Efficient dynamic modal load reconstruction using physics-informed Gaussian processes based on frequency-sparse Fourier basis functions

TL;DR

The paper addresses the challenge of reconstructing external dynamic loads from structural responses when direct force measurements are difficult. It introduces a physics-informed Gaussian-process framework whose prior for measurements is built from frequency-sparse Fourier basis functions and analytically related derivatives to connect displacement, velocity, and acceleration, with efficient training enabled by spectral sparsity and the Woodbury identity. The trained response model is integrated with the harmonic-oscillator equation to yield a probabilistic load model that predicts dynamic forces without force data during training, demonstrated on a wind-excited 76-story building and a scale Lillebælt Bridge model. Results show accurate reconstruction of modal forces for low- to mid-frequency modes and robust performance under noise, highlighting applicability to structural health monitoring, damage prognosis, and load model validation. Limitations include linear-system assumptions and sensitivities to modal mass, damping, and sensor quality, with future work proposed for non-linear and non-stationary scenarios and uncertainty propagation analysis.

Abstract

Knowledge of the force time history of a structure is essential to assess its behaviour, ensure safety and maintain reliability. However, direct measurement of external forces is often challenging due to sensor limitations, unknown force characteristics, or inaccessible load points. This paper presents an efficient dynamic load reconstruction method using physics-informed Gaussian processes (GP) based on frequency-sparse Fourier basis functions. The GP's covariance matrices are built using the description of the system dynamics, and the model is trained using structural response measurements. This provides support and interpretability to the machine learning model, in contrast to purely data-driven methods. In addition, the model filters out irrelevant components in the Fourier basis function by leveraging the sparsity of structural responses in the frequency domain, thereby reducing computational complexity during optimization. The trained model for structural responses is then integrated with the differential equation for a harmonic oscillator, creating a probabilistic dynamic load model that predicts load patterns without requiring force data during training. The model's effectiveness is validated through two case studies: a numerical model of a wind-excited 76-story building and an experiment using a physical scale model of the Lillebælt Bridge in Denmark, excited by a servo motor. For both cases, validation of the reconstructed forces is provided using comparison metrics for several signal properties. The developed model holds potential for applications in structural health monitoring, damage prognosis, and load model validation.

Figures (25)

• Figure 1: Schematic of the physics-informed Gaussian process model for dynamic load prediction. Using the frequency content of the measurement data (blue box) as inputs, a GP model for the measurements (green box, Sec. \ref{['sec:GPMeasurementPrior']}) is created using Fourier basis functions ${\psi}_i(t,f)$ for $i \in \lbrace u, \dot{u}, \Ddot{u}\rbrace$ and a Gaussian noise kernel. The measurement model is reduced and optimised (orange box, Sec. \ref{['sec:GPModelTraining']}), by filtering out low-contributing frequency bands and identifying optimal values for the free parameters $\sigma_s$ and $\sigma_{n,i}$. Further, physics-informed basis functions are created for loads (red box, Sec. \ref{['sec:GPModelPrediction']}), yielding a joint load-measurement model. Conditioning the model on the measurement data $\bm{u}_r$ yields probabilistic load predictions (violet box, Sec. \ref{['sec:GPModelPrediction']}).
• Figure 2: Top: covariance matrices $\bm{\Sigma}_{uu}$ generated with the squared exponential kernel (left), and with the basis functions model from Sec. \ref{['sec:GPMeasurementPrior']} with $f=0.3$ Hz (centre) and $f=0.6$ Hz (right). Colours in the top plots indicate the covariance magnitudes, which are normalised to one. Bottom: zero-mean sample signals and 95% confidence interval for each respective covariance matrix.
• Figure 3: Influence of the ratios $r_{\sigma}$ of measurement variance to total variance in the prior model for displacements. The covariance matrices for the cases of $r_{\sigma} = 1.00$ (left), $r_{\sigma} = 0.75$ (centre) and $r_{\sigma} = 0.00$ (right) are shown on top (colours indicate the covariance magnitudes, which are normalised to one, see Fig. \ref{['fig:KernelFreqs']}), with zero-mean samples generated from each respective covariance at the bottom. The shaded area in the bottom plots represents the 95% confidence interval.
• Figure 4: Prior model's response covariance matrix based on a multiple-frequency spectrum. The dense covariance matrix (left) leads to zero-mean samples containing the target frequencies (centre), as shown by their power spectral density (right). In the left plot, colours indicate the covariance magnitudes, which are normalised to one (see Fig. \ref{['fig:KernelFreqs']}), and in the central plot, the shaded area represents the 95% confidence interval.
• Figure 5: Left: block-covariance matrix for a heterogeneous response case, where displacements $u$, velocities $\dot{u}$ and accelerations $\Ddot{u}$ are used as training data (colours indicate the covariance magnitudes, which are normalised to one, see Fig. \ref{['fig:KernelFreqs']}). Right: Time-domain samples for displacements (top), velocities (centre) and accelerations (bottom) sampled from the covariance matrix. The grey shaded area represents the signal's 95% confidence interval.