Physics-informed neural operators for predicting structural intensity from laser Doppler vibrometry measurements of plates

Structural intensity is a powerful diagnostic tool for characterizing vibrational energy flow and identifying dominant transfer paths in structures. For thin-walled plates, the structural intensity can be computed from the transverse displacement field and its spatial derivatives. However, accurate estimation from experimental data remains challenging, as numerical differentiation is highly sensitive to measurement noise, particularly when computing higher-order spatial gradients. This study proposes a machine learning-based framework employing a physics-informed deep operator network to predict structural intensity directly from noisy displacement measurements. Instead of differentiating raw measurement data, the method first learns a smooth, fully differentiable neural-network surrogate of the displacement field, enabling stable and accurate evaluation of higher-order spatial derivatives via automatic differentiation. This eliminates noise-induced instabilities of conventional numerical differentiation methods and enables robust structural intensity prediction from experimental data. The methodology is assessed using two case studies: an analytical benchmark problem of a simply supported plate, and experimental laser Doppler vibrometry measurements of a plate structure for practical validation. The results demonstrate that the physics-informed deep operator network accurately predicts the displacement and structural intensity fields over a broad frequency range, capturing both magnitude distributions and directional patterns with high fidelity. Across all studies, the proposed data-driven approach consistently outperforms numerical differentiation techniques, demonstrating enhanced robustness and accuracy while confirming its applicability to problems in experimental structural dynamics and noise control engineering.