Modal Physics-Informed Neural Networks for Forward and Inverse Structural Vibration Problems

Physics-informed neural networks face significant computational challenges when applied to structural dynamics problems due to spectral bias and high dimensional state spaces. This work proposes a modal physics-informed neural network framework that integrates classical modal decomposition with Fourier feature embeddings to enable efficient learning of high frequency structural responses. By transforming multi degree of freedom and continuous systems into uncoupled modal coordinates, the approach achieves accurate dynamic response prediction while preserving essential physical behavior. The framework is validated through numerical studies on discrete and continuous systems, achieving mean absolute errors of [Formula: see text] m for forward problems. Numerical studies on a cantilever beam with stiffness reductions ranging from 1% to 60% demonstrate stable convergence and consistent identification of modal characteristics. The natural frequency identification exhibits errors between 4% and 14% depending on damage severity, with higher errors corresponding to severe damage cases where absolute frequency values are lower. Validation on a Warren truss with single and two-site damage scenarios yields frequency identification errors of 6.65% to 10.87% and Modal Assurance Criterion values above 0.93, confirming generalisation beyond beam structures. Experimental validation on an aluminium cantilever beam with piezoelectric accelerometers confirms the practical applicability of the method, achieving 4.3% identification error using only five sensors. The modal approach achieves 53% reduction in training time compared to direct physics informed neural network formulations while maintaining comparable accuracy. The decoupled modal structure enables parallel training and improves scalability for large scale structural systems. The proposed framework offers a computationally efficient and physically interpretable strategy for structural vibration analysis and damage identification with potential applications in structural health monitoring.