Physics-informed neural networks (PINNs) have emerged as a powerful paradigm in scientific machine learning by embedding governing physical laws into neural network training through loss functions. They have demonstrated remarkable success in solving various forward and inverse problems governed by partial differential equations (PDEs). However, in practical applications, purely physics-constrained PINNs that rely solely on PDE residuals often suffer from slow or non-convergence and limited prediction accuracy, particularly when modeling high-order dynamical systems (e.g., second-order and above). Moreover, conventional PINNs struggle to effectively capture high-frequency components in complex physical fields, which further limits their generalization and representational capability.

To address these challenges, this study proposes a data-guided physics-informed neural network with Fourier feature enhancement. In the proposed framework, a small amount of high-fidelity measurement or simulation data is incorporated to guide the training process, providing explicit guidance that complement the physics-based constraints. Meanwhile, Fourier feature embeddings are introduced into the input layer of the network to enhance its ability to represent high-frequency variations and multi-scale solution structures. This synergistic integration of data guidance and Fourier-enhanced representations accelerates convergence, improves robustness, and enhances the accuracy of PDE solutions.

The effectiveness of the proposed PINN model is validated through numerical and simulation studies on Euler-Bernoulli beam vibration problems, which serve as representative examples of high-order mechanical systems. The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes. These findings highlight the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.

Load identification is a crucial topic in structural health monitoring (SHM). Existing approaches involve a trade-off between the amount of data required and the fidelity of available parametric physical models. Purely data-driven methods require extensive labeled training data for reliable identification and lack physical interpretability. Methods based on physical model inversion techniques are interpretable, but their accuracy depends on precise knowledge of the structural model and its parameters, which requires extensive model-tuning procedures.

This contribution presents and discusses a physics-augmented deep learning (PADL) framework, intended for identification of structural dynamic excitations and aimed at striking the balance between data and model precision demands. A long short-term memory (LSTM) neural network is used as the basic identification tool. However, in contrast to typical data-driven methods, its input consists of not only measured structural responses. It includes also the results of FRF-based inverse processing performed with a simplified, reduced-order model of the monitored structure, yielding inexact estimates of the unknown input and state vector. In this way, the typical data-based input is augmented with a physically consistent data that indirectly embeds and provides information about the physical structure and its equations of motion. Such a dual-input design effectively addresses the trade-off noted above: on the one hand, the required structural model can be very coarse, inexact and reduced-order; on the other, the physical consistency of the information it provides (even if it is strongly simplified and approximate) significantly reduces the amount of training data required. Moreover, the employed FRF-based approximate model inversion is sensor-agnostic, which enables applications with heterogeneous sensor networks. A specific data normalization scheme ensures applicability across a range of excitation amplitudes.

The proposed PADL framework is verified in numerical simulations and laboratory experiments using random and impulsive excitation profiles. The tests demonstrate that PADL outperforms purely physics-based and purely data-driven baseline methods and that high identification accuracy can be achieved even with very small training sets and highly simplified structural models and inaccurate parameters. An ablation study confirms that including the approximate state vector in the input data, in addition to the computed approximate excitation, provides additional physically consistent information and is important for the accuracy of the obtained identification results.

A neural calibration framework is presented for estimating the nonlinear hysteretic behavior of jointed structures using the parameters of the Bouc-Wen model, which serves as a reduced-order representation of bolted joints under dynamic excitation. This methodology integrates data-driven modeling, physics-based representation, and ensemble-based variability analysis to achieve robust parameter estimation and to propagate parameter dispersion into response-level uncertainty envelopes. A feed-forward neural network is employed to minimize the discrepancy between simulated and measured responses, with multiple random initializations producing an ensemble of calibrations. This ensemble quantifies variability resulting from both neural initialization and experimental repetitions under identical boundary conditions. Experimental validation was performed on a beam testbed subjected to different vibration regimes and various torque-tightening levels. The results demonstrate that the calibrated models capture key hysteretic features, such as stiffness degradation and energy dissipation, while yielding physically admissible parameter sets and consistent uncertainty envelopes across repeated measurements. The evolution of these parameters may serve as health indicators for structural health monitoring (SHM), facilitating the development of uncertainty-aware digital shadows for nonlinear structural systems.

System identification (SI) is critical for ensuring the reliability of structural and mechanical components across engineering applications. Traditional model-based SI methods often struggle with complex dynamics and the scarcity of accurate physical models, while purely data-driven, model-free approaches though simple and fast lack physical interpretability and suffer from poor generalization. Recent advances such as physics-informed neural networks (PINNs) combine data and physics to overcome these limitations and have shown strong promise for structural health monitoring (SHM). However, most existing methods still require knowledge of the input force, which limits their practical use for parameter estimation. To address this challenge, an input-robust hybrid physics informed neural network (rHPINN) framework is proposed that integrates physics-based system dynamics with the temporal learning capability of HPINN. An output-injection strategy enables rejection of unknown input forces, allowing accurate estimation of system states and spatial health parameters without input force measurements. By explicitly preserving temporal dependencies often overlooked in conventional PINNs the method achieves stable, physics-consistent system identification. Numerical simulations on systems demonstrate that rHPINN remains robust under unknown excitation, measurement noise, sparse data, and varying damage scenarios, highlighting its potential for real-world SHM under uncertain conditions.