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A machine-learned probability distribution approximates the natural distribution of turbulent channel flow, enabling realistic synthetic fields and probabilistic reconstruction.

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2026-08-01 16:13 UTC pith:Z3Q7YK6X

load-bearing objection A solid ML-for-turbulence paper with a genuinely new construction; the conditional sampling derivation leans on an unproven independence assumption that deserves a close look, but the empirical results are encouraging. the 2 major comments →

arxiv 2607.18058 v1 pith:Z3Q7YK6X submitted 2026-07-20 physics.flu-dyn

A machine-learned probability distribution in the phase space of turbulent channel flow for synthetic turbulence and flow reconstruction

Frederik Aerts , Dirk Nuyens , Johan Meyers This is my paper
classification physics.flu-dyn PACS 47.27.-i47.27.E-
keywords turbulent channel flownatural distributioninvariant measureflow-based generative modelflow matchingconditional samplingsynthetic turbulenceflow reconstruction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that a machine-learned probability distribution can approximate the physical invariant distribution—the natural distribution—of turbulent channel flow at Re_tau=180. If true, one can sample realistic turbulent fields directly, bypassing the long spin-up needed to reach statistical stationarity, and can reconstruct flow fields from sparse observations with quantified uncertainty. The authors train a flow-based generative model on a minimal conditional flow unit and introduce a conditional sampling procedure that reuses the same network. They show the synthetic fields match DNS statistics up to fourth order, including intermittency, and that ensembles remain statistically stationary when used to initialize DNS, supporting the claim of dynamical invariance.

Core claim

The central claim is that the learned distribution provides a good approximation to the natural distribution of the turbulent dynamical system (Abstract, §4). Three properties are demonstrated: physical ensemble statistics (mean profile, Reynolds stresses, energy spectra, skewness, flatness, heavy-tailed velocity increments), consistent conditional sampling (reconstruction errors match an LSE-based estimate, with ensemble variance as uncertainty), and dynamical invariance (synthetic fields used as DNS initial conditions show stationary energy spectra and TKE budget terms after a brief Kolmogorov-timescale transient). The minimal conditional flow unit is the key enabling object, making large-

What carries the argument

The flow-based generative model trained via conditional flow matching, with a Gaussian conditional optimal transport path, maps random Gaussian fields to synthetic turbulence through straight-line trajectories. For conditional sampling, the modified ODE system (2.23) pins observed components to their measurements along the path while letting unobserved components evolve under the marginal generator. The minimal conditional flow unit—the smallest domain in homogeneous directions outside which conditioning on a central velocity component is indistinguishable from unconditional sampling—carries the argument, since it permits sequential conditional sampling to arbitrarily large domains under a l

Load-bearing premise

The load-bearing premise is that the unobserved part of the guided generator network can be treated as independent of the observed components, so that the previously trained marginal generator can be reused in Eq. (2.23); the paper does not prove this independence holds for a general trained network.

What would settle it

For a mask that observes a large fraction of the domain (e.g., 75% of the velocity components), compute the ensemble variance of the reconstructions produced by Eq. (2.23) and compare it to the true conditional variance estimated from DNS snapshots; a systematic discrepancy that grows with the observed fraction would indicate the independence assumption is violated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Synthetic initial conditions can be generated that skip the long transient to statistical stationarity in DNS, because samples already live on the attractor.
  • Flow reconstruction from sparse observations becomes a sampling problem, producing an ensemble of realistic fields with quantified uncertainty rather than a single conditional average.
  • The learned distribution can be reused for arbitrary masks of observed/unobserved variables without retraining, making the method flexible for varying sensor placements.
  • Non-Gaussian statistics such as intermittency, skewness, and nonlinear energy transfer are preserved, which Gaussian-based synthetic generators cannot capture.
  • The minimal conditional flow unit provides a domain-size-independent representation: sequential sampling extends fields to domains 24 times the training unit with adequate spectral fidelity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The local-conditioning approximation (conditioning only on the neighboring subdomain) relies on a screening effect; at higher Reynolds numbers or in flows with very long coherent structures, this screening may weaken, and the approximation may need larger units.
  • The independence assumption behind the guided generator (unobserved part independent of observations) is untested for general masks; if it fails, the conditional sampling ensembles could under- or over-estimate reconstruction uncertainty.
  • A natural extension is to condition on derived quantities such as wall shear stress or pressure, which the paper suggests but does not demonstrate; the same conditional-sampling mechanism should apply.
  • Because the model learns incompressibility implicitly, explicitly enforcing a divergence-free constraint at generation time could reduce small-scale discrepancies at higher Reynolds numbers.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper trains a flow-matching generative model on a 'minimal conditional flow unit' (MCFU) to approximate the invariant phase-space distribution of turbulent channel flow at Re_tau=180, and evaluates the approximation through three properties: physical ensemble statistics, consistent conditional sampling, and dynamical invariance. The authors define the MCFU as the smallest domain outside which conditional fields are indistinguishable from unconditional fields in mean-square sense, estimate its size from two-point correlations, and use sequential conditional sampling to generate fields on larger domains. For conditional sampling they derive a guided flow-matching loss and propose an approximate minimizer that reuses the unconditional generator. Comparisons with DNS are reported for one-point, two-point, and higher-order statistics, for a flow-reconstruction problem, and for DNS initialized with synthetic fields. The paper concludes that the learned distribution is a good approximation to the natural distribution.

Significance. If the claims hold, this is a notable step: it is one of the first explicit attempts to approximate the full three-dimensional invariant phase-space distribution of a wall-bounded turbulent flow rather than only selected statistics. The unconditional generation reproduces higher-order moments (skewness, flatness, increment distributions) and the dynamical-invariance test via DNS initialization is a meaningful and nonstandard validation. The study also has concrete strengths: a held-out test set, validation-based model selection, comparison against a carefully derived LSE-based error estimate, and DNS-based scrutiny of the generated fields. These elements make the empirical core credible. The main risk is the unproven conditional-independence approximation in the conditional sampling procedure, which is load-bearing for the reconstruction-uncertainty interpretation and for the sequential generation of large domains.

major comments (2)
  1. [§2.4, Eq. (2.22)–(2.24)] The derivation of the conditional sampling ODE rests on an unproven independence restriction. The paper states that if the unobserved part M⊥ f(v,ξ|u_o) is 'restricted to be independent of u_o', then the guided conditional flow-matching loss (2.22) reduces to a selection of terms from the unconditional loss (2.18), allowing reuse of the marginal generator. However, the resulting ODE (2.23) has unobserved update M⊥ f*_θ(v,ξ), and v depends on u_o through the observed-part integration (2.24a), i.e., v = ξu_o + (1−ξ)(observed components of v0). Thus the approximation is effectively E[u_u | v] ≈ E[u_u | v, u_o]. The marginal generator was trained on the unconditional loss (2.18), not on the guided loss (2.22), so there is no training pressure enforcing this. If the approximation fails, Eq. (2.23) samples a distribution different from π(u_u|u_o), invalidating the uncertainty quantification in
  2. [§2.2, Eq. (2.10) and §3.2, Figs. 9–11] The MCFU construction and the sequential sampling rely on two approximations whose quantitative validity is not established: (i) the dimensions R_x≈πδ and R_z≈πδ/4 are inferred from a visually assessed threshold of squared correlations, with no explicit criterion for 'sufficiently small'; (ii) the local-conditioning approximation π(u_C|u_A,u_B)≈π(u_C|u_B) in Eq. (2.10) is motivated by the screening effect, but the footnote in §2.2 correctly notes that marginal decorrelation does not imply conditional independence and the relay effect can transmit information. The large-domain results in Figs. 9–11 show precisely the kind of artifacts one would expect if this approximation is imperfect: cut-through structures at the interfaces between conditioned blocks and underrepresentation of the largest streamwise/spanwise scales. Since the paper claims the MCFU representation enables sampling on 'ar
minor comments (5)
  1. [§2.2, Fig. 2] The definition of R_h would benefit from a quantitative threshold, e.g., the distance beyond which ρ² drops below a stated value, rather than an informal visual estimate.
  2. [§3.1, Fig. 6] The sentence 'Only the largest spanwise scales of the streamwise velocity component are underrepresented as seen in figure 6(e)' is not easy to verify from the plotted curves; consider adding an inset or arrow.
  3. [§3.3, Figs. 12–13] The color legend is labeled '0.1 2 4 6 8 10'; it would be clearer to use a continuous colorbar with a title such as 't u_τ/δ'.
  4. [Eq. (3.1)] The numerator in the integrand appears to omit an ensemble-average bracket; please check the notation so that the equation matches the described quantity.
  5. [§2.5] The acronym MCFU is used in the abstract and in the text but is not defined at first use; define it when the minimal conditional flow unit is introduced.

Circularity Check

0 steps flagged

No significant circularity: the derivation is a standard flow-matching construction, and the validation uses held-out DNS data and DNS evolution rather than re-stating the training statistics.

full rationale

The paper's derivation chain is not circular. The flow-matching loss (2.18) and the guided conditional loss (2.20) are standard mathematical constructions; the approximate conditional sampler (2.23) is obtained by explicitly reusing the marginal generator under an independence restriction, not by defining the target distribution to equal the trained generator. The central claim—that the learned distribution approximates the natural distribution—is tested against held-out DNS snapshots (10% test set), against DNS evolution in the dynamical-invariance test, and against an LSE-based reconstruction-error benchmark computed from DNS data (Appendix B). These are external checks, not identities with the training objective. The paper explicitly acknowledges the local-conditioning approximation is not guaranteed by marginal decorrelation alone, and the independence restriction in §2.4 is unproven; these are honest limitations rather than circular reductions. The self-citations (SPWind code, shifted periodic boundary conditions) are methodological references and are not load-bearing evidence for the main claim. No uniqueness theorem or ansatz is imported from the authors' prior work, and no fitted parameter is renamed as a prediction. Therefore the appropriate finding is no significant circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 1 invented entities

The central claim rests on several hand-chosen structural parameters (MCFU size, coarsening factor, integration steps) and on approximations in conditional sampling that are not rigorously derived. The flow-matching math is standard, but the physical-natural-distribution assumptions and the local-conditioning/screening approximations are domain assumptions specific to this paper.

free parameters (3)
  • MCFU half-extent R_x≈πδ, R_z≈πδ/4 = R_x≈πδ, R_z≈πδ/4
    Chosen by eye from squared cross-correlations with an unspecified 'sufficiently small' threshold (§2.2, Fig. 2). Determines the domain of the learned distribution and the validity of sequential conditional sampling.
  • Coarsening factor 2 (field resolution 23.6×[0.34-6.2]×11.8) = Δx+≈23.6, Δy+∈[0.34,6.2], Δz+≈11.8
    The target distribution is learned on DNS fields filtered by a factor 2 and downsampled in y (§2.5); this truncates the small scales and introduces spectral leakage for non-periodic units.
  • Number of RK4 integration steps = 20
    Sampling integrates the learned ODE with 20 timesteps (§3, App. A); accuracy of the flow map at ξ=1 depends on this choice.
axioms (6)
  • domain assumption A coarse-grained natural density π exists via noise kernel K_ε (Eq. 2.4)
    The paper notes rigorous construction of SRB measures for turbulence is open, but assumes kernel smoothing yields a well-defined density for training.
  • ad hoc to paper Local conditioning approximation π(u_C|u_A,u_B)≈π(u_C|u_B) (Eq. 2.10)
    Used in sequential conditional sampling; the paper acknowledges the relay effect can violate it and invokes screening (§2.2).
  • ad hoc to paper Unobserved part of the guided generator can be treated as independent of u_o (§2.4)
    Allows reuse of the marginal generator in Eq. (2.23); not shown to hold for the trained network.
  • ad hoc to paper MCFU defined by single-component conditioning is adequate for half-domain or multi-component conditioning
    The reconstruction test conditions on an entire upstream half, and Fig. 8(b) shows imperfect decorrelation; the paper admits larger units might be needed.
  • domain assumption DNS snapshots are samples of the coarse-grained natural distribution
    Assumes the reference trajectory is stationary, ergodic, and validated against Kim et al. (§2.5); used as training target.
  • standard math Flow-matching/COT path theory (Lipman et al.)
    Marginal/conditional flow-matching loss equivalence is taken as established; standard math in generative modeling.
invented entities (1)
  • Minimal conditional flow unit (MCFU) no independent evidence
    purpose: A small non-periodic domain of size 2πδ×2δ×πδ/2 that serves as the training domain, enabling large-domain synthesis via sequential conditional sampling.
    Not a physical entity but a new methodological construct; its dimensions are inferred from DNS correlations and empirically supported by the paper's own validation, but it is not an independently testable physical hypothesis.

pith-pipeline@v1.3.0-alltime-deepseek · 31763 in / 14964 out tokens · 158718 ms · 2026-08-01T16:13:54.529037+00:00 · methodology

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read the original abstract

Although a complete characterisation of the probability distribution in the phase space of turbulent flows remains elusive, accurately sampling this distribution is essential for both synthetic turbulence generation and turbulent flow reconstruction. Motivated by these applications, we examine to what extent a machine-learned distribution can approximate the physical invariant distribution of turbulent channel flow at $\mathrm{Re}_\tau=180$. We assess three important properties of the approximation: physical ensemble statistics, consistent conditional sampling, and dynamical invariance. To this end, a flow-based generative model is trained on a minimal conditional flow unit, which we define as the smallest domain outside which conditional fields, given a single observation at the domain centre, are indistinguishable from unconditional fields in terms of mean-square discrepancy to other conditional fields. We also introduce a consistent procedure for sampling from the conditional learned distribution. Comparisons with direct numerical simulation show that synthetic turbulent fields reproduce key statistical and dynamical features of turbulence, including intermittency and nonlinear energy transfer. The consistency of conditional sampling is demonstrated in a flow reconstruction problem, and subsequently used to generate synthetic turbulent velocity fields on a large domain. When adopted as initial conditions in direct numerical simulations, these fields yield physical and statistically stationary ensemble statistics, indicating that the learned distribution provides a good approximation to the natural distribution of the turbulent dynamical system.

Figures

Figures reproduced from arXiv: 2607.18058 by Dirk Nuyens, Frederik Aerts, Johan Meyers.

Figure 1
Figure 1. Figure 1: Illustration of the natural distribution for the dynamical system representing cellular convection by Lorenz (1963): (a) invariant distribution obtained by the push forward of typical initial conditions, (b) marginal distribution 𝑝(𝑥1, 𝑥2) derived from the natural distribution, (c) Gaussian approximation to this marginal distribution, (d) reconstruction of 𝑥2 given 𝑥1 = 10 based on samples of the natural d… view at source ↗
Figure 2
Figure 2. Figure 2: Definition of the minimal conditional flow unit in a turbulent channel flow: (a) the minimal conditional flow unit in red, (b) squared streamwise correlation at different wall-normal heights, (c) squared spanwise correlation at different wall-normal heights. The correlations are taken from the database by Kim et al. (1987). In (a) the subdomains 𝐴, 𝐵, and 𝐶 used in the sequential conditional sampling argum… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the sampling process with the flow-based generative model: (a) The flow map 𝒗𝜉 = 𝛹 𝜉 𝝑 (𝒗0) deterministically maps an initial condition 𝒗0, sampled from a standard Gaussian, to a turbulent channel flow field 𝒗1 at fictitious time 𝜉 = 1. (b) Through sequential conditional sampling with minimal conditional flow units, turbulent channel flow fields on arbitrarily large domains can … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the one-point first- and second-order ensemble statistics between the synthetic and the DNS velocity fields: (a) mean velocity profile, (b) root-mean-square velocity fluctuations, and (c) Reynolds shear stress. 3. Applications in synthetic turbulence and turbulent flow reconstruction Three properties of the machine-learned distribution are examined. First, the physicality of its ensemble stat… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the one-point third- and fourth-order ensemble statistics between the synthetic and the DNS velocity fields: (a,c) skewness 𝑆(𝜑) = ⟨𝜑 3 ⟩/⟨𝜑 2 ⟩ 3/2 , (b,d) flatness 𝐹(𝜑) = ⟨𝜑 4 ⟩/⟨𝜑 2 ⟩ 2 ; (a,b) velocity fluctuations, (c,d) turbulent shear stress. The dashed grey lines denote the Gaussian values of 𝑆 = 0 and 𝐹 = 3. turbulent kinetic energy by sweeps of high-speed fluid towards the wall (𝑢 ′… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the one-dimensional energy spectra ⟨𝐸𝑢𝑢⟩, ⟨𝐸𝑣𝑣⟩, and ⟨𝐸𝑤𝑤⟩ between the synthetic and the DNS velocity fields on the minimal conditional channel flow unit of 2𝜋𝛿×2𝛿×𝜋𝛿/2 at three wall-normal heights: (a,d) 𝑦 + = 5.2, (b,e) 𝑦 + = 29, (c,f) 𝑦 + = 178. −5 0 5 10−4 10−3 10−2 10−1 100 δ∥u/⟨(δ∥u) 2 ⟩ 1/2 PDF (a) y + = 178 y + = 29 −5 0 5 10−4 10−3 10−2 10−1 100 δ⊥u/⟨(δ⊥u) 2 ⟩ 1/2 PDF (b) DNS Synthet… view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reconstruction error in a flow reconstruction problem: (a) schematic representation of the problem, (b-d) mean-square error on the streamwise, wall-normal, and spanwise velocity component. The mean ± one standard deviation for the wall-normal average are shown, both for the mean and members of the ensemble of reconstructed velocity fields. The orange lines denote estimates based on Linear Stochastic Estima… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the streamwise velocity fluctuation field between the DNS (left) and the synthetic (right) velocity fields at different wall-normal heights: (a-b) 𝑦 + = 178, (c-d) 𝑦 + = 59.5, (e-f) 𝑦 + = 29, (g-h) 𝑦 + = 10, (i-j) 𝑦 + = 5.2. The procedure of generating half of a minimal conditional flow unit conditioned on an observed or known other half may be applied sequentially, as depicted in figure 3(b)… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the wall-normal velocity fluctuation field between the DNS (left) and the synthetic (right) velocity fields at different wall-normal heights: (a-b) 𝑦 + = 178, (c-d) 𝑦 + = 59.5, (e-f) 𝑦 + = 29, (g-h) 𝑦 + = 10, (i-j) 𝑦 + = 5.2. the streamwise length of the minimal conditional flow unit. At the centre of the channel (𝑦 + = 178), as shown in figure 9(a-b), the largest spanwise scales appear to b… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the one-dimensional energy spectra ⟨𝐸𝑢𝑢⟩, ⟨𝐸𝑣𝑣⟩, and ⟨𝐸𝑤𝑤⟩ between the synthetic and the DNS velocity fields on a domain of 8𝜋𝛿 × 2𝛿 × 3𝜋𝛿 at three wall-normal heights: (a,d) 𝑦 + = 5.2, (b,e) 𝑦 + = 29, (c,f) 𝑦 + = 178. The scales larger than those represented on the minimal conditional flow unit are shaded. shown). The wall-normal velocity component of the synthetic velocity field is shown i… view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of the premultiplied one-dimensional energy spectra averaged over an ensemble of 10 trajectories starting from synthetic initial conditions on a domain of 8𝜋𝛿 × 2𝛿 × 3𝜋𝛿. The spectra are shown at three wall-normal heights: (a,d) 𝑦 + = 5.2, (b,e) 𝑦 + = 29, (c,f) 𝑦 + = 178. The line colour indicates the time of evaluation, with intervals of 𝛥𝑡𝑢𝜏 /𝛿 = 0.1. The scales larger and smaller than th… view at source ↗
Figure 13
Figure 13. Figure 13: Time evolution of turbulent kinetic energy budget terms from a DNS on a domain of 8𝜋𝛿 ×2𝛿 ×3𝜋𝛿 initialised with a synthetic turbulent velocity field: (a) production, (b) turbulent transport, (c) dissipation. The line colour indicates the time of evaluation, with intervals of 𝛥𝑡𝑢𝜏 /𝛿 = 0.1. The grey dashed lines indicate the 0.5th and 99.5th percentiles from the reference DNS trajectory. fluctuations, it v… view at source ↗

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Reference graph

Works this paper leans on

87 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Ergodic theory of chaos and strange attractors , author =. Rev. Mod. Phys. , volume =. 1985 , month =. doi:10.1103/RevModPhys.57.617 , url =

    doi:10.1103/revmodphys.57.617 1985

  2. [2]

    Synchronization of turbulence in channel flow , volume =. J. Fluid Mech. , author =. 2022 , pages =. doi:10.1017/jfm.2022.397 , language =

    work page doi:10.1017/jfm.2022.397 2022

  3. [3]

    Regeneration of. Phys. Rev. Lett. , author =. 2005 , pages =. doi:10.1103/PhysRevLett.94.014501 , language =

    work page doi:10.1103/physrevlett.94.014501 2005

  4. [4]

    Logarithmic-layer turbulence: a view from the wall , volume =. Phys. Rev. Fluids , author =. 2019 , pages =. doi:10.1103/PhysRevFluids.4.114603 , language =

    work page doi:10.1103/physrevfluids.4.114603 2019

  5. [5]

    Stochastic estimation of organized turbulent structure: homogeneous shear flow , volume =. J. Fluid Mech. , author =. 1988 , pages =. doi:10.1017/S0022112088001442 , language =

    doi:10.1017/s0022112088001442 1988

  6. [6]

    Robust flow reconstruction from limited measurements via sparse representation , volume =. Phys. Rev. Fluids , author =. 2019 , pages =. doi:10.1103/PhysRevFluids.4.103907 , language =

    work page doi:10.1103/physrevfluids.4.103907 2019

  7. [7]

    Unsupervised deep learning for super-resolution reconstruction of turbulence , volume =. J. Fluid Mech. , author =. 2021 , pages =. doi:10.1017/jfm.2020.1028 , language =

    arXiv 2021

  8. [8]

    High-fidelity reconstruction of turbulent flow from spatially limited data using enhanced super-resolution generative adversarial network , volume =. Phys. Fluids , author =. 2021 , pages =. doi:10.1063/5.0066077 , language =

    work page doi:10.1063/5.0066077 2021

  9. [9]

    Symmetry-preserving discretization of turbulent flow , volume =. J. Comput. Phys. , author =. 2003 , pages =. doi:10.1016/S0021-9991(03)00126-8 , language =

    doi:10.1016/s0021-9991(03)00126-8 2003

  10. [10]

    Synchronization of. Phys. Rev. Lett. , author =. 2013 , pages =. doi:10.1103/PhysRevLett.110.084102 , language =

    work page doi:10.1103/physrevlett.110.084102 2013

  11. [11]

    and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , year =

    Song, Yang and Sohl-Dickstein, Jascha and Kingma, Diederik P. and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , year =. Score-. doi:10.48550/arXiv.2011.13456 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2011.13456 2011

  12. [12]

    Generative modeling of turbulence , volume =. Phys. Fluids , author =. 2022 , pages =. doi:10.1063/5.0082562 , language =

    doi:10.1063/5.0082562 2022

  13. [13]

    A physics-informed diffusion model for high-fidelity flow field reconstruction , volume =. J. Comput. Phys. , author =. 2023 , pages =. doi:10.1016/j.jcp.2023.111972 , language =

    arXiv 2023

  14. [14]

    and Martínez-Tossas, Luis A

    Rybchuk, Alex and Hassanaly, Malik and Hamilton, Nicholas and Doubrawa, Paula and Fulton, Mitchell J. and Martínez-Tossas, Luis A. , title =. Phys. Fluids , volume =. 2023 , month =. doi:10.1063/5.0172559 , url =

    work page doi:10.1063/5.0172559 2023

  15. [15]

    Pope, Stephen B. , doi =. Turbulent flows , publisher=

  16. [16]

    Synthetic turbulence inflow conditions for large-eddy simulation , volume =. Phys. Fluids , author =. 2006 , pages =. doi:10.1063/1.2130744 , language =

    work page doi:10.1063/1.2130744 2006

  17. [17]

    A synthetic-eddy-method for generating inflow conditions for large-eddy simulations , volume =. Int. J. Heat Fluid Flow , author =. 2006 , pages =. doi:10.1016/j.ijheatfluidflow.2006.02.006 , language =

    work page doi:10.1016/j.ijheatfluidflow.2006.02.006 2006

  18. [18]

    Shifted periodic boundary conditions for simulations of wall-bounded turbulent flows , volume =. Phys. Fluids , author =. 2016 , pages =. doi:10.1063/1.4941912 , language =

    work page doi:10.1063/1.4941912 2016

  19. [19]

    Conditional neural field latent diffusion model for generating spatiotemporal turbulence , volume =. Nat. Commun. , author =. 2024 , pages =. doi:10.1038/s41467-024-54712-1 , language =

    doi:10.1038/s41467-024-54712-1 2024

  20. [20]

    Turbulence:

    Frisch, Uriel , month = nov, year =. Turbulence:

  21. [21]

    and Azizpour, Hossein and Vinuesa, Ricardo , month = may, year =

    Vishwasrao, Abhijeet and Gutha, Sai Bharath Chandra and Cremades, Andres and Wijk, Klas and Patil, Aakash and Gorle, Catherine and McKeon, Beverley J. and Azizpour, Hossein and Vinuesa, Ricardo , month = may, year =. Diff-. doi:10.48550/arXiv.2506.00214 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2506.00214

  22. [22]

    Atmosphere , author =

    Multi-. Atmosphere , author =. 2023 , pages =. doi:10.3390/atmos15010060 , language =

    work page doi:10.3390/atmos15010060 2023

  23. [23]

    A deep-learning approach for reconstructing. Sci. Rep. , author =. 2023 , pages =. doi:10.1038/s41598-023-29525-9 , language =

    doi:10.1038/s41598-023-29525-9 2023

  24. [24]

    The spatial structure of neutral atmospheric surface-layer turbulence , volume =. J. Fluid Mech. , author =. 1994 , pages =. doi:10.1017/S0022112094001886 , language =

    work page doi:10.1017/s0022112094001886 1994

  25. [25]

    A minimal multiscale. Phys. Fluids , author =. 2006 , pages =. doi:10.1063/1.2227003 , language =

    work page doi:10.1063/1.2227003 2006

  26. [26]

    Denoising

    Ho, Jonathan and Jain, Ajay and Abbeel, Pieter , month = dec, year =. Denoising. doi:10.48550/arXiv.2006.11239 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2006.11239 2006

  27. [27]

    Diffusion

    Dhariwal, Prafulla and Nichol, Alex , month = jun, year =. Diffusion. doi:10.48550/arXiv.2105.05233 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2105.05233

  28. [28]

    Lipman, Yaron and Havasi, Marton and Holderrieth, Peter and Shaul, Neta and Le, Matt and Karrer, Brian and Chen, Ricky T. Q. and Lopez-Paz, David and Ben-Hamu, Heli and Gat, Itai , month = dec, year =. Flow. doi:10.48550/arXiv.2412.06264 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2412.06264

  29. [29]

    Lipman, Yaron and Chen, Ricky T. Q. and Ben-Hamu, Heli and Nickel, Maximilian and Le, Matt , month = feb, year =. Flow. doi:10.48550/arXiv.2210.02747 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2210.02747

  30. [30]

    Hierarchy of minimal flow units in the logarithmic layer , volume =. Phys. Fluids , author =. 2010 , pages =. doi:10.1063/1.3464157 , language =

    work page doi:10.1063/1.3464157 2010

  31. [31]

    Turbulence statistics in fully developed channel flow at low. J. Fluid Mech. , author =. 1987 , pages =. doi:10.1017/S0022112087000892 , language =

    doi:10.1017/s0022112087000892 1987

  32. [32]

    Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David , month = dec, year =. Neural. doi:10.48550/arXiv.1806.07366 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.1806.07366

  33. [33]

    Esser, Patrick and Kulal, Sumith and Blattmann, Andreas and Entezari, Rahim and Müller, Jonas and Saini, Harry and Levi, Yam and Lorenz, Dominik and Sauer, Axel and Boesel, Frederic and Podell, Dustin and Dockhorn, Tim and English, Zion and Lacey, Kyle and Goodwin, Alex and Marek, Yannik and Rombach, Robin , month = mar, year =. Scaling. doi:10.48550/arXi...

    Pith/arXiv arXiv doi:10.48550/arxiv.2403.03206

  34. [34]

    Effect of the computational domain on direct simulations of turbulent channels up to. Phys. Fluids , author =. 2014 , pages =. doi:10.1063/1.4862918 , language =

    doi:10.1063/1.4862918 2014

  35. [35]

    Three-dimensional generative adversarial networks for turbulent flow estimation from wall measurements , volume =. J. Fluid Mech. , author =. 2024 , pages =. doi:10.1017/jfm.2024.432 , language =

    work page doi:10.1017/jfm.2024.432 2024

  36. [36]

    On spectral linear stochastic estimation , volume =. Exp. Fluids , author =. 2006 , pages =. doi:10.1007/s00348-006-0199-5 , language =

    doi:10.1007/s00348-006-0199-5 2006

  37. [37]

    The principles of diffusion models , copyright =

    Lai, Chieh-Hsin and Song, Yang and Kim, Dongjun and Mitsufuji, Yuki and Ermon, Stefano , year =. The principles of diffusion models , copyright =. doi:10.48550/arXiv.2510.21890 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.2510.21890

  38. [38]

    Cvitanovi\'c, Predrag and Artuso, Roberto and Mainieri, Ronnie and Tanner, Gregor and Vattay, Gábor , year =. Chaos:

  39. [39]

    Inflow. Annu. Rev. Fluid Mech. , author =. 2017 , pages =. doi:10.1146/annurev-fluid-010816-060322 , language =

    work page doi:10.1146/annurev-fluid-010816-060322 2017

  40. [40]

    Diffusion by a. Phys. Fluids , author =. 1970 , pages =. doi:10.1063/1.1692799 , number =

    work page doi:10.1063/1.1692799 1970

  41. [41]

    Flow Turbul

    Efficient. Flow Turbul. Combust. , author =. 2008 , keywords =. doi:10.1007/s10494-008-9151-5 , language =

    work page doi:10.1007/s10494-008-9151-5 2008

  42. [42]

    Random. J. Fluids Eng. , author =. 2001 , pages =. doi:10.1115/1.1369598 , number =

    work page doi:10.1115/1.1369598 2001

  43. [43]

    Dissipation in. Annu. Rev. Fluid Mech. , author =. 2015 , pages =. doi:10.1146/annurev-fluid-010814-014637 , language =

    work page doi:10.1146/annurev-fluid-010814-014637 2015

  44. [44]

    Generation of. J. Comput. Phys. , author =. 1998 , pages =. doi:10.1006/jcph.1998.5882 , language =

    arXiv 1998

  45. [45]

    A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations , volume =. J. Comput. Phys. , author =. 2003 , pages =. doi:10.1016/S0021-9991(03)00090-1 , language =

    work page doi:10.1016/s0021-9991(03)00090-1 2003

  46. [46]

    Indiana Univ

    Statistical. Indiana Univ. Math. J. , author =. 1952 , pages =. doi:10.1512/iumj.1952.1.51004 , language =

    work page doi:10.1512/iumj.1952.1.51004 1952

  47. [47]

    Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics , volume =. Proc. Natl. Acad. Sci. U.S.A. , author =. 2012 , pages =. doi:10.1073/pnas.1218747109 , language =

    work page doi:10.1073/pnas.1218747109 2012

  48. [48]

    Monin, A. S. and Yaglom, A. M. , year =. Statistical fluid mechanics: mechanics of turbulence , isbn =

  49. [49]

    A mathematical example displaying features of turbulence , volume =. Commun. Pure Appl. Math. , author =. 1948 , pages =. doi:10.1002/cpa.3160010401 , language =

    doi:10.1002/cpa.3160010401 1948

  50. [50]

    What. Prog. Theor. Phys. Suppl. , author =. 1978 , pages =. doi:10.1143/PTPS.64.339 , language =

    work page doi:10.1143/ptps.64.339 1978

  51. [51]

    Flow Turbul

    A. Flow Turbul. Combust. , author =. 2013 , keywords =. doi:10.1007/s10494-013-9488-2 , language =

    work page doi:10.1007/s10494-013-9488-2 2013

  52. [52]

    Adrian, R. J. , editor =. Stochastic. Eddy. 1996 , doi =

    1996

  53. [53]

    Stochastic estimation and proper orthogonal decomposition:. Exp. Fluids , author =. 1994 , keywords =. doi:10.1007/BF01874409 , language =

    work page doi:10.1007/bf01874409 1994

  54. [54]

    Aerodynamic. AIAA J. , author =. 2004 , pages =. doi:10.2514/1.2159 , number =

    doi:10.2514/1.2159 2004

  55. [55]

    Karhunen–. J. Opt. Soc. Am. A , author =. 1995 , pages =. doi:10.1364/JOSAA.12.001657 , language =

    doi:10.1364/josaa.12.001657 1995

  56. [56]

    AIAA AVIATION 2021 FORUM , year =

    Anna Bertram and Philipp Bekemeyer and Matthias Held , title =. AIAA AVIATION 2021 FORUM , year =. doi:10.2514/6.2021-2602 , URL =

    work page doi:10.2514/6.2021-2602 2021

  57. [57]

    What. J. Stat. Phys. , author =. 2002 , pages =. doi:10.1023/A:1019762724717 , language =

    doi:10.1023/a:1019762724717 2002

  58. [58]

    Deterministic. J. Atmos. Sci. , author =. 1963 , pages =. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 , language =

    doi:10.1175/1520-0469(1963)020 1963

  59. [59]

    Multi-scale reconstruction of turbulent rotating flows with proper orthogonal decomposition and generative adversarial networks , volume =. J. Fluid Mech. , author =. 2023 , pages =. doi:10.1017/jfm.2023.573 , language =

    work page doi:10.1017/jfm.2023.573 2023

  60. [60]

    The minimal flow unit in near-wall turbulence , volume =. J. Fluid Mech. , author =. 1991 , pages =. doi:10.1017/S0022112091002033 , language =

    doi:10.1017/s0022112091002033 1991

  61. [61]

    Villani, Cédric , year =. Optimal

  62. [62]

    Conditional flow matching for generative modelling of near-wall turbulence with quantified uncertainty , volume =. J. Fluid Mech. , author =. 2026 , pages =. doi:10.1017/jfm.2026.11193 , language =

    arXiv 2026

  63. [63]

    Super-resolution reconstruction of turbulent velocity fields using a generative adversarial network-based artificial intelligence framework , volume =. Phys. Fluids , author =. 2019 , pages =. doi:10.1063/1.5127031 , number =

    doi:10.1063/1.5127031 2019

  64. [64]

    From coarse wall measurements to turbulent velocity fields through deep learning , volume =. Phys. Fluids , author =. 2021 , pages =. doi:10.1063/5.0058346 , number =

    doi:10.1063/5.0058346 2021

  65. [65]

    ACM Trans. Graph. , author =. 2018 , pages =. doi:10.1145/3197517.3201304 , language =

    arXiv 2018

  66. [66]

    Deep unsupervised learning of turbulence for inflow generation at various. J. Comput. Phys. , author =. 2020 , keywords =. doi:10.1016/j.jcp.2019.109216 , urldate =

    arXiv 2020

  67. [67]

    Bayesian conditional diffusion models for versatile spatiotemporal turbulence generation , volume =. Comput. Methods Appl. Mech. Eng. , author =. 2024 , pages =. doi:10.1016/j.cma.2024.117023 , language =

    arXiv 2024

  68. [68]

    Generative learning for forecasting the dynamics of high-dimensional complex systems , volume =. Nat. Commun. , author =. 2024 , keywords =. doi:10.1038/s41467-024-53165-w , language =

    doi:10.1038/s41467-024-53165-w 2024

  69. [69]

    Generative adversarial networks , volume =. Commun. ACM , author =. 2020 , pages =. doi:10.1145/3422622 , number =

    doi:10.1145/3422622 2020

  70. [70]

    Gappy data and reconstruction procedures for flow past a cylinder , volume =. J. Fluid Mech. , author =. 2004 , pages =. doi:10.1017/S0022112004001338 , language =

    work page doi:10.1017/s0022112004001338 2004

  71. [71]

    Physica D: Nonlinear Phenomena , author =

    Chaotic dynamics of coherent structures , volume =. Physica D: Nonlinear Phenomena , author =. 1989 , pages =. doi:10.1016/0167-2789(89)90123-1 , language =

    work page doi:10.1016/0167-2789(89)90123-1 1989

  72. [72]

    Quadrant. Annu. Rev. Fluid Mech. , author =. 2016 , pages =. doi:10.1146/annurev-fluid-122414-034550 , language =

    work page doi:10.1146/annurev-fluid-122414-034550 2016

  73. [73]

    Turbulence generation and data assimilation in wall-bounded flows with a latent diffusion model

    Steinbrenner, Fabian and Turan, Baris and Teng, Hao and Xiao, Heng , year =. Turbulence generation and data assimilation in wall-bounded flows with a latent diffusion model , copyright =. doi:10.48550/arXiv.2603.02143 , language =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.02143

  74. [74]

    Amorós-Trepat, Marc and Medrano-Navarro, Luis and Liu, Qiang and Guastoni, Luca and Thuerey, Nils , title =. Phys. Fluids , volume =. 2026 , month =. doi:10.1063/5.0304492 , url =

    doi:10.1063/5.0304492 2026

  75. [75]

    Is plane-channel flow a friendly case for the testing of large-eddy simulation subgrid-scale models? , volume =. Phys. Fluids , author =. 2007 , pages =. doi:10.1063/1.2722422 , number =

    work page doi:10.1063/1.2722422 2007

  76. [76]

    The screening effect in. Ann. Stat. , author =. 2002 , pages =. doi:10.1214/aos/1015362194 , language =

    arXiv 2002

  77. [77]

    2010. Ann. Stat. , author =. 2011 , pages =

    2010

  78. [78]

    Geostatistics:

    Chilès, Jean-Paul and Delfiner, Pierre , year =. Geostatistics:

  79. [79]

    Liu, Liyuan and Jiang, Haoming and He, Pengcheng and Chen, Weizhu and Liu, Xiaodong and Gao, Jianfeng and Han, Jiawei , month = oct, year =. On the. doi:10.48550/arXiv.1908.03265 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.1908.03265 1908

  80. [80]

    and Maheswaranathan, Niru and Ganguli, Surya , month = nov, year =

    Sohl-Dickstein, Jascha and Weiss, Eric A. and Maheswaranathan, Niru and Ganguli, Surya , month = nov, year =. Deep. doi:10.48550/arXiv.1503.03585 , language =

    Pith/arXiv arXiv doi:10.48550/arxiv.1503.03585

Showing first 80 references.