scaling laws for turbulent boundary layers across flow regimes , author=. arXiv preprint arXiv:2604.09947 , year=

arXiv
[93]

Nature Communications , volume=

Dimensionless learning based on information , author=. Nature Communications , volume=. 2025 , publisher=

2025
[94]

Journal of computational physics , volume=

Generation of turbulent inflow data for spatially-developing boundary layer simulations , author=. Journal of computational physics , volume=. 1998 , publisher=

1998
[95]

2025 , publisher=

Franz, Aleksandra and Wei, Hao and Guastoni, Luca and Thuerey, Nils , journal=. 2025 , publisher=

2025
[96]

Computer Methods in Applied Mechanics and Engineering , volume=

Differentiability in unrolled training of neural physics simulators on transient dynamics , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2025 , publisher=

2025
[97]

A differentiable wall-modeled large-eddy simulation method for high-

Zhang, Fengshun and Yang, Xiaolei and He, Guowei , journal=. A differentiable wall-modeled large-eddy simulation method for high-. 2026 , publisher=

2026
[98]

2025 , eprint=

Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics , author=. 2025 , eprint=

2025
[99]

Physics of Fluids , volume=

Near-wall turbulent fluctuations and coherent structures in wall-modeled large-eddy simulation , author=. Physics of Fluids , volume=. 2025 , publisher=

2025
[100]

Journal of Fluid Mechanics , volume=

A Lagrangian relaxation towards equilibrium wall model for large eddy simulation , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

2022
[101]

Journal of the Aeronautical Sciences , volume=

On turbulent flow near a wall , author=. Journal of the Aeronautical Sciences , volume=
[102]

arXiv preprint arXiv:2312.01184 , year=

Reynolds number required to accurately discriminate between proposed trends of skin friction and normal stress in wall turbulence , author=. arXiv preprint arXiv:2312.01184 , year=

arXiv
[103]

Journal of Fluid Mechanics , volume=

Assessment of direct numerical simulation data of turbulent boundary layers , author=. Journal of Fluid Mechanics , volume=. 2010 , publisher=

2010
[104]

Large-eddy simulation of high-

Nabae, Yusuke and Inagaki, Kazuhiro and Kobayashi, Hiromichi and Gotoda, Hiroshi and Fukagata, Koji , journal=. Large-eddy simulation of high-. 2025 , publisher=

2025
[105]

1988 , school=

Models for large eddy simulations of turbulent channel flows including transpiration , author=. 1988 , school=

1988
[106]

Physics of Fluids A: Fluid Dynamics , volume=

Subgrid-scale backscatter in turbulent and transitional flows , author=. Physics of Fluids A: Fluid Dynamics , volume=. 1991 , publisher=

1991
[107]

Physics of Fluids , volume=

Subgrid-scale energy transfer and near-wall turbulence structure , author=. Physics of Fluids , volume=. 1996 , publisher=

1996
[108]

Characterisation and modelling of interscale energy transfers in high

Chedevergne, Fran. Characterisation and modelling of interscale energy transfers in high. Journal of Fluid Mechanics , volume=. 2025 , publisher=

2025
[109]

Journal of Fluid Mechanics , volume=

Energy transfer structures associated with large-scale motions in a turbulent boundary layer , author=. Journal of Fluid Mechanics , volume=. 2021 , publisher=

2021
[110]

Annual Review of Fluid Mechanics , volume=

Wall-layer models for large-eddy simulations , author=. Annual Review of Fluid Mechanics , volume=. 2002 , publisher=

2002
[111]

Computers & Fluids , volume=

Systematic study of accuracy of wall-modeled large eddy simulation using uncertainty quantification techniques , author=. Computers & Fluids , volume=. 2019 , publisher=

2019
[112]

A hybrid two-layer

Temmerman, L and Had. A hybrid two-layer. International Journal of Heat and Fluid Flow , volume=. 2005 , publisher=

2005
[113]

Approximate near-wall treatments based on zonal and hybrid

Tessicini, F and Temmerman, L and Leschziner, MA , journal=. Approximate near-wall treatments based on zonal and hybrid. 2006 , publisher=

2006
[114]

International Journal of Heat and Fluid Flow , volume=

The inner--outer layer interface in large-eddy simulations with wall-layer models , author=. International Journal of Heat and Fluid Flow , volume=. 2003 , publisher=

2003
[115]

Progress in Aerospace Sciences , volume=

Wall-layer models for large-eddy simulations , author=. Progress in Aerospace Sciences , volume=. 2008 , publisher=

2008
[116]

Physics of Fluids , volume=

Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy , author=. Physics of Fluids , volume=. 2012 , publisher=

2012
[117]

Physics of Fluids , volume=

Influence of atmospheric stability on wind-turbine wakes: A large-eddy simulation study , author=. Physics of Fluids , volume=. 2015 , publisher=

2015
[118]

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=

Approach to an asymptotic state for zero pressure gradient turbulent boundary layers , author=. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=. 2007 , publisher=

2007
[119]

, journal =

Cantwell, Brian J. , journal =. Integral measures of the zero pressure gradient boundary layer over the. 2021 , publisher =

2021
[120]

AIAA Journal , volume=

Computational aerodynamics development and outlook , author=. AIAA Journal , volume=
[121]

Approximate wall boundary conditions in the large-eddy simulation of high

Cabot, William and Moin, Parviz , journal=. Approximate wall boundary conditions in the large-eddy simulation of high. 2000 , publisher=

2000
[122]

Annual Review of Fluid Mechanics , volume=

Detached-eddy simulation , author=. Annual Review of Fluid Mechanics , volume=. 2009 , publisher=

2009
[123]

Journal of Computational Physics , volume=

Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli , author=. Journal of Computational Physics , volume=. 1975 , publisher=

1975
[124]

Journal of Fluid Mechanics , volume=

A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer , author=. Journal of Fluid Mechanics , volume=. 2000 , publisher=

2000
[125]

Mechanical Engineering Reviews , volume=

Large eddy simulation with modeled wall-stress: recent progress and future directions , author=. Mechanical Engineering Reviews , volume=. 2016 , publisher=

2016
[126]

13th Fluid and Plasmadynamics Conference , pages=

Improved subgrid-scale models for large-eddy simulation , author=. 13th Fluid and Plasmadynamics Conference , pages=
[127]

Journal of Fluid Mechanics , volume=

Evaluation of subgrid-scale models using an accurately simulated turbulent flow , author=. Journal of Fluid Mechanics , volume=. 1979 , publisher=

1979
[128]

International Journal of Heat and Fluid Flow , volume=

A mixed one-equation subgrid model for large-eddy simulation , author=. International Journal of Heat and Fluid Flow , volume=. 2002 , publisher=

2002
[129]

Physics of Fluids , volume=

Differential subgrid stress models in large eddy simulations , author=. Physics of Fluids , volume=. 1997 , publisher=

1997
[130]

Wyngaard, JC and Peltier, LJ and Khanna, S , journal=
[131]

Journal of Fluid Mechanics , volume=

Large-eddy simulation and wall modelling of turbulent channel flow , author=. Journal of Fluid Mechanics , volume=. 2009 , publisher=

2009
[132]

Annual Review of Fluid Mechanics , volume=

Scale-invariance and turbulence models for large-eddy simulation , author=. Annual Review of Fluid Mechanics , volume=. 2000 , publisher=

2000
[133]

50 Years of CFD in Engineering Sciences: A Commemorative Volume in Memory of D

Large eddy simulation of flows of engineering interest: A Review , author=. 50 Years of CFD in Engineering Sciences: A Commemorative Volume in Memory of D. Brian Spalding , pages=. 2020 , publisher=

2020
[134]

Journal of Fluid Mechanics , volume=

Convolutional-network models to predict wall-bounded turbulence from wall quantities , author=. Journal of Fluid Mechanics , volume=. 2021 , publisher=

2021
[135]

arXiv preprint arXiv:2010.08895 , year=

Fourier neural operator for parametric partial differential equations , author=. arXiv preprint arXiv:2010.08895 , year=

2010 arXiv
[136]

Long-term predictions of turbulence by implicit

Li, Zhijie and Peng, Wenhui and Yuan, Zelong and Wang, Jianchun , journal=. Long-term predictions of turbulence by implicit. 2023 , publisher=

2023
[137]

Journal of Fluid Mechanics , volume=

Reynolds averaged turbulence modelling using deep neural networks with embedded invariance , author=. Journal of Fluid Mechanics , volume=. 2016 , publisher=

2016
[138]

Weatheritt, Jack and Sandberg, Richard D , journal=. Hybrid. 2017 , publisher=

2017
[139]

Journal of Fluid Mechanics , volume=

Subgrid modelling for two-dimensional turbulence using neural networks , author=. Journal of Fluid Mechanics , volume=. 2019 , publisher=

2019
[140]

Wall model based on neural networks for

Zhou, Zhideng and He, Guowei and Yang, Xiaolei , journal=. Wall model based on neural networks for. 2021 , publisher=

2021
[141]

Journal of Fluid Mechanics , volume=

Machine learning building-block-flow wall model for large-eddy simulation , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

2023
[142]

Reynolds-averaged

Wu, Jinlong and Xiao, Heng and Sun, Rui and Wang, Qiqi , journal=. Reynolds-averaged. 2019 , publisher=

2019
[143]

Stable a posteriori

Guan, Yifei and Chattopadhyay, Ashesh and Subel, Adam and Hassanzadeh, Pedram , journal=. Stable a posteriori. 2022 , publisher=

2022
[144]

Scientific Data , volume=

A curated dataset for data-driven turbulence modelling , author=. Scientific Data , volume=. 2021 , publisher=

2021
[145]

Ensemble

Zhang, Xin-Lei and Xiao, Heng and Luo, Xiaodong and He, Guowei , journal=. Ensemble. 2022 , publisher=

2022
[146]

Journal of Fluid Mechanics , volume=

Machine-learning wall-model large-eddy simulation accounting for isotropic roughness under local equilibrium , author=. Journal of Fluid Mechanics , volume=. 2025 , publisher=

2025
[147]

Annual Review of Fluid Mechanics , volume=

Turbulence modeling in the age of data , author=. Annual Review of Fluid Mechanics , volume=. 2019 , publisher=

2019
[148]

Physical Review Fluids , volume=

Embedded training of neural-network subgrid-scale turbulence models , author=. Physical Review Fluids , volume=. 2021 , publisher=

2021
[149]

Journal of Fluid Mechanics , volume=

Deep learning closure models for large-eddy simulation of flows around bluff bodies , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

2023
[150]

Physics of Fluids , volume=

Toward discretization-consistent closure schemes for large eddy simulation using reinforcement learning , author=. Physics of Fluids , volume=. 2023 , publisher=

2023
[151]

Nature Communications , volume=

Scientific multi-agent reinforcement learning for wall-models of turbulent flows , author=. Nature Communications , volume=. 2022 , publisher=

2022
[152]

Physical Review Fluids , volume=

Consistency requirement of data-driven subgrid-scale modeling in large-eddy simulation , author=. Physical Review Fluids , volume=. 2026 , publisher=

2026
[153]

Computer Physics Communications , volume=

JAX-Fluids: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows , author=. Computer Physics Communications , volume=. 2023 , publisher=

2023
[154]

Proceedings of the National Academy of Sciences , volume=

Machine learning--accelerated computational fluid dynamics , author=. Proceedings of the National Academy of Sciences , volume=. 2021 , publisher=

2021
[155]

Weymouth and B

G.D. Weymouth and B. Font , title =. 2025 , volume =. doi:10.1016/j.cpc.2025.109748 , journal =

2025
[156]

Solver-in-the-loop: Learning from differentiable physics to interact with iterative

Um, Kiwon and Brand, Robert and Fei, Yun Raymond and Holl, Philipp and Thuerey, Nils , journal=. Solver-in-the-loop: Learning from differentiable physics to interact with iterative
[157]

Journal of Fluid Mechanics , volume=

Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

2022
[158]

Journal of Fluid Mechanics , volume=

What is observable from wall data in turbulent channel flow? , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

2022
[159]

arXiv preprint arXiv:2601.20786 , year=

Machine-learning wall model of large-eddy simulation for low-and high-speed flows over rough surfaces , author=. arXiv preprint arXiv:2601.20786 , year=
[160]

AIAA SCITECH 2025 Forum , pages=

Numerically consistent data-driven subgrid-scale model via data assimilation and machine learning , author=. AIAA SCITECH 2025 Forum , pages=

2025
[161]

Journal of Computational Physics , volume=

Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations , author=. Journal of Computational Physics , volume=. 2014 , publisher=

2014
[162]

Physical Review Fluids , volume=

Ensemble-variational assimilation of statistical data in large-eddy simulation , author=. Physical Review Fluids , volume=. 2021 , publisher=

2021
[163]

Physics of Fluids , volume=

A dynamic slip boundary condition for wall-modeled large-eddy simulation , author=. Physics of Fluids , volume=. 2014 , publisher=

2014
[164]

Journal of Fluid Mechanics , volume=

Dynamic slip wall model for large-eddy simulation , author=. Journal of Fluid Mechanics , volume=. 2019 , publisher=

2019
[165]

Requirements and sensitivity analysis of

Whitmore, Michael and Lozano-Dur. Requirements and sensitivity analysis of. arXiv preprint arXiv:2012.14900 , year=

2012 arXiv
[166]

Journal of Computational Physics , volume=

Sensitivity analysis of wall-modeled large-eddy simulation for separated turbulent flow , author=. Journal of Computational Physics , volume=. 2024 , publisher=

2024
[167]

arXiv preprint arXiv:2511.18566 , year=

Effect of subgrid-scale anisotropy on wall-modeled large-eddy simulation of separated flow , author=. arXiv preprint arXiv:2511.18566 , year=

arXiv
[168]

Physics of Fluids , volume=

Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation , author=. Physics of Fluids , volume=. 2021 , publisher=

2021
[169]

NCAR Report , year=

On the application of the eddy viscosity concept in the inertial sub-range of turbulence , author=. NCAR Report , year=
[170]

arXiv preprint arXiv:2607.00460 , year=

A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction , author=. arXiv preprint arXiv:2607.00460 , year=

arXiv
[171]

arXiv preprint arXiv:2606.31025 , year=

Offline Reinforcement Learning for Fluid Controls: Data-based Multi-observational Policy Extraction , author=. arXiv preprint arXiv:2606.31025 , year=

arXiv
[172]

arXiv preprint arXiv:2606.22084 , year=

Patched Flow Matching: Generative Wall-Pressure Reconstruction Beyond Training-Domain Scales from Sparse Sensors , author=. arXiv preprint arXiv:2606.22084 , year=

arXiv
[173]

Liu, Yi and Fan, Xiantao and Wang, Jian-Xun , journal=
[174]

Du, Pan and Li, Yongqi and Xu, Mingqi and Wang, Jian-Xun , journal=
[175]

Parikh, Meet Hemant and Chen, Yaqin and Wang, Jian-Xun , journal=
[176]

arXiv preprint arXiv:2511.12455 , year=

Generative Reconstruction of Spatiotemporal Wall-Pressure in Turbulent Boundary Layers via Patchwise Latent Diffusion , author=. arXiv preprint arXiv:2511.12455 , year=
[177]

Du, Pan and Xu, Mingqi and Zhu, Xiaozhi and Wang, Jian-xun , journal=
[178]

arXiv preprint arXiv:2508.16640 , year=

Generative latent diffusion model for inverse modeling and uncertainty analysis in geological carbon sequestration , author=. arXiv preprint arXiv:2508.16640 , year=

arXiv
[179]

arXiv preprint arXiv:2504.05588 , year=

Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systems , author=. arXiv preprint arXiv:2504.05588 , year=

arXiv
[180]

ReALM-GEN workshop at the ICLR , year=

Disentangled Compositional Diffusion for Controllable Scientific Data Generation , author=. ReALM-GEN workshop at the ICLR , year=
[181]

Physics of Fluids , volume =

Guo, Junyi and Du, Pan and Fan, Xiantao and Li, Yahui and Wang, Jian-Xun , title =. Physics of Fluids , volume =. 2026 , month =. doi:10.1063/5.0310238 , url =

2026 doi
[182]

Journal of Fluid Mechanics , volume=

Conditional flow matching for generative modelling of near-wall turbulence with quantified uncertainty , author=. Journal of Fluid Mechanics , volume=. 2026 , pages=

2026
[183]

APL Electronic Devices , volume=

Scalable non-diffusive thermal analysis of field-effect transistors using neural operators , author=. APL Electronic Devices , volume=. 2026 , publisher=

2026
[184]

2026 , issn =

Computer Methods in Applied Mechanics and Engineering , volume =. 2026 , issn =. doi:https://doi.org/10.1016/j.cma.2025.118455 , author =

2026
[185]

2025 , publisher=

An, Delin and Du, Pan and Wang, Jian-Xun and Wang, Chaoli , journal=. 2025 , publisher=

2025
[186]

Materials Today Physics , pages=

Physics-informed neural networks with hard-encoded angle-dependent boundary conditions for phonon Boltzmann transport equation , author=. Materials Today Physics , pages=. 2025 , publisher=

2025
[187]

Predicting Stress and Damage in Carbon Fiber-Reinforced Composites Deformation Process using Composite

Chen, Zeping and Yacouti, Marwa and Shakiba, Maryam and Wang, Jian-Xun and Luo, Tengfei and Varshney, Vikas , journal=. Predicting Stress and Damage in Carbon Fiber-Reinforced Composites Deformation Process using Composite. 2025 , publisher=

2025
[188]

2025 , publisher=

Du, Pan and An, Delin and Wang, Chaoli and Wang, Jian-Xun , journal=. 2025 , publisher=

2025
[189]

2025 , issn =

Implicit neural differentiable model for spatiotemporal dynamics , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.cma.2025.118280 , url =

2025
[190]

2025 , publisher=

Akhare, Deepak and Luo, Tengfei and Wang, Jian-Xun , journal=. 2025 , publisher=

2025
[191]

2025 , publisher=

Shang, Wenjie and Zhou, Jiahang and Panda, JP and Xu, Zhihao and Liu, Yi and Du, Pan and Wang, Jian-Xun and Luo, Tengfei , journal=. 2025 , publisher=

2025
[192]

2025 , publisher=

Liu, Xin-Yang and Parikh, Meet Hemant and Fan, Xiantao and Du, Pan and Wang, Qing and Chen, Yi-Fan and Wang, Jian-Xun , journal=. 2025 , publisher=

2025
[193]

Theoretical and Applied Mechanics Letters , volume=

Gradient-free optimization of non-differentiable hybrid neural solvers for spatially heterogeneous composites , author=. Theoretical and Applied Mechanics Letters , volume=. 2025 , publisher=

2025
[194]

Direct medical image to simulation using auto-segmentation and point cloud-based

Corpuz, Ashton M and Jaiswal, Monu and Du, Pan and Ramachandra, Abhay B and Wang, Jian-Xun and Hsu, Ming-Chen , journal=. Direct medical image to simulation using auto-segmentation and point cloud-based. 2025 , publisher=

2025
[195]

Physics-Informed Bayesian Neural Networks for Solving Phonon

Li, Ruiyang and Zhou, Jiahang and Wang, Jian-Xun and Luo, Tengfei , journal=. Physics-Informed Bayesian Neural Networks for Solving Phonon. 2025 , publisher=

2025
[196]

Engineering with Computers , pages=

Asynchronous parallel reinforcement learning for optimizing propulsive performance in fin ray control , author=. Engineering with Computers , pages=. 2024 , publisher=

2024
[197]

Computer Methods in Applied Mechanics and Engineering , volume=

Neural differentiable modeling with diffusion-based super-resolution for two-dimensional spatiotemporal turbulence , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2025 , publisher=

2025
[198]

Nature Communications , year=

Conditional Neural Field Latent Diffusion Model for Generating Spatiotemporal Turbulence , author=. Nature Communications , year=
[199]

Journal of Computing and Information Science in Engineering , volume=

Scientific Machine Learning for Manufacturing Processes and Material Systems , author=. Journal of Computing and Information Science in Engineering , volume=. 2024 , publisher=

2024
[200]

Computer Methods in Applied Mechanics and Engineering , volume=

Bayesian conditional diffusion models for versatile spatiotemporal turbulence generation , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2024 , publisher=

2024
[201]

npj Computational Materials , volume=

Probabilistic physics-integrated neural differentiable modeling for isothermal chemical vapor infiltration process , author=. npj Computational Materials , volume=. 2024 , publisher=

2024
[202]

Review of Scientific Instruments , volume=

A neural network-based algorithm for the reconstruction and filtering of single particle trajectory in magnetic particle tracking , author=. Review of Scientific Instruments , volume=. 2024 , publisher=

2024
[203]

2024 , publisher=

Ren, Pu and Rao, Chengping and Chen, Su and Wang, Jian-Xun and Sun, Hao and Liu, Yang , journal=. 2024 , publisher=

2024
[204]

Communications Physics , volume=

Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics , author=. Communications Physics , volume=. 2024 , publisher=

2024
[205]

Journal of Fluid Mechanics , volume=

Effects of antagonistic muscle actuation on the bilaminar structure of ray-finned fish in propulsion , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

2023
[206]

2023 , publisher=

Ren, Pu and Rao, Chengping and Liu, Yang and Ma, Zihan and Wang, Qi and Wang, Jian-Xun and Sun, Hao , journal=. 2023 , publisher=

2023
[207]

Journal of Computational Physics , volume=

Differentiable hybrid neural modeling for fluid-structure interaction , author=. Journal of Computational Physics , volume=. 2024 , publisher=

2024
[208]

Neural Computing and Applications , pages=

An advanced spatio-temporal convolutional recurrent neural network for storm surge predictions , author=. Neural Computing and Applications , pages=. 2023 , publisher=

2023
[209]

Predicting

Movahhedi, Mohammadreza and Liu, Xin-Yang and Geng, Biao and Elemans, Coen and Xue, Qian and Wang, Jian-Xun and Zheng, Xudong , journal=. Predicting. 2023 , publisher=

2023
[210]

Physics-integrated neural differentiable

Akhare, Deepak and Luo, Tengfei and Wang, Jian-Xun , journal=. Physics-integrated neural differentiable. 2023 , publisher=

2023
[211]

Numerical Algebra, Control and Optimization , pages=

Group sparse Bayesian learning for data-driven discovery of explicit model forms with multiple parametric datasets , author=. Numerical Algebra, Control and Optimization , pages=. 2022 , publisher=

2022
[212]

Materials Today Physics , volume=

Machine learning-assisted exploration of thermally conductive polymers based on high-throughput molecular dynamics simulations , author=. Materials Today Physics , volume=. 2022 , publisher=

2022
[213]

Reducing Geometric Uncertainty in Computational Hemodynamics by Deep Learning-Assisted Parallel-Chain

Du, Pan and Wang, Jian-Xun , journal=. Reducing Geometric Uncertainty in Computational Hemodynamics by Deep Learning-Assisted Parallel-Chain. 2022 , publisher=

2022
[214]

Physics of Fluids , volume=

Deep learning-based surrogate model for three-dimensional patient-specific computational fluid dynamics , author=. Physics of Fluids , volume=. 2022 , publisher=

2022
[215]

Annals of Biomedical Engineering , volume=

Machine learning for cardiovascular biomechanics modeling: challenges and beyond , author=. Annals of Biomedical Engineering , volume=. 2022 , publisher=

2022
[216]

A Deep-Learning Based Generalized Empirical Flow Model of Glottal Flow During Normal Phonation

Zhang, Yang and Jiang, Weili and Sun, Luning and Wang, Jian-Xun and Zheng, Xudong and Xue, Qian , journal =. A Deep-Learning Based Generalized Empirical Flow Model of Glottal Flow During Normal Phonation. 2022 , month =. doi:10.1115/1.4053862 , eprint =

2022 doi
[217]

Physics-informed deep learning for solving phonon

Li, Ruiyang and Wang, Jian-Xun and Lee, Eungkyu and Luo, Tengfei , journal=. Physics-informed deep learning for solving phonon. 2022 , publisher=

2022
[218]

Physics-informed graph neural

Gao, Han and Zahr, Matthew J and Wang, Jian-Xun , journal=. Physics-informed graph neural. 2022 , publisher=

2022
[219]

2022 , publisher=

Ren, Pu and Rao, Chengping and Liu, Yang and Wang, Jian-Xun and Sun, Hao , journal=. 2022 , publisher=

2022
[220]

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =

Liu, Xin-Yang and Wang, Jian-Xun , title =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2021 , doi =

2021
[221]

, title =

Arzani,Amirhossein and Wang,Jian-Xun and D'Souza,Roshan M. , title =. Physics of Fluids , volume =. 2021 , doi =

2021
[222]

A semi-analytical solution and

Wu, Huixuan and Du, Pan and Kokate, Rohan and Wang, Jian-Xun , journal=. A semi-analytical solution and. 2021 , publisher=

2021
[223]

Physics of Fluids , volume=

Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels , author=. Physics of Fluids , volume=. 2021 , publisher=

2021
[224]

2021 , publisher=

Zhang, Jun and Tao, Jun and Wang, Jian-Xun and Wang, Chaoli , journal=. 2021 , publisher=

2021
[225]

A Bi-fidelity ensemble kalman method for

Gao, Han and Wang, Jian-Xun , journal=. A Bi-fidelity ensemble kalman method for. 2021 , publisher=

2021
[226]

2021 , publisher=

Gao, Han and Sun, Luning and Wang, Jian-Xun , journal=. 2021 , publisher=

2021
[227]

Physica D: Nonlinear Phenomena , volume=

Non-intrusive model reduction of large-scale, nonlinear dynamical systems using deep learning , author=. Physica D: Nonlinear Phenomena , volume=. 2020 , publisher=

2020
[228]

Computer Methods in Applied Mechanics and Engineering , volume=

A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2020 , publisher=

2020
[229]

Theoretical and Applied Mechanics Letters , volume=

Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data , author=. Theoretical and Applied Mechanics Letters , volume=. 2020 , publisher=

2020
[230]

Computer Methods in Applied Mechanics and Engineering , volume=

Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2020 , publisher=

2020
[231]

Annals of biomedical engineering , volume=

Data-augmented modeling of intracranial pressure , author=. Annals of biomedical engineering , volume=. 2019 , publisher=

2019
[232]

Theoretical and Computational Fluid Dynamics , volume=

Prediction of Reynolds stresses in high-Mach-number turbulent boundary layers using physics-informed machine learning , author=. Theoretical and Computational Fluid Dynamics , volume=. 2019 , publisher=

2019
[233]

Physical Review Fluids , volume=

Predictive large-eddy-simulation wall modeling via physics-informed neural networks , author=. Physical Review Fluids , volume=. 2019 , publisher=

2019
[234]

2018 , publisher=

Tang, Hui and Wang, Jianxun and Weiss, Robert and Xiao, Heng , journal=. 2018 , publisher=

2018
[235]

Geophysical Journal International , volume=

Inferring tsunami flow depth and flow speed from sediment deposits based on ensemble Kalman filtering , author=. Geophysical Journal International , volume=. 2018 , publisher=

2018
[236]

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering , volume=

Propagation of input uncertainty in presence of model-form uncertainty: a multifidelity approach for computational fluid dynamics applications , author=. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering , volume=. 2018 , publisher=

2018
[237]

Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on

Wang, Jian-Xun and Wu, Jin-Long and Xiao, Heng , journal=. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on. 2017 , publisher=

2017
[238]

arXiv preprint arXiv:1701.07102 , year=

A comprehensive physics-informed machine learning framework for predictive turbulence modeling , author=. arXiv preprint arXiv:1701.07102 , year=

arXiv
[239]

Flow, Turbulence and Combustion , volume=

A priori assessment of prediction confidence for data-driven turbulence modeling , author=. Flow, Turbulence and Combustion , volume=. 2017 , publisher=

2017
[240]

An implicitly consistent formulation of a dual-mesh hybrid

Xiao, Heng and Wang, Jian-Xun and Jenny, Patrick , journal=. An implicitly consistent formulation of a dual-mesh hybrid. 2017 , publisher=

2017
[241]

Computer Methods in Applied Mechanics and Engineering , volume=

A random matrix approach for quantifying model-form uncertainties in turbulence modeling , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2017 , publisher=

2017
[242]

Proceedings of the 2017 AIAA SciTech , year=

Physics-informed machine learning for predictive turbulence modeling: Progress and perspectives , author=. Proceedings of the 2017 AIAA SciTech , year=

2017
[243]

Wu, Jin-Long and Wang, Jian-Xun and Xiao, Heng , journal=. A. 2016 , publisher=

2016
[244]

International Journal for Uncertainty Quantification , volume=

Incorporating prior knowledge for quantifying and reducing model-form uncertainty in RANS simulations , author=. International Journal for Uncertainty Quantification , volume=. 2016 , publisher=

2016
[245]

Data-driven

Wang, Jian-Xun and Xiao, Heng , journal=. Data-driven. 2016 , publisher=

2016
[246]

International Journal of Heat and Fluid Flow , volume=

Quantification of uncertainties in turbulence modeling: A comparison of physics-based and random matrix theoretic approaches , author=. International Journal of Heat and Fluid Flow , volume=. 2016 , publisher=

2016
[247]

Quantifying and reducing model-form uncertainties in

Xiao, Heng and Wu, J-L and Wang, J-X and Sun, Rui and Roy, CJ , journal=. Quantifying and reducing model-form uncertainties in. 2016 , publisher=

2016
[248]

Dynamic evaluation of mesh resolution and its application in hybrid

Xiao, Heng and Wang, Jianxun and Jenny, Patrick , journal=. Dynamic evaluation of mesh resolution and its application in hybrid. 2014 , publisher=

2014
[249]

2025 IEEE International Symposium on Biomedical Imaging , pages=

Hierarchical LoG Bayesian Neural Network for Enhanced Aorta Segmentation , author=. 2025 IEEE International Symposium on Biomedical Imaging , pages=. 2025 , organization=

2025
[250]

Wang, Qi and Ren, Pu and Zhou, Hao and Liu, Xin-Yang and Deng, Zhiwen and Zhang, Yi and Chengze, Ruizhi and Liu, Hongsheng and Wang, Zidong and Wang, Jian-Xun and others , booktitle=. P \^. 2025 , organization=

2025
[251]

Advances in Neural Information Processing Systems , volume=

Unifying predictions of deterministic and stochastic physics in mesh-reduced space with sequential flow generative model , author=. Advances in Neural Information Processing Systems , volume=
[252]

Symbolic physics learner: Discovering governing equations via

Sun, Fangzheng and Liu, Yang and Wang, Jian-Xun and Sun, Hao , booktitle=. Symbolic physics learner: Discovering governing equations via. 2023 , url=

2023
[253]

Learning on Graphs Conference , pages=

PatchGT: Transformer over Non-trainable Clusters for Learning Graph Representations , author=. Learning on Graphs Conference , pages=. 2022 , organization=

2022
[254]

NeurIPS , year=

Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty , author=. NeurIPS , year=
[255]

International Conference on Learning Representations , year=

Predicting Physics in Mesh-reduced Space with Temporal Attention , author=. International Conference on Learning Representations , year=
[256]

2020 , organization=

Guo, Li and Ye, Shaojie and Han, Jun and Zheng, Hao and Gao, Han and Chen, Danny Z and Wang, Jian-Xun and Wang, Chaoli , booktitle=. 2020 , organization=

2020
[257]

Adding constraints to

Wu, Jiacheng and Wang, Jian-Xun and Shadden, Shawn C , booktitle=. Adding constraints to
[258]

Prashanth, Mohit and Du, Pan and Wang, Jianxun and Wu, Huixuan , booktitle=
[259]

Visualization of high dimensional turbulence simulation data using

Wu, Jinlong and Wang, Jianxun and Xiao, Heng and Ling, Julia , booktitle=. Visualization of high dimensional turbulence simulation data using
[260]

55th AIAA Aerospace Sciences Meeting , pages=

High-Mach-number turbulence modeling using machine learning and direct numerical simulation database , author=. 55th AIAA Aerospace Sciences Meeting , pages=
[261]

55th AIAA aerospace sciences meeting , pages=

A physics-informed machine learning approach of improving RANS predicted reynolds stresses , author=. 55th AIAA aerospace sciences meeting , pages=
[262]

2016 , institution=

Towards a complete framework of physics-informed machine learning for predictive turbulence modeling , author=. 2016 , institution=

2016

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Physics of Fluids 27 (3)

Abkar, Mahdi & Port \'e -Agel, Fernando 2015 Influence of atmospheric stability on wind-turbine wakes: A large-eddy simulation study . Physics of Fluids 27 (3)

2015

[2] [2]

Theoretical and Applied Mechanics Letters p

Akhare, Deepak , Luo, Tengfei & Wang, Jian-Xun 2025 HybridNDiff-UQ : Uncertainty quantification for hybrid neural differentiable modeling . Theoretical and Applied Mechanics Letters p. 100609

2025

[3] [3]

Fluids 6 (3), 112

Bae, H Jane & Lozano-Dur \'a n, Adri \'a n 2021 Effect of wall boundary conditions on a wall-modeled large-eddy simulation in a finite-difference framework . Fluids 6 (3), 112

2021

[4] [4]

In 13th Fluid and Plasmadynamics Conference\/ , p

Bardina, Jorge , Ferziger, J & Reynolds, WC 1980 Improved subgrid-scale models for large-eddy simulation. In 13th Fluid and Plasmadynamics Conference\/ , p. 1357

1980

[5] [5]

Computer Physics Communications 282 , 108527

Bezgin, Deniz A , Buhendwa, Aaron B & Adams, Nikolaus A 2023 Jax-fluids: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows . Computer Physics Communications 282 , 108527

2023

[6] [6]

Annual Review of Fluid Mechanics 50 , 535--561

Bose, Sanjeeb T & Park, George Ilhwan 2018 Wall-modeled large-eddy simulation for complex turbulent flows . Annual Review of Fluid Mechanics 50 , 535--561

2018

[7] [7]

2021 Integral measures of the zero pressure gradient boundary layer over the R eynolds number range 0 R_ <

Cantwell, Brian J. 2021 Integral measures of the zero pressure gradient boundary layer over the R eynolds number range 0 R_ < . Physics of Fluids 33 (8), 085108

2021

[8] [8]

AIAA Journal 17 (12), 1293--1313

Chapman, Dean R 1979 Computational aerodynamics development and outlook . AIAA Journal 17 (12), 1293--1313

1979