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Aerodynamic force reconstruction using physics-informed Gaussian processes

This paper introduces a probabilistic physics-informed machine learning approach that accurately reconstructs aerodynamic loads from noisy structural response data without overfitting or regularization, demonstrating its efficacy on the Great Belt East Bridge for applications like model validation and structural damage prognosis.

Original authors: Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Published 2026-05-22

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CC BY 4.0

Original authors: Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out exactly how hard the wind is hitting a giant suspension bridge. You can't stick a giant wind sensor directly onto the invisible air currents, but you can see how the bridge moves, shakes, and sways in response.

This paper presents a clever new way to work backward: taking the noisy, shaky movements of the bridge and mathematically "reverse-engineering" the invisible wind forces that caused them.

Here is how the authors did it, explained simply:

1. The Problem: The "Noisy Phone Call"

Imagine trying to hear a friend's voice on a phone call, but the line is full of static and crackling noise. If you try to write down exactly what they said based on that bad connection, you might guess wrong.

In engineering, sensors on bridges are like that bad phone line. They measure the bridge's movement (displacement, speed, and acceleration), but the data is always "noisy." Traditional methods to figure out the wind force from this noisy data often require the engineer to manually tune knobs and settings (called "regularization") to stop the math from getting confused. It's like trying to clean a muddy photo by manually erasing pixels one by one—it's tedious and prone to human error.

2. The Solution: The "Physics-Savvy Detective"

The authors created a smart computer model called a Physics-Informed Gaussian Process. Think of this model as a detective who knows two things very well:

The Rules of the Game (Physics): They know the exact laws of physics that govern how a bridge swings (like a pendulum). They know that if you push a bridge, it moves in a specific, smooth way.
The Clues (Data): They look at the messy, noisy sensor data from the bridge.

Instead of guessing, this detective uses the laws of physics to draw a "map" of what the wind force should look like. Because the model knows the bridge must move smoothly according to physics, it can ignore the random "static" (noise) in the sensor data and focus on the real story.

3. How It Works: The "Smoothie" Analogy

Imagine the true wind force is a smooth, delicious smoothie. The sensors give you a smoothie that has been mixed with gravel and sand (noise).

Old methods tried to pick out the gravel by hand, often accidentally throwing away some of the fruit (the real data) or leaving some gravel behind.
This new method knows that a smoothie should be smooth. It uses a mathematical filter that says, "I know this bridge can't vibrate like a jagged saw blade; it has to flow like water." So, it naturally smooths out the gravel without needing a human to tell it how much gravel to remove.

4. The Test: The Great Belt East Bridge

To prove this worked, the authors tested it on a famous bridge in Denmark called the Great Belt East Bridge.

They created a perfect computer simulation of the bridge and the wind (the "truth").
They then took the "perfect" movement data and added artificial static noise to it, mimicking real-world sensors.
They fed this noisy data into their new model.

The Result: The model successfully reconstructed the wind force. It matched the "true" wind force very closely, especially for the big, slow sways of the bridge.

Where it shined: It got the overall strength (magnitude) and timing (phase) of the wind right.
Where it struggled: When the wind had very fast, high-pitched vibrations (high frequencies), the model was a bit less precise. However, it was smart enough to admit, "I'm not 100% sure about these fast parts," and showed a "confidence interval" (a range of possibilities) rather than making up a fake answer.

5. Why This Matters

The paper claims this method is a powerful tool for:

Checking Models: It helps engineers see if their wind models are accurate by comparing the "reconstructed" wind to what they expected.
Estimating Future Loads: It can help predict what kind of wind forces a bridge might face later.
Checking for Damage: If a bridge starts behaving differently than the physics model predicts, it could be a sign of structural damage.

In short: The authors built a smart, physics-aware AI that can listen to a noisy, shaking bridge and accurately tell you exactly how hard the wind is blowing, without needing a human to manually tune the settings. It turns a messy, confusing signal into a clear picture of the invisible forces at play.

Technical Summary: Aerodynamic Force Reconstruction Using Physics-Informed Gaussian Processes

Problem Statement
Accurate modeling of aerodynamic loads is fundamental to predicting the responses of complex structural systems, such as long-span bridges. However, existing models often rely on simplifications of true physical forces, introducing assumptions that limit accuracy. Validating these models is particularly difficult when measurement data is noisy or incomplete. Traditional methods for reconstructing aerodynamic loads from dynamic responses—such as impulse response matrices or augmented Kalman filters—typically require user-defined regularization parameters to handle noise. Furthermore, methods like latent force models often rely on constant sampling intervals and can only process a single dataset at a time, limiting their flexibility with heterogeneous data.

Methodology
The authors propose a probabilistic, physics-informed machine learning approach using Gaussian Processes (GP) to reconstruct underlying aerodynamic loads from noisy structural dynamic responses. The methodology integrates mathematical formulations of physical system behavior directly into the machine learning framework:

Physical Formulation: The dynamic response of a harmonic oscillator under an arbitrary time-varying load $F$ is defined by the scalar equation $m\ddot{z} + 2m\zeta\omega_n \dot{z} + m\omega_n^2 z = F$ .
Kernel Construction: The displacement response is modeled as a zero-mean Gaussian process. By exploiting the linear time-derivative relationships between displacement, velocity, and acceleration, the authors derive covariance kernels that relate different physical measurement types. These kernels are constructed by applying time derivatives to a base squared exponential (SE) kernel.
Physics-Informed Training: The GP model is trained using maximum likelihood estimation to identify optimal parameters (variance scales, length scales, and noise levels) based solely on measurement data (displacements, velocities, and accelerations). Crucially, no force information is used during the training stage.
Force Reconstruction: Once parameters are identified, a joint distribution is built between the training data and the dynamic forces. The force model is conditioned on the data to yield a stochastic representation of the aerodynamic load, providing both a mean prediction and uncertainty quantification.

Key Contributions

Automatic Regularization: The Bayesian nature of the GP approach provides an automatic trade-off between data fitting and model complexity, eliminating the need for user-defined regularization schemes and preventing overfitting.
Heterogeneous Data Handling: The method supports the use of heterogeneous and multi-fidelity data (e.g., combining displacement, velocity, and acceleration) and accommodates irregular sampling intervals.
Uncertainty Quantification: Due to its probabilistic framework, the method naturally provides prediction uncertainty, allowing the model to distinguish between physical signal content and measurement noise.
Analytical Kernels: The covariance kernels are analytically derived by applying linear differential operators to a base kernel, ensuring the model remains physically consistent with the governing equations of motion.

Results
The efficacy of the approach was demonstrated by reconstructing aerodynamic loads on the Great Belt East Bridge in Denmark, simulated under a linear unsteady (LU) assumption. The structural responses were synthetically contaminated with white noise (SNR = 20) to simulate realistic sensor readings.

Signal Agreement: The reconstructed forces showed strong agreement with the true LU model loads regarding root mean squared errors (RMS), magnitude, phase angle, and peak values.
Frequency Response: The model accurately captured low-frequency content. In regions dominated by high-frequency content, the GP model treated these as part of its uncertainty range rather than forcing a fit, resulting in a smoothed prediction.
Performance Metrics: Evaluation using four normalized metrics (RMS, magnitude, phase, and peak) indicated good agreement across nearly all 22 modal components.
Limitations: The worst performance was observed in the third mode shape (lateral sway, $f = 0.123$ Hz). In this mode, the regressed force showed increased spectral amplitudes at frequencies above 0.200 Hz compared to the LU model. This was attributed to the noise in the global coordinate system having a higher relative impact on modes with low mechanical admittance and minimal contribution to global responses.

Significance and Claims
The paper claims that this physics-informed approach serves as a robust tool for model validation, future load estimation, and structural damage prognosis. By combining machine learning with mathematical models of structural behavior, the method allows for the use of diverse data types without the need for regularization.

The authors emphasize that while the model effectively regresses underlying forces and incorporates high-frequency noise into its uncertainty bounds, its performance is dependent on the signal-to-noise ratio. In scenarios where structural motion amplitudes are minimal compared to signal noise, the model maintains favorable agreement in low-frequency ranges but tends to align with inherent signal noise in high-frequency ranges. The study concludes that precise procedures for regressing dynamic forces are effective for investigating model accuracy in complex structures, particularly in the context of climate change where wind field characteristics may deviate from initial design assumptions.