Stochastic Inference of Plate Bending from Heterogeneous Data: Physics-informed Gaussian Processes via Kirchhoff-Love Theory

TL;DR

The paper tackles inferring the flexural rigidity $D$ of Kirchhoff-Love plates from heterogeneous, noisy measurements by embedding the plate PDEs into a physics-informed Gaussian Process. By placing a GP prior on the deflection $w$ and deriving cross-covariances using operators like $\mathcal{L}_{q}^D=D\nabla^4$, the authors construct a multi-output GP that links deflection, load, rotations, curvatures, and internal forces within a unified probabilistic framework. They perform Bayesian inference on $D$ and kernel hyperparameters via MCMC and MAP, and demonstrate accuracy in learning $D$ and predicting both observed and unobserved quantities for two boundary-condition scenarios. The approach offers principled uncertainty quantification and data fusion across sensor types, with potential applications in structural health monitoring of plate-like structures; it also highlights avenues for kernel design and integration of boundary conditions directly into the kernel.

Abstract

Advancements in machine learning and an abundance of structural monitoring data have inspired the integration of mechanical models with probabilistic models to identify a structure's state and quantify the uncertainty of its physical parameters and response. In this paper, we propose an inference methodology for classical Kirchhoff-Love plates via physics-informed Gaussian Processes (GP). A probabilistic model is formulated as a multi-output GP by placing a GP prior on the deflection and deriving the covariance function using the linear differential operators of the plate governing equations. The posteriors of the flexural rigidity, hyperparameters, and plate response are inferred in a Bayesian manner using Markov chain Monte Carlo (MCMC) sampling from noisy measurements. We demonstrate the applicability with two examples: a simply supported plate subjected to a sinusoidal load and a fixed plate subjected to a uniform load. The results illustrate how the proposed methodology can be employed to perform stochastic inference for plate rigidity and physical quantities by integrating measurements from various sensor types and qualities. Potential applications of the presented methodology are in structural health monitoring and uncertainty quantification of plate-like structures.

Figures (11)

• Figure 1: Schematic for a physics-informed GP model: The model is derived leveraging the Kirchhoff-Love theory for the assumed GP prior on the deflection $w$. The flexural rigidity $D$ (green operators) is part of the model formulation and it can be inferred from noisy observations $\boldsymbol{z}$ (red crosses). The boundary conditions are imposed as noiseless observations $\boldsymbol{z}_\mathrm{BC}$ (black crosses).
• Figure 2: Numerical experiments: i) simply-supported plate subjected to sinusoidal load (left); ii) fixed plate subjected to uniform load (right). The red crosses represent the observation locations $\boldsymbol{X}$.
• Figure 3: Simply-supported plate with sinusoidal loading: Histogram of MCMC samples for the flextural rigidity based on a single set of measurements ($N_o$=1). The dashed lines represent the MLE estimates.
• Figure 4: Simply-supported plate with sinusoidal loading: Monte Carlo analysis for $N_o$=1000 sets of measurements for each signal-to-noise ratio. Mean of MCMC means $\overline{D}$ (opaque; cf. \ref{['eq:MCMC']}) and the MLE estimates $\hat{D}$ (transparent; cf. \ref{['eq:MLE']}) of the learned flexural rigidities, and their corresponding 25% / 75% quantiles and minimum/maximum values.
• Figure 5: Simply-supported plate with sinusoidal loading: Normalised mean deflection $w$ (top), curvatures $\kappa_x$ (centre) and $\kappa_{xy}$ (bottom) - True values (left), prediction for L1 (center) and prediction for L3 (right). The red crosses represent the training points for the corresponding learning cases.