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

This study demonstrates that Fourier Neural Operators (FNOs) outperform Physics-Informed Neural Networks (PINNs) in predicting the multi-scale dynamic wakes of floating offshore wind turbines by more accurately capturing high-frequency turbulent structures and higher-order harmonics while offering significantly faster training speeds.

The Gist

The Tale of Two Digital Wind Experts: Predicting the "Wobble" of Floating Wind Turbines

Imagine you are standing on a pier, watching a massive, spinning wind turbine floating out in the deep ocean. Because it’s floating, it doesn’t just sit still; it dances. It surges forward and pitches (tilts) back and forth with the waves.

This "dance" creates a massive, swirling mess of wind behind it—a chaotic, turbulent wake. For engineers, this wake is a nightmare. If a second turbine is placed too close to that messy wind, it won't produce much power, and the constant "buffeting" from the turbulence can shake the turbine apart like a loose tooth.

To run a wind farm efficiently, we need a "Digital Crystal Ball" that can predict exactly how that messy wind will swirl and move in real-time. This paper compares two different types of Artificial Intelligence (AI) to see which one is the better fortune teller.

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The Two Contenders

1. The PINN: "The Strict Professor"

Think of the PINN (Physics-Informed Neural Network) as a very smart, very strict professor. When you ask the professor to predict the wind, they don't just look at patterns; they insist on following the "Laws of Physics" (mathematical equations) every single step of the way.

• The Good: They are very disciplined and try to make sure their predictions don't break the rules of nature.
• The Bad: Because they are so obsessed with following the rules, they tend to be "over-cautious." They smooth everything out. If the wind has a sharp, sudden, chaotic gust, the Professor says, "That's too messy; let's just call it a gentle breeze to keep things orderly."

2. The FNO: "The Master Impressionist"

Think of the FNO (Fourier Neural Operator) as a master painter who looks at the world through "frequencies." Instead of looking at every tiny individual drop of water, they see the big waves, the medium ripples, and the tiny splashes all at once. They use a mathematical trick (the Fourier Transform) to see the "rhythm" of the wind.

• The Good: They are incredibly fast and can see the "texture" of the chaos. They don't just see the big wave; they see the tiny, high-speed ripples on top of it.
• The Bad: They don't "force" the physics rules as strictly as the Professor does, but they are so good at recognizing the patterns of physics that they often get it right anyway.

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The Great Race: Who Won?

The researchers put both AIs to the test using supercomputer data (the "Gold Standard") of a floating turbine. Here is how the battle went down:

1. The Speed Test (The FNO wins by a landslide!)
The Professor (PINN) took over 2 hours to finish his homework. The Impressionist (FNO) finished in just 15 minutes. In the world of real-time wind farm control, being 8 times faster is the difference between reacting to a storm and being hit by it.

2. The Detail Test (The FNO is a high-def camera; the PINN is a blurry photo)
When looking at the swirling wind, the PINN (the Professor) acted like a "Low-Pass Filter." Imagine looking at a beautiful, jagged mountain range through a foggy window—you see the general shape, but you lose all the sharp peaks and crags. The PINN smoothed out the turbulence.
The FNO, however, captured the "sharpness." It saw the tiny, high-frequency swirls and the complex "harmonics" (the secondary rhythms) caused by the turbine's tilting.

3. The Long-Term Prediction (The FNO stays on track)
When asked to predict what the wind would look like much later in time, the PINN's prediction started to fall apart and become a blurry mess. The FNO stayed sharp, accurately predicting where the "center" of the wind mess would be and how wide it would spread.

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The Bottom Line

If you want to manage a floating wind farm in the middle of a stormy ocean, you don't want a "Strict Professor" who smooths over the dangerous, chaotic details. You want the "Master Impressionist" (the FNO).

The FNO is faster, sharper, and better at seeing the "rhythm of the chaos." It gives engineers a high-definition, real-time view of the wind, allowing them to position turbines perfectly and keep them safe from the turbulent "dance" of the floating giants.

Technical Summary

Technical Summary: Multi-scale Dynamic Wake Modeling of FOWTs via FNO and PINNs

1. Problem Statement

Floating Offshore Wind Turbines (FOWTs) are subject to complex six-degree-of-freedom (6-DOF) motions, specifically coupled surge and pitch. These motions induce highly turbulent, multi-scale wake interactions that significantly impact downstream turbine performance (causing up to 40% power loss) and structural integrity (increasing fatigue loads by up to 80%).

Traditional modeling approaches face a critical trade-off:

• Analytical Models: Computationally efficient but fail to capture multi-scale turbulent structures.
• Computational Fluid Dynamics (CFD): High-fidelity (e.g., LES) but computationally prohibitive for real-time control or large-scale wind farm optimization.
• Existing Data-Driven/ML Methods: Reduced-order models (ROM) often struggle with generalization, while standard Physics-Informed Neural Networks (PINNs) face high training complexity and struggle to resolve high-frequency turbulent details.

2. Methodology

The study proposes a comparative framework to evaluate two advanced deep learning paradigms for reconstructing and predicting FOWT wakes: Fourier Neural Operators (FNOs) and Physics-Informed Neural Networks (PINNs).

• Data Generation (Ground Truth): High-fidelity data was generated using Large-Eddy Simulation (LES) combined with the Actuator Line Method (ALM). The simulation modeled an NREL 5MW turbine undergoing coupled surge and pitch motions across various Strouhal numbers ($St \in [0, 0.6]$).
• PINN Architecture: A multi-layer fully connected neural network that incorporates the 2D non-dimensionalized Navier-Stokes equations into its loss function. It attempts to learn the mapping from $(x, y, t)$ to $(u, v, p)$ by minimizing both data discrepancy and physical residual errors.
• FNO Architecture: A neural operator that learns mappings between infinite-dimensional function spaces. It utilizes the Fast Fourier Transform (FFT) to parameterize the integral kernel in the Fourier domain, allowing it to capture multi-scale features through spectral convolutions.

3. Key Contributions

1. First Comparative Application: This is the first study to systematically compare FNOs and PINNs for modeling the multi-scale dynamic wakes of FOWTs under coupled 6-DOF motions using high-fidelity LES data.
2. Identification of the "Low-Pass Filter" Effect: The research identifies a fundamental physical limitation in PINNs, demonstrating that they act as spatiotemporal low-pass filters, smoothing out critical high-frequency turbulent structures.
3. Spectral Fidelity Analysis: The study provides a rigorous spectral evaluation, proving that FNOs can resolve higher-order harmonics and small-scale coherent structures that PINNs miss.

4. Results and Discussion

• Computational Efficiency: FNO is significantly more efficient, achieving a training speed approximately 8 times faster than PINN (approx. 15 minutes vs. >2 hours) and converging in far fewer epochs (500 vs. 20,000).
• Reconstruction Accuracy:
  • FNO produced streamwise velocity fields virtually indistinguishable from CFD ground truth, maintaining a stable and low Root Mean Square Error (RMSE $\approx 0.01$).
  • PINN exhibited a "smoothing effect," failing to resolve sharp velocity gradients and high-frequency oscillations, with an RMSE five times higher than FNO.
• Wake Parameter Tracking: FNO accurately tracked the non-monotonic variations in wake center ($y_c$), wake half-width ($R_{1/2}$), and velocity deficit ($\Delta U_c$). PINN consistently underestimated the intensity of temporal fluctuations and the magnitude of the velocity deficit.
• Spectral Performance: Power Spectral Density (PSD) analysis revealed that while both models capture primary wake meandering frequencies ($St$), only FNO successfully resolved the higher-order harmonics ($2St, 3St$) and the energy in the high-frequency regime ($St > 1$).

5. Significance

The findings establish FNO as a superior framework for FOWT wake modeling compared to PINNs. By effectively capturing both large-scale meandering and small-scale turbulence with high computational efficiency, FNO provides a robust pathway for:

• Real-time wake monitoring and active wake control.
• High-fidelity wind farm layout optimization.
• Reducing the computational burden of wind energy design without sacrificing the physical accuracy required to manage fatigue and power losses.