Towards Generalisable Virtual Sensing: A Physics-Informed Graph Neural Network Approach On Cantilever Beams

Virtual sensing reconstructs structural responses at uninstrumented locations from a sparse set of physical measurements, enabling structural health monitoring where dense sensor networks are impractical. Conventional model-based methods depend on an accurate representation of the structure. When this representation does not match reality the reconstruction degrades, and restoring accuracy requires recalibration for each unit of a population, making the methodology time consuming. This work investigates whether graph neural networks can instead provide a single model that requires no per-structure recalibration and generalises to unseen geometries and loads, by learning relational dependencies directly from data and structural information. We propose a virtual sensing framework that combines graph attention networks for spatial information propagation with long short-term memory networks for temporal modelling. The structural topology and mechanical properties are encoded directly from a finite element mesh, while a physics-informed regularisation term, based on the governing equations of motion, is introduced to improve extrapolation robustness. The framework is trained on a population of simulated cantilever beams with varying geometries, tip masses and loading conditions, and evaluated on both in-distribution and out-of-distribution scenarios involving unseen beam lengths, force amplitudes and excitation types. Compared with modal expansion, the proposed approach consistently achieves lower reconstruction errors while maintaining a single model that generalises across previously unseen structures. On in-distribution data it reaches a mean absolute percentage error of about 8.40%, against 19.93% for modal expansion. Relative to an otherwise identical purely data-driven graph neural network, the physics-informed formulation improves extrapolation performance, reducing error from 11.17% to 9.50% when inferring on beams longer than those seen during training. These results demonstrate that graph neural networks provide a promising framework for generalisable virtual sensing, while physics-informed regularisation further improves robustness under unseen geometries and operating conditions.