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arxiv: 2604.23937 · v2 · pith:LYOY4KQKnew · submitted 2026-04-27 · ⚛️ physics.flu-dyn · cs.LG

Multi-scale Dynamic Wake Modeling and Prediction of Floating Offshore Wind Turbines via Physics-Informed Neural Networks and Fourier Neural Operators

Guodan Dong , Jianhua Qin , Chang Xu This is my paper

Pith reviewed 2026-05-21 09:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LG
keywords floating offshore wind turbineswake meanderingFourier neural operatorsphysics-informed neural networkslarge-eddy simulationmulti-scale modelingStrouhal number
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The pith

Fourier neural operators predict floating wind turbine wakes more efficiently and with greater structural fidelity than physics-informed networks.

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

This paper establishes that Fourier neural operators offer a superior method for modeling the dynamic, multi-scale wakes generated by floating offshore wind turbines undergoing coupled surge and pitch motions. These motions cause the wake to meander, which impacts power production and structural loads in wind farms. A sympathetic reader would care because real-time, accurate wake predictions could enable better control strategies for these turbines, improving efficiency and reducing costs in offshore wind energy. The study trains both PINNs and FNOs on data from high-fidelity large-eddy simulations and shows FNOs achieve faster computation and better long-term accuracy.

Core claim

The central discovery is that while both physics-informed neural networks and Fourier neural operators can model the dominant large-scale meandering structures in FOWT wakes, the FNO framework provides an 8-fold speedup in computation, 40-fold faster convergence, superior long-term predictive capability, and better fidelity in capturing multi-scale coherent structures including higher-order harmonics of the meandering frequency and the energy cascade.

What carries the argument

Fourier neural operators, which learn the integral operator in Fourier space to map the turbine motion parameters to the time-evolving wake flow fields.

If this is right

  • Real-time modeling becomes feasible for operational control of floating wind turbines.
  • High-frequency turbulent fluctuations and coherent structures are better resolved, leading to more accurate load predictions.
  • The primary meandering frequency and its harmonics are preserved with energy levels matching high-fidelity simulations.
  • The energy cascade in the wake is sustained rather than dissipating rapidly at high frequencies.

Where Pith is reading between the lines

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

  • These models could be extended to predict wakes under more complex environmental conditions like waves and currents.
  • Combining FNOs with existing control algorithms might optimize power output in floating wind farms.
  • The approach may apply to other fluid-structure interaction problems involving moving bodies.

Load-bearing premise

The large-eddy simulations with actuator line model provide an accurate representation of the wake meandering caused by the turbine's surge and pitch motions.

What would settle it

If new simulations or field measurements show that the FNO-predicted wake spectra deviate significantly from observations at Strouhal numbers above 1.0 over extended time periods, the claimed long-term predictive advantage would be falsified.

Figures

Figures reproduced from arXiv: 2604.23937 by Chang Xu, Guodan Dong, Jianhua Qin.

Figure 1
Figure 1. Figure 1: FIG. 1: Computational setup. (a) the computational domain and (b) the detailed 3 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The PINNs architecture used in the present work view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The FNO architecture used in the present work view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Learning loss for PINN view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Learning loss for FNO view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of the reconstruction capability of PINNs with the original wake view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Comparison of the reconstruction capability of FNO with the original wake view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of the prediction capability of PINNs with the original wake obtained view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison of the prediction capability of FNO with the original wake obtained view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: MAE of PINNs and FNO for the streamwise ( view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: RMSE of PINNs and FNO for the streamwise ( view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Comparison of line profiles for the streamwise velocity ( view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Comparison of temporal evolution of the streamwise velocity ( view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Comparison of fitted wake center ( view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Comparison of fitted wake half-width ( view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Comparison of fitted velocity deficit (∆ view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Comparison of standard deviation of the fitted wake center ( view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Comparison of the standard deviation of the fitted wake half-width ( view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Comparison of Power Spectral Density (PSD) among CFD (ground truth), FNO, view at source ↗
read the original abstract

Multi-scale dynamic wake modeling and prediction are essential for the real-time control and optimization of floating offshore wind turbines (FOWTs). In this study, wakes of FOWTs under coupled surge and pitch motions across a range of Strouhal numbers (St), which can induce wake meandering, are modeled via two novel deep-learning frameworks: physics-informed neural networks (PINNs) and Fourier neural operators (FNOs). The high-fidelity dataset is obtained from large-eddy simulations with the actuator line model (LES-AL). The results demonstrate that the dominant large-scale dynamic structures, such as meandering, can be well modeled by both frameworks; however, FNOs exhibit significant advantages over the PINN model in terms of efficiency (8-fold computational speedup and 40-fold faster convergence), long-term predictive capability, and multi-scale coherent structural fidelity. Furthermore, the wakes predicted by the PINN model exhibit a smoothing effect that limits the resolution of high-frequency coherent structures and underestimates turbulent fluctuations in both the wake center and half-width. Spectral analysis reveals that FNOs resolve the primary meandering frequency (where Stp denotes the frequency induced by the coupled surge and pitch motions), its corresponding higher-order harmonics (2Stp, 3Stp), and the energy cascade. In contrast, the energy cascade in the PINN predictions dissipates more rapidly in the high-frequency regime (St > 1.0). Additionally, the pre-multiplied power spectral density indicates that the energy contained in meandering and the corresponding harmonic frequencies modeled by PINNs is relatively low compared to that in CFD and FNOs. These findings suggest that FNOs are promising for the high-fidelity, real-time modeling of FOWT wakes.

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

2 major / 2 minor

Summary. The manuscript introduces physics-informed neural networks (PINNs) and Fourier neural operators (FNOs) for modeling multi-scale dynamic wakes of floating offshore wind turbines (FOWTs) subjected to coupled surge and pitch motions. Using data from large-eddy simulations with actuator line model (LES-AL), the study finds that both methods capture dominant large-scale structures like wake meandering, but FNOs provide significant advantages in computational efficiency (8-fold speedup and 40-fold faster convergence), long-term predictive capability, and fidelity to multi-scale coherent structures and spectral content compared to PINNs, which exhibit smoothing and underestimation of high-frequency fluctuations.

Significance. If the results hold, this work advances real-time high-fidelity wake modeling for FOWTs, which is critical for control and optimization in offshore wind energy. The direct comparison of PINN and FNO frameworks on multi-scale wake dynamics, including spectral analysis of meandering frequencies and harmonics, provides useful insights into operator-learning methods for capturing energy cascades.

major comments (2)
  1. [Abstract] Abstract: The central claims of FNO superiority in long-term prediction and multi-scale spectral fidelity (including resolution of Stp, 2Stp, 3Stp harmonics and energy cascade) are benchmarked directly against LES-AL data. However, the actuator-line forcing approximates blade loads and omits detailed blade-wake and platform-motion coupling; this can systematically bias low-frequency wake meandering trajectories and harmonic energy distribution at the tested Strouhal numbers, undermining the quantitative comparisons.
  2. [Methodology] Methodology/Results: Training details (data partitioning, hyperparameter selection, exact definitions of the error metrics used for the reported 8-fold speedup and 40-fold convergence gains) are unspecified. Without these, the performance edges and long-term predictive claims cannot be independently verified or reproduced.
minor comments (2)
  1. [Abstract] Abstract: Specify the exact range of Strouhal numbers examined and the precise definition of the coupled surge-pitch frequency Stp.
  2. [Results] Figures: Ensure spectral plots include consistent axis scaling and explicit legends distinguishing CFD, PINN, and FNO curves for pre-multiplied PSD.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment below and have made revisions to strengthen the presentation of our methods and results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claims of FNO superiority in long-term prediction and multi-scale spectral fidelity (including resolution of Stp, 2Stp, 3Stp harmonics and energy cascade) are benchmarked directly against LES-AL data. However, the actuator-line forcing approximates blade loads and omits detailed blade-wake and platform-motion coupling; this can systematically bias low-frequency wake meandering trajectories and harmonic energy distribution at the tested Strouhal numbers, undermining the quantitative comparisons.

    Authors: We acknowledge that the actuator-line model approximates blade loads and does not capture every detail of blade-wake interactions or full platform-motion coupling. This is an inherent limitation of the LES-AL framework. However, both the PINN and FNO models are trained and evaluated on exactly the same LES-AL dataset, so any systematic bias affects the reference data equally for both approaches. The reported advantages of FNOs are therefore relative performance differences within this established modeling framework. We have added a dedicated paragraph in the revised manuscript that explicitly discusses the limitations of the actuator-line approach and their potential influence on low-frequency meandering and harmonic content. revision: yes

  2. Referee: [Methodology] Methodology/Results: Training details (data partitioning, hyperparameter selection, exact definitions of the error metrics used for the reported 8-fold speedup and 40-fold convergence gains) are unspecified. Without these, the performance edges and long-term predictive claims cannot be independently verified or reproduced.

    Authors: We agree that the original manuscript did not provide sufficient detail on the training procedure. In the revised version we have expanded the Methodology section to include explicit information on data partitioning, the hyperparameter selection process, and the precise definitions of the error metrics underlying the reported computational speedup and convergence improvements. These additions are intended to enable independent verification and reproduction of the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation or validation chain

full rationale

The paper trains PINNs and FNOs on an external high-fidelity LES-AL dataset and evaluates long-term predictions, efficiency, and spectral fidelity by direct quantitative comparison against the same independent LES-AL simulations used as ground truth. No load-bearing step reduces to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the central claims rest on external data benchmarks rather than internal tautologies. This is the standard non-circular ML validation pattern.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the accuracy of the LES-AL ground-truth data and on the ability of the neural networks to generalize from the simulated Strouhal range; network weights are fitted parameters.

free parameters (2)
  • neural network weights and biases
    Fitted during supervised training to minimize discrepancy with LES-AL velocity fields.
  • FNO and PINN hyperparameters
    Chosen to achieve the reported convergence and accuracy on the training dataset.
axioms (1)
  • domain assumption LES-AL simulations provide a sufficiently accurate representation of the true fluid dynamics for the purpose of training and validation.
    Invoked when the abstract states that the high-fidelity dataset is obtained from LES-AL and used to benchmark both models.

pith-pipeline@v0.9.0 · 5864 in / 1324 out tokens · 42461 ms · 2026-05-21T09:05:05.761763+00:00 · methodology

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Reference graph

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