VibPINNs:Physics-informed Neural Networks for Forward and Inverse Problems in Structural Vibrations guided by Sparse Measurements

Physics-informed neural networks (PINNs) provide a principled route for solving partial differential equations (PDEs) by integrating governing laws directly into the learning process, providing a compelling paradigm for modelling dynamical systems in structural, thermal and fluid mechanics. Despite this promise, PINNs often exhibit slow convergence and reduced accuracy when applied to high-order dynamics, particularly in vibration problems governed by second-order (and higher) PDEs. Moreover, many existing vibration-oriented PINN formulations rely on fully specified boundary conditions (BCs), which are rarely available in practical sensing and monitoring settings. Here we show that limited measurement guidance is essential for training PINNs in high-order dynamical systems, and we propose a physics-informed neural network framework for vibration modelling guided by sparse time-and frequency-domain data (VibPINNs). The framework incorporates only a small set of sparse reference measurements, obtained from either sensor measurements or numerical simulations, to regularize optimization, while preserving PDE residuals as the dominant constraint on the solution manifold. To further address the spectral bias of the network and imbalance among loss terms, we developed physics-consistent Fourier feature embedding (FFE) and conjugate-kernel (CK) enhancement strategies, in which embedding hyperparameters are inferred from modal characteristics extracted from the reference data. Benchmarks on the Euler-Bernoulli equation and experimentally relevant beam dynamics show the reliable, accurate forward predictions at unmeasured locations can be obtained using only three measurement points. It also enables reliable structural parameter identification in inverse problems. These results highlight the broader potential of VibPINNs for physics-guided learning of complex dynamical systems.<br/>