Sparse measurement correction for full-field displacement prediction of nonlinear structures using reduced-order models and Gaussian process regression

Online full-field displacement reconstruction from sparse measurements is an important task in nonlinear structural analysis, monitoring, and digital-twin applications. Direct finite element simulations are often too expensive for repeated online use, whereas input-only reduced-order model–Gaussian process regression (ROM-GPR) surrogate models may produce biased full-field predictions when the target structural state is insufficiently represented by offline training data. Sparse displacement measurements provide current-state information, but observation-only reconstruction is usually underdetermined or noise-sensitive under limited sensing conditions. To address this issue, this work develops a probabilistic correction strategy for ROM-GPR full-field displacement predictions by statistically weighting the input-informed surrogate estimate and sparse measurements according to their respective uncertainties. The Gaussian process regression (GPR) prediction provides a low-dimensional estimate with quantified regression uncertainty, while the measurement constraint uses an observation error covariance that accounts for sensor noise and the observation-space projection of reduced-order reconstruction residuals. Sparse measurements are incorporated through an error-covariance-weighted correction in the reduced-order space, with the ensemble Kalman filter (EnKF) and unscented Kalman filter (UKF) adopted as sampling-based implementations. Tests on two nonlinear 2D finite element examples and a 3D liver model show that the proposed correction reduces the mean reconstruction error by 55.92%–73.51% for autoencoder-based ROM-GPR and 72.48%–93.10% for proper orthogonal decomposition-based ROM-GPR.