# Mode-Shape Expansion Using Physics-Constrained Gaussian Process Regression

arXiv: https://arxiv.org/abs/2605.23101 · HTML: https://mlsanity.com/paper/2605.23101/

## Key facts

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- First posted on arXiv 21 May 2026.

## Abstract (arXiv)

This paper addresses the challenge of reconstructing full-field structural mode shapes from sparse sensor data. While Gaussian Process Regression (GPR) offers a robust non-parametric framework for spatial interpolation and uncertainty quantification, standard formulations often yield physically inconsistent mode-shape reconstructions under sparse sensing conditions. A Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework is derived that utilizes independent modal kernels while coupling the optimization via a mass-orthogonality penalty. The paper presents derivations for the marginal likelihood, hyperparameter gradients, and penalty coupling. Numerical verification on a multi-degree-of-freedom structure demonstrates that the proposed method overcomes existing limitations in GP-based prediction, providing more accurate and reliable expanded mode shapes.