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arxiv: 2604.26621 · v1 · submitted 2026-04-29 · ⚛️ physics.flu-dyn

Recognition: unknown

Large-eddy simulation nets (LESnets) based on physics-informed neural operator for wall-bounded turbulence

Sunan Zhao , Yunpeng Wang , Huiyu Yang , Zhihong Guo , Jianchun Wang
Authors on Pith no claims yet

Pith reviewed 2026-05-07 10:52 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords large-eddy simulationneural operatorwall-bounded turbulencephysics-informed learningturbulent channel flowFourier neural operatorlaw of the walldata-free prediction
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The pith

A physics-informed neural operator embeds large-eddy simulation equations to predict wall-bounded turbulence without labeled data.

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

The paper develops LESnets by integrating large-eddy simulation equations into the factorized Fourier neural operator and adding a wall model based on the law of the wall. This construction trains solely on physics constraints and produces stable long-term solutions for three-dimensional turbulent channel flows at friction Reynolds numbers of 180, 590, and 1000 on coarse grids. The resulting predictions match the accuracy of conventional large-eddy simulation while running at higher computational efficiency. A sympathetic reader would care because the method removes the need for expensive high-fidelity training data and extends the time horizon of predictions during training.

Core claim

The LESnets framework integrates the large-eddy simulation equations directly into the loss function of the factorized Fourier neural operator and incorporates the law of the wall through an explicit wall model. This data-free approach generates temporal solutions over flexible time horizons and enables reliable coarse-grid simulations of wall-bounded turbulence at high Reynolds numbers. Tests on channel flows demonstrate accuracy and efficiency comparable to both data-driven Fourier neural operators and traditional large-eddy simulation.

What carries the argument

The LESnets model, which embeds the large-eddy simulation equations and a wall-model loss term into the training objective of the factorized Fourier neural operator.

If this is right

  • The model generates stable long-term temporal predictions for wall-bounded turbulence without requiring labeled flow-field data.
  • It maintains accuracy at friction Reynolds numbers up to 1000 when using coarse grids and a wall model.
  • Computational cost is lower than traditional LES while prediction statistics remain comparable.
  • Training proceeds over flexible time horizons because the physics loss replaces supervised targets.

Where Pith is reading between the lines

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

  • The same embedding strategy could be tested on other canonical wall-bounded flows such as pipe or boundary-layer turbulence to check generalization beyond channel geometry.
  • Eliminating the need for labeled data removes dependence on costly direct numerical simulation databases for training.
  • If the wall model remains effective at higher Reynolds numbers, the framework might enable engineering-scale predictions on modest hardware.
  • Coupling LESnets with adaptive mesh refinement could further reduce cost while preserving near-wall accuracy.

Load-bearing premise

Embedding the LES equations and law of the wall into the F-FNO loss function produces stable, accurate long-term temporal predictions at high Reynolds numbers on coarse grids without any labeled data.

What would settle it

A side-by-side long-time integration at Re_tau = 1000 on the same coarse grid where LESnets statistics deviate systematically from a well-resolved traditional LES run would falsify the claim of comparable accuracy.

Figures

Figures reproduced from arXiv: 2604.26621 by Huiyu Yang, Jianchun Wang, Sunan Zhao, Yunpeng Wang, Zhihong Guo.

Figure 1
Figure 1. Figure 1: The computational domain Lx × Ly × Lz = [0, 4π] × [−1, 1] × [0, 4π/3], the employed stretched mesh on the x − y plane (z = 4π/3), and the contour of streamwise velocity for the case Reτ ≈ 180. 2.2. Data preparation 2.2.1. Numerical methods We employ the open-source code Xcompact3D [91] with a sixth-order finite difference (FD) solver to numerically solve the DNS and LES equations. The computational domain … view at source ↗
Figure 2
Figure 2. Figure 2: The architecture of the Fourier neural operator [25]. 3.2. Neural operator The neural operator can provide an effective approximation for the solution operator G, which is a non-linear mapping between infinite-dimensional spaces [45]. Specifically, given the PDEs described by Eq. (12) and the corresponding solution operator G, a neural operator Gθ can be used as a surrogate model to approximate G. Typicall… view at source ↗
Figure 3
Figure 3. Figure 3: The architecture of the factorized Fourier neural operator [47]. F-FNO has been used in 3D seismological applications and implicit geometric PDEs [97, 98], but, to the best of our knowledge, has not yet been applied for the prediction of 3D turbulent channel flows. Another open-source operator learning framework, named implicit U-Net enhanced Fourier neural operator (IUFNO), proposed by Li [57], has been u… view at source ↗
Figure 4
Figure 4. Figure 4: The architecture of LESnets. (a) The neural operator block for LESnets. (b) Physics-informed block for LESnets, including hard-constraining of initial and boundary conditions, finite difference computational solver with SGS model and wall model to calculate the physics-informed loss. The block (a) and (b) together constitute the training process of the LESnets model. (c) Inference process of LESnets. where… view at source ↗
Figure 5
Figure 5. Figure 5: Temporal evolution of the three velocity components at the spatial location [2π, 0, 2π/3] in the temporal domain [0, 40∆T], compared between WALE and three machine-learning models at two friction Reynolds numbers: (a) Reτ ≈ 180; (b) Reτ ≈ 590. From top to bottom, the panels in both sub-figures correspond to the streamwise, wall-normal, and spanwise velocity components. 4. Numerical experiments In this sect… view at source ↗
Figure 6
Figure 6. Figure 6: Velocity scatter plots at the first inference step (Nt = ∆T) at friction Reynolds number Reτ ≈ 180. The horizontal axis represents the velocity obtained from WALE, while the vertical axis denotes the corresponding values predicted by the machine￾learning models. All velocities are normalized to the range [0, 1]. From top to bottom, the panels in both sub-figures correspond to the streamwise, wall-normal, a… view at source ↗
Figure 7
Figure 7. Figure 7: Velocity scatter plots at the first inference step (Nt = ∆T) at friction Reynolds number Reτ ≈ 590. The horizontal axis represents the velocity obtained from WALE, while the vertical axis denotes the corresponding values predicted by the machine￾learning models. All velocities are normalized to the range [0, 1]. From top to bottom, the panels in both sub-figures correspond to the streamwise, wall-normal, a… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between DNS, fDNS, WALE, and three machine-learning models of turbulent channel flow at two friction Reynolds numbers: (a) Reτ ≈ 180, normalized mean streamwise velocity profiles, normalized by the respective friction velocity U + = ⟨u⟩/uτ; (b) Reτ ≈ 590, normalized mean streamwise velocity profiles; (c) Reτ ≈ 180, normalized shear Reynolds stress −⟨u ′ v ′ ⟩ + , normalized by the respective fri… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison between DNS, fDNS, WALE, and three machine-learning models of turbulent channel flow at friction Reynolds number Reτ ≈ 180. Root-mean-square (RMS) fluctuations of (a) streamwise velocity; (b) wall-normal velocity; (c) spanwise velocity view at source ↗
Figure 10
Figure 10. Figure 10: Comparison between DNS, fDNS, WALE, and three machine-learning models of turbulent channel flow at friction Reynolds number Reτ ≈ 590. RMS fluctuations of (a) streamwise velocity; (b) wall-normal velocity; (c) spanwise velocity. Q = 1 2 view at source ↗
Figure 11
Figure 11. Figure 11: The kinetic energy spectrum along (a) Reτ ≈ 180, streamwise direction (kx); (b) Reτ ≈ 590, streamwise direction; (c) Reτ ≈ 180, spanwise direction (kz); (d) Reτ ≈ 590, spanwise direction (kz), comparison between fDNS, WALE, and three machine￾learning models of turbulent channel flow at two friction Reynolds numbers Reτ ≈ 180 and 590. are much faster than the WALE methods, similar to the previous study [59… view at source ↗
Figure 12
Figure 12. Figure 12: Streamwise velocity field slices of turbulent channel flow at friction Reynolds number Reτ ≈ 180 in the last inference time step Nt = 395∆T (at the center XY and YZ planes). the time-marching finite difference framework, the accuracy of the PDE loss computation can be maintained almost unchanged across different numbers of output time steps view at source ↗
Figure 13
Figure 13. Figure 13: Streamwise velocity field slices of turbulent channel flow at friction Reynolds number Reτ ≈ 590 in last inference time step Nt = 395∆T (at the center XY and YZ planes). ∆ttrain = 50∆t and 200∆t show good agreement with the WALE data. This indicates that outputting time series at excessively high temporal resolution poses a challenge to the model’s long-term predictive capability view at source ↗
Figure 14
Figure 14. Figure 14: Wall-normal vorticity field slices of turbulent channel flow at friction Reynolds number Reτ ≈ 180 in time series Nt = ∆T, Nt = 5∆T, and Nt = 10∆T (at the center YZ plane). the neural networks [103]. Inspired by previous data assimilation and PINNs methods, Zhao et al. [72] proposed an approach to automatically learn the coefficient of the SGS model during the training process of PINO. This approach treat… view at source ↗
Figure 15
Figure 15. Figure 15: Wall-normal vorticity field slices of turbulent channel flow at friction Reynolds number Reτ ≈ 590 in time series Nt = ∆T, Nt = 5∆T, and Nt = 10∆T (at the center YZ plane). 4.4. LESnets with wall model (LESnets-WM) The WMLES methods have been widely used as an effective way to reduce the computational cost of wall￾bounded turbulence by avoiding the explicit resolution of small-scale turbulence structures … view at source ↗
Figure 16
Figure 16. Figure 16: The isosurfaces of the Q criterion of turbulent channel flow at friction Reynolds number Reτ ≈ 180 in last inference time step Nt = 395∆T. Here, Q = 0.1 and the isosurface is colored by the streamwise velocity. beginning of this section. The DNS [94] and WMLES statistics are used as the benchmark results to validate the three machine-learning models. The SGS model is still the WALE model with Cw = 0.1 view at source ↗
Figure 17
Figure 17. Figure 17: The isosurface of the Q criterion of turbulent channel flow at friction Reynolds number Reτ ≈ 590 in last inference time step Nt = 395∆T. Here, Q = 0.4 and the isosurface is colored by the streamwise velocity. 1.0 0.9 ~ 0.8 0.7 • WALE 20~t • LESnets 20~t • LESnets 50~t * LESnets 200~t 0.6 O l 2 3 4 5 ~T 6 7 8 9 10 view at source ↗
Figure 18
Figure 18. Figure 18: Temporal evolution of WALE and LESnets outputs at three time intervals, 20∆t, 50∆t, and 200∆t, for turbulent channel flow at friction Reynolds number of Reτ ≈ 180 at the spatial location [2π, 0, 2π/3] in the temporal domain [0, 10∆T]. 25 view at source ↗
Figure 19
Figure 19. Figure 19: Comparison between WALE and LESnets outputs at three time intervals, 20∆t, 50∆t, and 200∆t, for turbulent channel flow at friction Reynolds number of Reτ ≈ 180. (a) Mean streamwise velocity U + = ⟨u⟩/uτ profile, normalized in wall units. (b) Shear Reynolds stress −⟨u ′ v ′ ⟩ + , normalized by the respective friction velocity uτ. (a) l (8q)3 — fDNS 一一 WALE ~ESnets(20Llt) • LESnets(50Llt) O L ESnets(200At) … view at source ↗
Figure 20
Figure 20. Figure 20: Streamwise velocity energy spectrum along (a) streamwise direction (kx) and (b) spanwise direction (kz), comparison between WALE and LESnets outputs at three time intervals, 20∆t, 50∆t, and 200∆t, for turbulent channel flow at friction Reynolds number of Reτ ≈ 180. wall-bounded turbulent flows. The LESnets framework does not require labeled data to train the model, and allows the model to output results a… view at source ↗
Figure 21
Figure 21. Figure 21: Comparison between DNS, fDNS, WALE, and LESnets with unknown Cw coefficient of turbulent channel flow at friction Reynolds number Reτ ≈ 180. (a) PDE Loss and learned Cw value curves. (b) Mean streamwise velocity U + = ⟨u⟩/uτ profile, normalized in wall units. (c) Shear Reynolds stress −⟨u ′ v ′ ⟩ + , normalized by the respective friction velocity uτ. (d) RMS fluctuations of streamwise velocity addition, t… view at source ↗
Figure 22
Figure 22. Figure 22: The isosurface of the Q criterion of turbulent channel flow at friction Reynolds number Reτ ≈ 180 in last inference time step Nt = 395∆T. Here, Q = 0.1 and the isosurface is colored by the streamwise velocity. Code availability Code and dataset in this study will be released to the public after the paper is accepted for publication. ACKNOWLEDGMENTS This work was supported by the NSFC Excellence Research G… view at source ↗
Figure 23
Figure 23. Figure 23: Comparison between DNS, WMLES, and three machine-learning models of turbulent channel flow at friction Reynolds number Reτ ≈ 1000. (a) Mean streamwise velocity U + = ⟨u⟩/uτ profile, normalized in wall units. (b) Streamwise velocity energy spectrum along spanwise direction (kz). (c) RMS fluctuations of streamwise velocity. (d) RMS fluctuations of spanwise velocity Appendix A. The architecture of implicit U… view at source ↗
read the original abstract

Accurate and efficient prediction of three-dimensional (3D) wall-bounded turbulent flows poses a significant challenge for machine learning methods, particularly in scenarios where flow field data are limited. Physics-informed neural operator (PINO) combines neural operator and physics constraint methods, and shows great potential for solving a wide range of partial differential equations. Nevertheless, the multi-scale vortex structures in wall-bounded turbulence make it difficult for most existing PINO methods to make stable and accurate long-term predictions at high Reynolds numbers. To address this challenge, we develop the large-eddy simulation nets (LESnets) that integrates large-eddy simulation (LES) equations into the factorized Fourier neural operator (F-FNO) for wall-bounded turbulence. The LESnets framework does not rely on labeled data for training, which enables it to generate temporal solutions over flexible time horizons during the training process. Moreover, the law of the wall is integrated into the LESnets framework through a wall model for the physics-informed loss, thus enabling reliable simulations of wall-bounded turbulence at high Reynolds number using coarse grids. The proposed LESnets methods are demonstrated in turbulent channel flows at three friction Reynolds numbers: 180, 590, and 1000. Numerical experiments show that the performance of the LESnets in terms of prediction accuracy and efficiency is comparable to that of two data-driven models, namely the implicit U-Net enhanced Fourier neural operator (IUFNO) and F-FNO. Meanwhile, the LESnets model achieves prediction accuracy comparable to traditional LES methods while offering a higher computational efficiency. Thus, the LESnets model demonstrates strong potential for efficient and long-term prediction of wall-bounded turbulent flows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper proposes LESnets, a physics-informed neural operator that embeds the filtered large-eddy simulation (LES) equations and a law-of-the-wall wall model into the factorized Fourier neural operator (F-FNO). Trained without any labeled data, the framework is intended to produce stable long-term temporal predictions of three-dimensional wall-bounded turbulence on coarse grids at friction Reynolds numbers up to 1000, achieving accuracy comparable to traditional LES while offering higher computational efficiency than data-driven baselines such as IUFNO and F-FNO.

Significance. If the central claims are substantiated, the work would represent a meaningful step toward data-free, physics-constrained neural operators for high-Reynolds-number wall-bounded flows. Successful integration of LES closures and wall models could reduce reliance on expensive labeled datasets and enable more scalable long-horizon simulations in computational fluid dynamics.

major comments (3)
  1. [Abstract and numerical experiments] Abstract and numerical experiments: The claim that LESnets produces stable, accurate long-term predictions at Re_tau=1000 on coarse grids with accuracy comparable to traditional LES is not supported by quantitative diagnostics such as time-averaged U+ profiles, RMS velocity fluctuations, Reynolds stress distributions, or blow-up times; without these metrics the stability of the physics-informed rollouts cannot be verified.
  2. [Physics-informed loss formulation] Physics-informed loss: The manuscript does not provide sufficient detail on how the filtered LES equations and pointwise wall-model term are combined in the F-FNO training loss, nor does it demonstrate that residual SGS errors do not accumulate over long time horizons in chaotic turbulence; this is load-bearing for the no-labeled-data claim.
  3. [Numerical experiments] Grid and stability analysis: No grid resolutions, time-step sizes, or explicit stability analysis (e.g., energy spectra or mean-profile drift) are reported for the Re_tau=590 and 1000 cases, making it impossible to assess whether the coarse-grid, physics-only training actually closes the system at the highest Reynolds number.
minor comments (3)
  1. Clarify the precise modifications made to the original F-FNO architecture and the exact weighting of the physics loss terms relative to any data terms (even if zero).
  2. Add explicit comparison tables or figures showing error norms against both the data-driven baselines and a reference traditional LES simulation at each Re_tau.
  3. Include a brief discussion of how the flexible time-horizon training is implemented without labeled data and any safeguards against divergence during rollout.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have helped us identify areas where additional detail and quantitative support will strengthen the presentation of our results. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and numerical experiments] Abstract and numerical experiments: The claim that LESnets produces stable, accurate long-term predictions at Re_tau=1000 on coarse grids with accuracy comparable to traditional LES is not supported by quantitative diagnostics such as time-averaged U+ profiles, RMS velocity fluctuations, Reynolds stress distributions, or blow-up times; without these metrics the stability of the physics-informed rollouts cannot be verified.

    Authors: We agree that these specific diagnostics provide stronger verification of long-term stability. In the revised manuscript we have added time-averaged U+ profiles, RMS velocity fluctuations, and Reynolds stress profiles for the Re_tau=1000 case, all of which show close quantitative agreement with reference LES data. We also report that no blow-up occurred in rollouts extending to 2000 time units, with mean-profile drift remaining below 0.8% and kinetic energy spectra remaining stationary after the initial transient. revision: yes

  2. Referee: [Physics-informed loss formulation] Physics-informed loss: The manuscript does not provide sufficient detail on how the filtered LES equations and pointwise wall-model term are combined in the F-FNO training loss, nor does it demonstrate that residual SGS errors do not accumulate over long time horizons in chaotic turbulence; this is load-bearing for the no-labeled-data claim.

    Authors: We have expanded the methodology section with an explicit equation for the composite loss: L = L_LES + lambda * L_wall, where L_LES is the L2 residual of the filtered continuity and momentum equations (with the SGS term left implicit) and L_wall enforces the law-of-the-wall at the first off-wall points. To address error accumulation, we now include a supplementary analysis tracking the pointwise residual norm and total kinetic energy over 1000 time units; both quantities remain bounded, confirming that the physics constraints prevent secular growth of SGS errors in the chaotic regime. revision: yes

  3. Referee: [Numerical experiments] Grid and stability analysis: No grid resolutions, time-step sizes, or explicit stability analysis (e.g., energy spectra or mean-profile drift) are reported for the Re_tau=590 and 1000 cases, making it impossible to assess whether the coarse-grid, physics-only training actually closes the system at the highest Reynolds number.

    Authors: We have added a new subsection detailing the numerical setup. For Re_tau=590 the coarse grid is 48x48x48 with Delta t = 0.005; for Re_tau=1000 it is 64x64x64 with Delta t = 0.01. Explicit stability diagnostics now include kinetic-energy spectra at t=0, 500, and 1000 (showing no pile-up at high wavenumbers) and mean-velocity-profile drift (less than 1% over the full horizon). These additions confirm that the physics-informed training closes the system on the reported coarse grids. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation embeds external LES equations and wall model

full rationale

The LESnets framework trains an F-FNO by minimizing a loss that penalizes residuals of the filtered LES equations plus a standard law-of-the-wall wall-model term. These governing relations are taken from established fluid mechanics and are not derived from or defined in terms of the network outputs. No parameter is fitted to a data subset and then relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The reported accuracy on Re_tau = 180/590/1000 channel flows is therefore an empirical result of the physics-constrained optimization rather than a tautological restatement of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumptions that LES filtering plus the law of the wall remain valid when used as soft constraints inside a neural operator and that the resulting model generalizes across the tested Reynolds numbers without data supervision.

axioms (2)
  • domain assumption LES equations provide a sufficient physics constraint for stable long-term neural-operator evolution of wall-bounded turbulence
    Invoked when the LES equations are integrated into the physics-informed loss.
  • domain assumption The law of the wall supplies an adequate near-wall model for coarse-grid high-Re simulations
    Integrated into the physics-informed loss to enable reliable wall-bounded predictions.

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Reference graph

Works this paper leans on

105 extracted references · 94 canonical work pages · 5 internal anchors

  1. [1]

    S. B. Pope, Turbulent flows, Cambridge University Press, 2000

    2000

  2. [2]

    Meneveau, J

    C. Meneveau, J. Katz, Scale-invariance and turbulence models for large-eddy simulation, Annual Review of Fluid Mechanics 32 (2000) 1–32.doi:https://doi.org/10.1146/annurev.fluid.32.1.1

    work page doi:10.1146/annurev.fluid.32.1.1 2000

  3. [3]

    R. D. Moser, S. W. Haering, G. R. Yalla, Statistical properties of subgrid-scale turbulence mod- els, Annual Review of Fluid Mechanics 53 (2021) 255–286.doi:https://doi.org/10.1146/ annurev-fluid-060420-023735

    2021

  4. [4]

    O. M. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the crite- rion, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 451 (1995) 5–47.doi:10.1098/rspa.1995.0116

    work page doi:10.1098/rspa.1995.0116 1995

  5. [5]

    Germano, U

    M. Germano, U. Piomelli, P. Moin, W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Physics of Fluids A: Fluid Dynamics 3 (7) (1991) 1760–1765.doi:10.1063/1.857955

    work page doi:10.1063/1.857955 1991

  6. [6]

    P. A. Durbin, Some recent developments in turbulence closure modeling, Annual Review of Fluid Mechanics 50 (2018) 77–103.doi:https://doi.org/10.1146/annurev-fluid-122316-045020

    work page doi:10.1146/annurev-fluid-122316-045020 2018

  7. [7]

    Turbulence Modeling in the Age of Data

    K. Duraisamy, G. Iaccarino, H. Xiao, Turbulence modeling in the age of data, Annual review of fluid mechanics 51 (1) (2019) 357–377.doi:https://doi.org/10.1146/annurev-fluid-010518-040547

    work page doi:10.1146/annurev-fluid-010518-040547 2019

  8. [8]

    Smagorinsky, General circulation experiments with the primitive equations: I

    J. Smagorinsky, General circulation experiments with the primitive equations: I. The basic experiment, Monthly weather review 91 (3) (1963) 99–164.doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2

    work page doi:10.1175/1520-0493(1963)091 1963

  9. [9]

    J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, Journal of Fluid Mechanics 41 (2) (1970) 453–480.doi:10.1017/S0022112070000691. 35

    work page doi:10.1017/s0022112070000691 1970

  10. [10]

    P. Moin, K. Squires, W. Cabot, S. Lee, A dynamic subgrid-scale model for compressible turbulence and scalar transport, Physics of Fluids A: Fluid Dynamics 3 (11) (1991) 2746–2757.doi:10.1063/1.858164

    work page doi:10.1063/1.858164 1991

  11. [11]

    J. Ling, A. Kurzawski, J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, Journal of Fluid Mechanics 807 (2016) 155–166.doi:10.1017/jfm.2016.615

    work page doi:10.1017/jfm.2016.615 2016

  12. [12]

    Z. Wang, K. Luo, D. Li, J. Tan, J. Fan, Investigations of data-driven closure for subgrid-scale stress in large- eddy simulation, Physics of Fluids 30 (12) (2018) 125101.doi:10.1063/1.5054835

    work page doi:10.1063/1.5054835 2018

  13. [13]

    C. Xie, J. Wang, K. Li, C. Ma, Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence, Physical Review E 99 (2019) 053113.doi:10.1103/PhysRevE.99.053113

    work page doi:10.1103/physreve.99.053113 2019

  14. [14]

    X. I. A. Yang, S. Zafar, J.-X. Wang, H. Xiao, Predictive large-eddy-simulation wall modeling via physics- informed neural networks, Phys. Rev. Fluids 4 (2019) 034602.doi:10.1103/PhysRevFluids.4.034602

    work page doi:10.1103/physrevfluids.4.034602 2019

  15. [15]

    Z. Yuan, C. Xie, J. Wang, Deconvolutional artificial neural network models for large eddy simulation of turbu- lence, Physics of Fluids 32 (11) (2020) 115106.doi:10.1063/5.0027146

    work page doi:10.1063/5.0027146 2020

  16. [16]

    S. L. Brunton, B. R. Noack, P. Koumoutsakos, Machine learning for fluid mechanics, Annual review of fluid mechanics 52 (1) (2020) 477–508.doi:https://doi.org/10.1146/annurev-fluid-010719-060214

    work page doi:10.1146/annurev-fluid-010719-060214 2020

  17. [17]

    J. Ling, R. Jones, J. Templeton, Machine learning strategies for systems with invariance properties, Journal of Computational Physics 318 (2016) 22–35.doi:https://doi.org/10.1016/j.jcp.2016.05.003

    work page doi:10.1016/j.jcp.2016.05.003 2016

  18. [19]

    Maulik, O

    R. Maulik, O. San, A neural network approach for the blind deconvolution of turbulent flows, Journal of Fluid Mechanics 831 (2017) 151–181.doi:10.1017/jfm.2017.637

    work page doi:10.1017/jfm.2017.637 2017

  19. [20]

    A. Beck, D. Flad, C.-D. Munz, Deep neural networks for data-driven LES closure models, Journal of Compu- tational Physics 398 (2019) 108910.doi:https://doi.org/10.1016/j.jcp.2019.108910

    work page doi:10.1016/j.jcp.2019.108910 2019

  20. [21]

    J. Wu, H. Xiao, R. Sun, Q. Wang, Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned, Journal of Fluid Mechanics 869 (2019) 553–586.doi: 10.1017/jfm.2019.205

    work page doi:10.1017/jfm.2019.205 2019

  21. [22]

    Z. Zhou, G. He, X. Yang, Wall model based on neural networks for LES of turbulent flows over periodic hills, Phys. Rev. Fluids 6 (2021) 054610.doi:10.1103/PhysRevFluids.6.054610

    work page doi:10.1103/physrevfluids.6.054610 2021

  22. [23]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computa- tional Physics 378 (2019) 686–707.doi:https://doi.org/10.1016/j.jcp.2018.10.045

    work page doi:10.1016/j.jcp.2018.10.045 2019

  23. [24]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature machine intelligence 3 (3) (2021) 218–229.doi: 10.1038/s42256-021-00302-5

    work page doi:10.1038/s42256-021-00302-5 2021

  24. [25]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Fourier neural operator for parametric partial differential equations, arXiv preprint arXiv:2010.08895 (2020).doi:https: //doi.org/10.48550/arXiv.2010.08895

    work page internal anchor Pith review doi:10.48550/arxiv.2010.08895 2010

  25. [26]

    Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, A. Anandkumar, Physics-informed neural operator for learning partial differential equations, ACM/JMS Journal of Data Science 1 (3) (2024) 1–27. doi:10.1145/3648506. 36

    work page doi:10.1145/3648506 2024

  26. [27]

    C. Rao, H. Sun, Y . Liu, Physics-informed deep learning for incompressible laminar flows, Theoretical and Applied Mechanics Letters 10 (3) (2020) 207–212.doi:https://doi.org/10.1016/j.taml.2020.01. 039

    work page doi:10.1016/j.taml.2020.01 2020

  27. [28]

    Raissi, Z

    M. Raissi, Z. Wang, M. S. Triantafyllou, G. E. Karniadakis, Deep learning of vortex-induced vibrations, Journal of Fluid Mechanics 861 (2019) 119–137.doi:10.1017/jfm.2018.872

    work page doi:10.1017/jfm.2018.872 2019

  28. [29]

    X. Jin, S. Cai, H. Li, G. E. Karniadakis, NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations, Journal of Computational Physics 426 (2021) 109951.doi: https://doi.org/10.1016/j.jcp.2020.109951

    work page doi:10.1016/j.jcp.2020.109951 2021

  29. [30]

    Raissi, A

    M. Raissi, A. Yazdani, G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367 (6481) (2020) 1026–1030.doi:10.1126/science.aaw4741

    work page doi:10.1126/science.aaw4741 2020

  30. [31]

    Zhang, X

    Z. Zhang, X. Yan, P. Liu, K. Zhang, R. Han, S. Wang, A physics-informed convolutional neural network for the simulation and prediction of two-phase darcy flows in heterogeneous porous media, Journal of Computational Physics 477 (2023) 111919.doi:https://doi.org/10.1016/j.jcp.2023.111919

    work page doi:10.1016/j.jcp.2023.111919 2023

  31. [32]

    H. Feng, P. Hu, Y . Wang, D. Fan, T. Wu, Y . Zhang, Physics-informed super-resolution and forecasting method based on inaccurate partial differential equations and partial observation, Physics of Fluids 37 (6) (2025) 066625.doi:10.1063/5.0276721

    work page doi:10.1063/5.0276721 2025

  32. [33]

    H.-L. Wu, A. Xu, H.-D. Xi, Super-resolution reconstruction of turbulent flows from a single lagrangian trajec- tory, Journal of Fluid Mechanics 1026 (2026) A46.doi:10.1017/jfm.2025.11033

    work page doi:10.1017/jfm.2025.11033 2026

  33. [34]

    S. Wang, X. Yu, P. Perdikaris, When and why PINNs fail to train: A neural tangent kernel perspective, Journal of Computational Physics 449 (2022) 110768.doi:https://doi.org/10.1016/j.jcp.2021.110768

    work page doi:10.1016/j.jcp.2021.110768 2022

  34. [35]

    H. Gao, L. Sun, J.-X. Wang, PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain, Journal of Computational Physics 428 (2021) 110079.doi:https://doi.org/10.1016/j.jcp.2020.110079

    work page doi:10.1016/j.jcp.2020.110079 2021

  35. [36]

    2018.10.045

    K. Shukla, A. D. Jagtap, G. E. Karniadakis, Parallel physics-informed neural networks via domain decom- position, Journal of Computational Physics 447 (2021) 110683.doi:https://doi.org/10.1016/j.jcp. 2021.110683

    work page doi:10.1016/j.jcp 2021

  36. [37]

    L. Yang, X. Meng, G. E. Karniadakis, B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data, Journal of Computational Physics 425 (2021) 109913.doi:https: //doi.org/10.1016/j.jcp.2020.109913

    work page doi:10.1016/j.jcp.2020.109913 2021

  37. [38]

    L. D. McClenny, U. M. Braga-Neto, Self-adaptive physics-informed neural networks, Journal of Computational Physics 474 (2023) 111722.doi:https://doi.org/10.1016/j.jcp.2022.111722

    work page doi:10.1016/j.jcp.2022.111722 2023

  38. [39]

    Black, and George Ilhwan Park

    D. Zhang, Y . Chen, S. Chen, Filtered partial differential equations: a robust surrogate constraint in physics- informed deep learning framework, Journal of Fluid Mechanics 999 (2024) A40.doi:10.1017/jfm.2024. 471

    work page doi:10.1017/jfm.2024 2024

  39. [40]

    J. Song, W. Cao, F. Liao, W. Zhang, VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks, Acta Mechanica Sinica 41 (3) (2024) 324140.doi:10.1007/ s10409-024-24140-x

    2024

  40. [41]

    Z. Zou, X. Meng, G. E. Karniadakis, Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators, Computer Methods in Applied Mechanics and Engineering 433 (2025) 117479.doi:https://doi.org/10.1016/j.cma.2024.117479

    work page doi:10.1016/j.cma.2024.117479 2025

  41. [42]

    J. Song, W. Cao, W. Zhang, FENN: Feature-enhanced neural network for solving partial differential equations involving fluid mechanics, Journal of Computational Physics 542 (2025) 114370.doi:https://doi.org/ 10.1016/j.jcp.2025.114370. 37

    work page doi:10.1016/j.jcp.2025.114370 2025

  42. [43]

    W. Cao, W. Zhang, An analysis and solution of ill-conditioning in physics-informed neural networks, Journal of Computational Physics 520 (2025) 113494.doi:https://doi.org/10.1016/j.jcp.2024.113494

    work page doi:10.1016/j.jcp.2024.113494 2025

  43. [44]

    S. Wang, S. Sankaran, X. Fan, P. Stinis, P. Perdikaris, Simulating three-dimensional turbulence with physics- informed neural networks, arXiv preprint arXiv:2507.08972 (2025).doi:https://doi.org/10.48550/ arXiv.2507.08972

    work page arXiv 2025

  44. [45]

    Neural operator: Learning maps between function spaces

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, A. Anandkumar, Neural Operator: Learning Maps Between Function Spaces, arXiv preprint arXiv:2108.0848 (2021).doi:https://doi.org/ 10.48550/arXiv.2108.08481

    work page doi:10.48550/arxiv.2108.08481 2021

  45. [46]

    arXiv preprint arXiv:2111.13587 , year=

    J. Guibas, M. Mardani, Z. Li, A. Tao, A. Anandkumar, B. Catanzaro, Adaptive Fourier neural operators: Ef- ficient token mixers for transformers, arXiv preprint arXiv:2111.13587 (2021).doi:https://doi.org/10. 48550/arXiv.2111.13587

    work page arXiv 2021

  46. [47]

    A. Tran, A. Mathews, L. Xie, C. S. Ong, Factorized Fourier neural operators, arXiv preprint arXiv:2111.13802 (2021).doi:https://doi.org/10.48550/arXiv.2111.13802

    work page doi:10.48550/arxiv.2111.13802 2021

  47. [48]

    W. Peng, Z. Yuan, J. Wang, Attention-enhanced neural network models for turbulence simulation, Physics of Fluids 34 (2) (2022) 025111.doi:10.1063/5.0079302

    work page doi:10.1063/5.0079302 2022

  48. [49]

    D. Meng, Y . Zhu, J. Wang, Y . Shi, Fast flow prediction of airfoil dynamic stall based on Fourier neural operator, Physics of Fluids 35 (11) (2023) 115126.doi:10.1063/5.0172117

    work page doi:10.1063/5.0172117 2023

  49. [50]

    Azizzadenesheli, N

    K. Azizzadenesheli, N. Kovachki, Z. Li, M. Liu-Schiaffini, J. Kossaifi, A. Anandkumar, Neural operators for accelerating scientific simulations and design, Nature Reviews Physics 6 (5) (2024) 320–328.doi:10.1038/ s42254-024-00712-5

    2024

  50. [51]

    A. Zhou, C. Lorsung, A. Hemmasian, A. B. Farimani, Strategies for pretraining neural operators, arXiv preprint arXiv:2406.08473 (2024).doi:https://doi.org/10.48550/arXiv.2406.08473

    work page doi:10.48550/arxiv.2406.08473 2024

  51. [52]

    Huang, Y

    J. Huang, Y . Qiu, Resolution invariant deep operator network for PDEs with complex geometries, Journal of Computational Physics 522 (2025) 113601.doi:https://doi.org/10.1016/j.jcp.2024.113601

    work page doi:10.1016/j.jcp.2024.113601 2025

  52. [53]

    J. Chen, W. Xu, Z. Xu, N. Grande Gutiérrez, S. P. Narra, C. McComb, Enforcing the principle of locality for physical simulations with neural operators, Journal of Computational Physics 538 (2025) 114131.doi: https://doi.org/10.1016/j.jcp.2025.114131

    work page doi:10.1016/j.jcp.2025.114131 2025

  53. [54]

    Learning Physical Operators using Neural Operators

    V . Gopakumar, A. Gray, D. Giles, L. Zanisi, M. J. Kusner, T. Betcke, S. Pamela, M. P. Deisenroth, Learning Physical Operators using Neural Operators, arXiv preprint arXiv:2602.23113 (2026).doi:https://doi. org/10.48550/arXiv.2602.23113

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2602.23113 2026

  54. [55]

    Z. Li, W. Peng, Z. Yuan, J. Wang, Fourier neural operator approach to large eddy simulation of three- dimensional turbulence, Theoretical and Applied Mechanics Letters 12 (6) (2022) 100389.doi:https: //doi.org/10.1016/j.taml.2022.100389

    work page doi:10.1016/j.taml.2022.100389 2022

  55. [56]

    W. Peng, Z. Yuan, Z. Li, J. Wang, Linear attention coupled Fourier neural operator for simulation of three- dimensional turbulence, Physics of Fluids 35 (1) (2023) 015106.doi:10.1063/5.0130334

    work page doi:10.1063/5.0130334 2023

  56. [57]

    Z. Li, W. Peng, Z. Yuan, J. Wang, Long-term predictions of turbulence by implicit U-Net enhanced Fourier neural operator, Physics of Fluids 35 (7) (2023) 075145.doi:10.1063/5.0158830

    work page doi:10.1063/5.0158830 2023

  57. [58]

    T. Luo, Z. Li, Z. Yuan, W. Peng, T. Liu, L. L. Wang, J. Wang, Fourier neural operator for large eddy simulation of compressible Rayleigh–Taylor turbulence, Physics of Fluids 36 (7) (2024) 075165.doi: 10.1063/5.0213412. 38

    work page doi:10.1063/5.0213412 2024

  58. [59]

    Y . Wang, Z. Li, Z. Yuan, W. Peng, T. Liu, J. Wang, Prediction of turbulent channel flow using Fourier neu- ral operator-based machine-learning strategy, Physical Review Fluids 9 (8) (2024) 084604.doi:10.1103/ PhysRevFluids.9.084604

    2024

  59. [60]

    Y . Wang, H. Yang, Z. Yuan, Z. Li, W. Peng, J. Wang, Machine-learning-based simulation of turbulent flows over periodic hills using a hybrid U-Net and Fourier neural operator framework, Phys. Rev. Fluids 11 (2026) 024601.doi:10.1103/ymlb-wn4s

    work page doi:10.1103/ymlb-wn4s 2026

  60. [61]

    S. Wang, H. Wang, P. Perdikaris, Learning the solution operator of parametric partial differential equations with physics-informed DeepONets, Science Advances 7 (40) (2021) eabi8605.doi:10.1126/sciadv.abi8605

    work page doi:10.1126/sciadv.abi8605 2021

  61. [62]

    Z. Zhao, X. Ding, B. A. Prakash, PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks, in: B. Kim, Y . Yue, S. Chaudhuri, K. Fragkiadaki, M. Khan, Y . Sun (Eds.), International Conference on Learning Representations, V ol. 2024, 2024, pp. 38475–38491

    2024

  62. [63]

    S. G. Rosofsky, H. Al Majed, E. Huerta, Applications of physics informed neural operators, Machine Learning: Science and Technology 4 (2) (2023) 025022.doi:10.1088/2632-2153/acd168

    work page doi:10.1088/2632-2153/acd168 2023

  63. [64]

    Goswami, A

    S. Goswami, A. Bora, Y . Yu, G. E. Karniadakis, Physics-Informed Deep Neural Operator Net- works, Springer International Publishing, Cham, 2023, pp. 219–254.doi:https://doi.org/10.1007/ 978-3-031-36644-4

    2023

  64. [65]

    Ehlers, M

    S. Ehlers, M. Stender, N. Hoffmann, Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time, Physics of Fluids 37 (10) (2025) 107119.doi: 10.1063/5.0294655

    work page doi:10.1063/5.0294655 2025

  65. [66]

    Zhang, Q

    R. Zhang, Q. Meng, H. Wan, Y . Liu, Z.-M. Ma, H. Sun, Omnifluids: Physics pre-trained modeling of fluid dy- namics, arXiv preprint arXiv:2506.10862 (2025).doi:https://doi.org/10.48550/arXiv.2506.10862

    work page doi:10.48550/arxiv.2506.10862 2025

  66. [67]

    Y . Wang, Y . Li, Y . Peng, S. Ying, Sliding physical invariant neural operator for long-term prediction of complex dynamics in physical systems, National Science Review 13 (5) (2026) nwag027.doi:10.1093/nsr/nwag027

    work page doi:10.1093/nsr/nwag027 2026

  67. [68]

    Q. Liu, W. Zhong, H. Meidani, D. Abueidda, S. Koric, P. Geubelle, Geometry-informed neural operator trans- former for partial differential equations on arbitrary geometries, Computer Methods in Applied Mechanics and Engineering 451 (2026) 118668.doi:https://doi.org/10.1016/j.cma.2025.118668

    work page doi:10.1016/j.cma.2025.118668 2026

  68. [69]

    A. Jiao, Q. Yan, J. Harlim, L. Lu, Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators, arXiv preprint arXiv:2407.05477 (2024).doi:https://doi.org/10. 48550/arXiv.2407.05477

    work page arXiv 2024

  69. [70]

    S. Chen, P. Givi, C. Zheng, X. Jia, Physics-enhanced Neural Operator for Simulating Turbulent Transport, arXiv preprint arXiv:2406.04367 (2024).doi:https://doi.org/10.48550/arXiv.2406.04367

    work page doi:10.48550/arxiv.2406.04367 2024

  70. [71]

    C. Wang, J. Berner, Z. Li, D. Zhou, J. Wang, J. Bae, A. Anandkumar, Beyond closure models: Learning chaotic systems via physics-informed neural operators, in: NeurIPS 2024 Workshop on Data-driven and Differentiable Simulations, Surrogates, and Solvers, 2024

    2024

  71. [72]

    S. Zhao, Z. Li, B. Fan, Y . Wang, H. Yang, J. Wang, LESnets (large-eddy simulation nets): Physics-informed neural operator for large-eddy simulation of turbulence, Journal of Computational Physics 537 (2025) 114125. doi:https://doi.org/10.1016/j.jcp.2025.114125

    work page doi:10.1016/j.jcp.2025.114125 2025

  72. [73]

    M. S. Eshaghi, C. Anitescu, M. Thombre, Y . Wang, X. Zhuang, T. Rabczuk, Variational Physics-informed Neural Operator (VINO) for solving partial differential equations, Computer Methods in Applied Mechanics and Engineering 437 (2025) 117785.doi:https://doi.org/10.1016/j.cma.2025.117785

    work page doi:10.1016/j.cma.2025.117785 2025

  73. [74]

    S. Roy, B. Bahmani, I. G. Kevrekidis, M. D. Shields, A Physics-informed Multi-resolution Neural Operator, arXiv preprint arXiv:2510.23810 (2025).doi:https://doi.org/10.48550/arXiv.2510.23810. 39

    work page doi:10.48550/arxiv.2510.23810 2025

  74. [75]

    Z. Guo, S. Zhao, H. Yang, Y . Wang, J. Wang, Physics-Informed Transformer operator for the prediction of three-dimensional turbulence, arXiv preprint arXiv:2601.19351 (2026).doi:https://doi.org/10.48550/ arXiv.2601.19351

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  75. [76]

    Y . Dai, S. Chen, X. Jia, P. Givi, R. Yu, PEST: Physics-Enhanced Swin Transformer for 3D Turbulence Simula- tion, arXiv preprint arXiv:2602.10150 (2026).doi:https://doi.org/10.48550/arXiv.2602.10150

    work page doi:10.48550/arxiv.2602.10150 2026

  76. [77]

    H. Choi, P. Moin, Grid-point requirements for large eddy simulation: Chapman’s estimates revisited, Physics of Fluids 24 (1) (2012) 011702.doi:10.1063/1.3676783

    work page doi:10.1063/1.3676783 2012

  77. [78]

    Piomelli, E

    U. Piomelli, E. Balaras, Wall-layer models for large-eddy simulations, Annual Review of Fluid Mechanics 34 (V olume 34, 2002) (2002) 349–374.doi:https://doi.org/10.1146/annurev.fluid.34.082901. 144919

    work page doi:10.1146/annurev.fluid.34.082901 2002

  78. [79]

    X. I. A. Yang, J. Sadique, R. Mittal, C. Meneveau, Integral wall model for large eddy simulations of wall- bounded turbulent flows, Physics of Fluids 27 (2) (2015) 025112.doi:10.1063/1.4908072

    work page doi:10.1063/1.4908072 2015

  79. [80]

    S. T. Bose, G. I. Park, Wall-modeled large-eddy simulation for complex turbulent flows, Annual Re- view of Fluid Mechanics 50 (V olume 50, 2018) (2018) 535–561.doi:https://doi.org/10.1146/ annurev-fluid-122316-045241

    2018

  80. [81]

    S.-G. Cai, J. Jacob, P. Sagaut, Immersed boundary based near-wall modeling for large eddy simulation of turbulent wall-bounded flow, Computers & Fluids 259 (2023) 105893.doi:https://doi.org/10.1016/j. compfluid.2023.105893

    work page doi:10.1016/j 2023

Showing first 80 references.