The augmented Kalman filter (AKF) is widely used for joint input-state estimation by augmenting the state vector with unknown inputs. However, its performance critically depends on the Gaussian white noise assumption and the proper selection of noise covariance matrices, making it sensitive to modeling errors in the state-space representation and hyperparameter tuning in practice. To address these challenges, this study proposes an AKF-inspired hybrid learned observer, termed AKFNet, for joint input-state estimation from sparse output-only measurements. Unlike the classical AKF, AKFNet no longer computes the Kalman gain analytically based on the minimum mean square error principle; instead, it is a hybrid data-driven and model-based algorithm that embeds a recurrent neural network (RNN) module into the AKF recursion and empirically learns the gain directly from measurements. The RNN module learns to map the information of real data within the state-space model to an effective Kalman gain, thereby improving the accuracy and robustness of joint input-state estimation in practical scenarios where model mismatches and complex noises are present. The proposed AKFNet is first validated on a numerical 9 DOF building for seismic input-state estimation, and compared with the classical AKF, dual Kalman filter (DKF) and an adaptive AKF with online covariance adjustment. Subsequently, an experimental shaker table test is conducted on a three-story frame to further validate its effectiveness. Finally, the AKFNet is used to estimate wind loads on a real-world large-span cable-stayed bridge using field data. In all tested scenarios, AKFNet provides competitive or improved accuracy, demonstrating its robustness and adaptive capability in the presence of modeling errors and non-white Gaussian noise.

In structural health monitoring, supervised learning methods were developed for damage identification. As the damage dataset and samples are unavailable in practice, the unsupervised methods are proposed to locate the damage. However, they cannot further quantify the structural damage due to the lack of physical mechanisms in the neural network. To overcome these limitations, an interpretable self-supervised learning strategy is proposed for damage identification. Specifically, a physics-encoded neural network (PENN) architecture is constructed on the data-driven and model-based modules. The data-driven module is used to estimate the unmeasurable load from the response data at discrete sensor locations. Subsequently, the structural full-field responses are reconstructed by multiplying the estimated load with the impulse response matrix in the model-based module. The difference between the measured and reconstructed responses at the sensor locations is formulated as the loss function to train the PENN. The PENN is first trained by using the data of the undamaged structure to establish a proxy baseline. After damage, the data-driven module is fixed and untrainable, and the structural parameters in the model-based module are updated by the backpropagation principle to locate and quantify structural damage. The proposed PENN works in a novel self-supervised way using output-only data. It is applied to the numerical simple-supported beam and laboratory shear frame models for verification. The results demonstrate that the developed PENN can locate and quantify damage with satisfactory accuracy without knowing the load location, and it applies to single and multiple damage cases.

Generally, finite element modelling often requires detailed drawings and a lot of time. However, in some special cases, modelers may not have available bridge drawings while need to conduct rapid modelling. In this regard, considering the similarities between bridge structures of the same type, it is necessary and beneficial to adopt a parametric modelling method. This paper introduces a parametric bridge modelling program using a cable-stayed bridge as an example. In the proposed method, the modelling procedures can be divided into five parts for each main component. Moreover, the bridge parameters are estimated based on limited inputs and known data from other similar bridges. The modelling results of the proposed method are compared with the finite element model using design drawings and experimental test results. From the perspective of dynamic characteristics, the proposed program has good accuracy.

Effective monitoring of bridge infrastructure is essential for ensuring safety, reliability, and longevity. Traditional Structural Health Monitoring (SHM) methods have been challenged by installation complexity, maintenance demands, and high costs, particularly in large-scale or long-span bridges. In response, a novel, non-intrusive drive-through method for extracting bridge eigenfrequencies has been developed using vehicle-multi-mounted Micro-Electro-Mechanical Systems (MEMS) sensors. A passing vehicle is thereby converted into a mobile sensing platform, and bridge dynamic properties are captured under normal traffic conditions. Experimental validation was conducted on Shanghai's Yangpu and Nanpu cable-stayed bridges, and eigenfrequency extraction from a single crossing was demonstrated, significantly reducing the need for multiple passes. The signal-to-noise ratio was improved by the fusion of data from multiple sensors, and frequency identification was achieved. A practical and cost-effective solution for large-scale bridge monitoring is thus offered, and further work is planned to extend the methodology to additional structural parameters and enhanced robustness.

In the realm of structural health monitoring (SHM) of bridge structures, the accurate reconstruction of girder‐end displacement (GED) is crucial for identifying potential structural damage and ensuring the monitoring system’s reliability. A novel fine‐grained spatial (FGS) attention mechanism, combined with efficient channel attention (ECA), has been proposed to effectively utilize multisource monitoring data. This hybrid attention mechanism has been integrated into an arithmetic optimization algorithm–bidirectional long short‐term memory (AOA–BiLSTM) framework for reconstructing GED using non‐GED data, including deflection, temperature, strain, and traffic data. Data are organized into a two‐dimensional array based on sensor types and spatial locations to capture interchannel and intrachannel correlations. ECA captures local correlations among different sensor types, while the proposed FGS enhances model interpretability by focusing on local dependencies within each sensor type. Huber loss is employed for robust performance, and AOA techniques are used for efficient hyperparameter optimization. Validation with real‐world data from a cable‐stayed bridge demonstrates the necessity and efficacy of considering multidimensional information correlations in response reconstruction for SHM applications. This work lays a theoretical foundation for improving safety assessments, anomaly detection, data recovery, and virtual sensing in bridge structures.

Structural displacement measurement techniques based on monocular vision have made great strides in the past decades, however, practical applications still face challenges. These challenges include the complexity and cost associated with long-time automatic monitoring, as well as limitations in dynamic displacement tracking. This study introduces an automatic visual method for measuring three-dimensional (3D) displacement using the monocular camera. The method leverages an improved feature points group tracking algorithm designed for long-time image sequences to achieve accurate measurements of both planar and 3D displacements. This technology addresses challenges associated with error accumulation due to extended time series and small sub-pixel vibrations in the tracking algorithm. In addition, the depth estimation of monocular vision is improved, thus improved the accuracy of displacement measurements. To validate the accuracy of the proposed method in displacement measurement, model experiments were conducted on a three-story frame model. In addition, a cable-stayed bridge model was tested to assess the robustness of the method for depth estimation. The results show that the proposed method achieves good results in different scenarios and is able to realize 3D spatial displacement measurement, and the measurement results are basically consistent with the actual values.

In system identification, augmenting time series data via time shifting and nonlinearisation can lead to both feature and sample redundancy. However, research has mainly focused on feature redundancy while largely ignoring the issue of sample redundancy. This paper proposes a novel data pruning method, called mini-batch FastCan, to reduce sample-wise redundancy based on dictionary learning. Time series data is represented by some representative samples via dictionary learning. The useful samples are selected based on their correlation with the representative samples. The method is tested on two simulated datasets and two benchmark datasets. The R-squared value between the coefficients of models trained on the full datasets and the coefficients of models trained on pruned datasets is adopted to evaluate the performance of data pruning methods. It is found that the proposed method significantly outperforms the random pruning method, with a higher median or mean and a lower variance of R-squared values.

The joint estimation of system states and unknown input loads in dynamic civil structures, based on limited observations, has garnered significant attention in recent years. A widely used method for this is the augmented Kalman filter (AKF), which works by modeling system identification errors and measurement noise as Gaussian processes. However, the AKF is highly sensitive to the tuning of hyperparameters and to inaccuracies in the state-space model, which hampers its accuracy and robustness in practical applications. To address these challenges, this study proposes a neural network-assisted AKF (AKFNet) for joint input-state estimation. The AKFNet combines data-driven and model-driven approaches by incorporating a recurrent neural network (RNN) module into the AKF's recursive framework. The RNN module learns to refine the computation of the Kalman gain from real data, thereby enabling this approach to mitigate the limitations of traditional AKF methods in civil engineering, particularly the challenges posed by unknown noise covariances and model errors.

An accurate understanding of anticipated deflection behavior in operational bridges allows for the early identification of potential structural anomalies. While structural monitoring systems collect extensive data, much of it remains underutilized for precise bridge health assessments. This paper proposes a deep learning method that leverages multi-source information fusion to enhance the analysis of bridge monitoring data. Firstly, the mutual information method is introduced for optimal selection of multi-source data, and then cyclic encoding is included in the multi-source data to further enhance the model's capability of learning periodic features. Then, the deep learning model Convolutional Neural Network-Bidirectional Long Short-Term Memory (CNN-BiLSTM) is employed in this study considering a mixed attention mechanism combining Squeeze-and-Excitation (SE) attention and soft attention, which is proposed to extract high-value information from the multi-source data, enabling the model to fully utilize the multi-source information and improve prediction accuracy. The Multi-Verse Optimizer method is employed to optimize the model parameters, and a quantile loss function is defined for objective interval prediction. Finally, the algorithm is validated using three sets of in-situ monitoring data from a cable-stayed bridge, and the performance of the algorithm is evaluated based on both point prediction and interval (quantile) prediction accuracies. The results show that the proposed method can make full use of the fused multi-source information, and has an evident advantage in prediction accuracy compared to other algorithms. Moreover, the predicted deflection intervals have the highest effective density, providing a theoretical basis for real-time warning for in-service bridges.

Accurate parameter identification is critical for the effective modeling and control of dynamic systems, especially those exhibiting complex, nonlinear behaviors such as non-smooth gap systems. These systems, characterized by abrupt changes in dynamics due to physical constraints, discontinuities, or contact phenomena, pose significant challenges for traditional parameter identification methods, often resulting in inaccurate models and suboptimal system performance. To address these challenges, this study introduces the Strong Tracking Square Root Spherical Simplex-Radial Cubature Quadrature Kalman Filter (STSR-SSRCQKF), an advanced filtering algorithm designed to enhance parameter identification accuracy in non-smooth gap systems. The STSR-SSRCQKF provides several key benefits, including improved numerical stability through the adoption of QR decomposition, which avoids the need for positive-definite matrices, rapid adaptation to sudden system changes via strong tracking techniques, increased accuracy through a two-fold increase in sampling points, and computational simulations by utilizing acceleration data for alignment with commonly available measurements. The effectiveness of this method is validated on both 1-DoF and 5-DoF non-smooth systems. Through extensive simulations and comparisons under varying noise levels, large initial errors and limited measurement, the proposed approach demonstrates good performance. The capability of the STSR-SSRCQKF to accurately identify unknown switching points and ensure reliable state tracking in complex, non-smooth systems highlight its potential for broader applications in structural health monitoring, robotics, and dynamic system analysis.

This paper proposes a hierarchical Bayesian model updating approach to quantify variability of aerodynamic stiffness and damping of an offshore wind turbine (OWT) under different environmental and operational conditions (EOCs) using in-situ vibration data and SCADA (supervisory control and data acquisition) over two months of continuous monitoring. The considered OWT is a Haliade 150, 6 MW GE turbine on a jacket substructure located at the Block Island Wind Farm in Rhode Island, USA. The OWT has been instrumented with a continuous monitoring system including an array of accelerometers and strain gauges. The modal parameters of the OWT are extracted using an automated system identification approach. These parameters exhibit significant variations under varying EOCs. This variation is more significant in natural frequencies and damping ratios of the first fore-aft bending mode due to the aerodynamic effects. In this paper, a modeling approach is proposed by introducing a spring and a damper at the nacelle level in the fore-aft direction to account for the observed aerodynamic effects. A hierarchical Bayesian model updating is formulated and implemented to update parameters representing the effects of aerodynamic stiffness and damping, as well as their statistical properties such as mean and covariance matrix which are updated as hyperparameters. To account for the correlation between aerodynamic effects and EOCs, the updating parameters are assumed to be functions of EOCs such as rpm and wind speed. Two levels of hierarchical Bayesian model updating are performed and compared. In level 1, only modal parameters are used in model updating, while in level 2, the assumed correlation between EOCs and modal parameters are accounted for to reduce the uncertainty of aerodynamic effects in the model. In addition to hyperparameters, the proposed hierarchical Bayesian approach provides statistical properties of error function by collecting the residual uncertainties that have not been accounted for, e.g., modeling errors, measurement noise and random disturbances. The predicted modal parameters using two levels of hierarchical Bayesian approach are compared with their identified counterparts, and the results indicate that level 2 approach significantly reduces estimation uncertainty in aerodynamic effects and produces model predictions consistent with observed values of natural frequencies and damping ratios.

The joint identification of the input-parameter-output from sparse measurements is essential to evaluate the safety condition of civil infrastructures. Existing physics-based methods require that the loading locations and covariances of process and observation noises are available. In contrast, pure data-driven approaches are limited in their generalization ability and interpretability. To address these issues, a physics-data-driven method is proposed for joint load-parameter-response identification. It consists of a data-driven and two physics-based modules. The former is a convolutional residual autoencoder (RAE) for load identification. In the physical modules, the finite element method is used for model updating, and the Newmark-beta algorithm is embedded to solve the structural dynamics. They are encoded into a deep learning architecture to realize the forward response calculation and backward system identification. Subsequently, the difference between the predicted and measured responses is formulated as the loss function, by minimizing which the RAE is trained, and the structural parameters are identified simultaneously. In such a process, structural property and load time history are learned in a novel self-supervised manner using output data only. The proposed method is finally applied to a numerical two-span beam and a laboratory-tested cantilever beam. Results show that the proposed approach can identify the structural load, parameters, damage, and responses accurately. The effect of observation noise, sensor placement, and load conditions on the identification results are discussed.

Updating Finite Element (FE) models based on modal data has become a standard tool for condition monitoring and damage detection in structures. The basis of this approach is to minimize the discrepancy between the experimental and analytical modal frequencies and shapes through a particular loss function derived through deterministic or probabilistic methods. However, the weighting of modal features can affect the identified structural model parameters, which requires proper treatment. This paper revisits this problem by proposing a novel hierarchical Bayesian formulation, wherein the covariance matrix of modal residuals is identified from the data, leading to automated weighting of the modal residuals. As the calculation of the posterior distribution encounters computational overheads, an efficient Expectation-Maximization algorithm is proposed to approximate the optimal values of the parameters. Bayesian information criterion is employed to select the most plausible layout for the prediction errors covariance matrix. A new strategy is also established for handling dynamical modes with close modal frequencies. Finally, the proposed approaches are examined using simulated and real data from FINO3 offshore platform. As a result, reasonable estimations with desirable uncertainty bounds are acquired. Although the weighting of modal features varies on a case-to-case basis, it is observed that the proposed method might lead to non-trivial and interesting weighting factors according to the data-driven uncertainty of modal residuals.

Predicting bridge deflection is crucial for identifying potential structural issues, as sustained deviations from the expected range may indicate stiffness degradation. To address the stochastic modeling errors often overlooked by existing methods, this paper proposes a Bayesian Dynamic Noise Model (BDNM) for predicting the daily average deflection of bridge structures. The dynamic noise equations are formulated based on measured deflection data and incorporate modeling errors. Using Bayes’ theorem, a recursive BDNM process for bridge deflection prediction is established. Within a Bayesian forecasting framework, key parameters, particularly the coefficient and variance of modeling errors, are estimated using the method of moments, while the Bayesian discount factor is determined using Bayesian optimization. In addition, a novel prediction interval formula is developed, considering both modeling errors and monitoring uncertainties, based on the additivity of the normal distribution. This prediction interval is used as an anomaly detection threshold, and the estimated modeling errors from within the model are employed as damage indicators. The model is validated using monitoring data from an in-service bridge and compared with several common methods. Results demonstrate that the proposed method achieves high prediction accuracy and provides reasonable prediction intervals. Simulated scenarios of increased response variability due to stiffness degradation further illustrate the model’s sensitivity to structural behavior anomalies. This method lays a theoretical foundation for developing real-time warning systems for in-service bridges.

Accurate deflection prediction of in-service bridges can be used to assess the overall structural stiffness and detect abnormal states in advance. The bridge structures, especially long-span bridges, experience varying environmental and operational conditions, including temperature, humidity, wind excitation, and traffic loads, as well as long-term material deterioration and stiffness degradation mechanisms, and therefore, their deformation behavior shows complex variation phenomena, which pose challenges to many current deflection prediction methods. To address this subject, a Bayesian dynamic difference model (BDDM) to predict bridge deflection behavior online is proposed in this paper, explicitly considering the effect of the nonstationarity of time series data under varying environmental and operational conditions. A novel dynamic difference model is first proposed to include the nonstationary residual term and provide a linear approximation of a complex nonlinear process. Then, the formulas for recursively updating the dynamic difference model based on Bayesian inference are proposed. The proposed method is first validated through a numerical application using simulated nonstationary time series data with a nonlinear trend, indicating that it can adaptively capture nonstationary variations, update noise variance estimations, and improve prediction accuracy. To further demonstrate its performance, the BDDM is employed to predict the daily maximum deflection of a real-world cable-stayed bridge using measured data, and its performance is compared with several existing methods. The findings reveal that the proposed method outperforms other methods in terms of prediction accuracy, and can be potentially implemented for an online monitoring and early warning system.

Operating in harsh and unsteady marine environment, offshore wind turbine (OWT) structures are exposed to unpredictable wind and wave loads. Identifying the structural loads and their effects on the OWTs allow for predicting the remaining fatigue life of these structures and improving the structural design procedure. In this paper, a finite element (FE) model inversion method is presented to estimate the unknown loads and model parameters of OWTs using sparse measurement data. A realistic FE model of an OWT structure with jacket substructure is created in the open-source simulation platform, OpenSees. A Bayesian inference framework is presented to integrate the measured data with the FE model to estimate unknown wind loads and mass of rotornacelle assembly. To evaluate the performance of this data assimilation framework, the effect of sensor type, number of sensors, and modeling errors on the estimation accuracy of wind loads and model parameters are investigated through different case studies where synthetic data are used as measurements. The results of this study are important to guide instrumentation of new OWT structures, and to understand the potential limitations and sources of error in the real-world application of this data assimilation framework for joint model parameter and input load estimation.

Structural Health Monitoring (SHM) systems provide extensive data on in-service bridges, which is crucial for evaluating structural performance. However, data loss frequently occurs due to environmental factors and technical failures. Although missing data reconstruction problem has been largely studied, the accuracy of response reconstruction in the frequency domain is often overlooked. To address this issue, this study proposes to integrate a time-frequency joint loss function within an Arithmetic Optimization Algorithm-Temporal Convolutional Network (AOA-TCN), which leverages temporal and spatial dependencies among measurement points to reconstruct missing dynamic deflection data. The time-frequency joint loss function is enhanced with linear decay weights to increase sensitivity to low-order modal contributions, with an adaptive weighting mechanism balancing loss contributions from time and frequency domains. These adaptive weight parameters are optimized through AOA to enhance the TCN's generalizability. The feasibility and effectiveness of the proposed method are first demonstrated using simulated dynamic deflection data of a finite element model of a continuous beam bridge, and then experimentally verified on a real-world simply-supported beam bridge. The results indicate that the proposed method achieves higher data reconstruction accuracy and more precise modal analysis results compared to existing methods. The Modal Assurance Criterion (MAC) between mode shapes identified using measured data and reconstructed response reached 96.9 %, proving that the proposed method is capable of reconstructing the missing structural response while retaining its frequency-domain information, which is valuable for structural integrity assessment based on modal parameter identification of operational bridges.

In structural health monitoring (SHM), an important issue is the limited availability of measurement data due to the spatial sparsity of sensors installed on the structure. These measurements are insufficient to accurately depict the actual dynamic behavior and response of the structure. Therefore, full-field (i.e., every degree of freedom) structural response reconstruction based on sparse measured data has drawn a lot of attention in recent years. Kalman filter (KF) is an effective technology for response reconstruction (also known as state estimation), providing an optimal solution for systems that can be well-represented by a fully known Gaussian linear state-space model. This implies that both the process noise and measurement noise follow known zero-mean Gaussian distribution, which is impractical in many civil engineering applications considering the unavoidable modeling errors and variations of environmental conditions. To address this challenge, a data-physics hybrid-driven method, i.e., KalmanNet, is proposed in this study for response reconstruction of partially known systems. By integrating a recurrent neural network (RNN) module into the KF framework, KalmanNet can efficiently learn and compute the Kalman gain using available monitoring data, without any Gaussian assumptions or explicit noise covariance specifications (e.g., covariance matrices of process and measurement noise). Both numerical and experimental investigations are conducted to validate this method. The results demonstrate that under the influence of non-Gaussian noise and modeling errors, KalmanNet can effectively and accurately reconstruct the structural response from sparse measurements in real-time and has higher accuracy and robustness compared to traditional KF even with optimal parameter settings.

Excluding the variations of structural dynamic characteristics due to operational loads in finite element model updating (FEMU) is essential for response prediction and structural condition assessment. However, existing FEMU techniques for long-span bridges usually directly use modal parameters in the objective function without considering the effects of various operational loads, resulting in high variations of the updated model. To this end, this paper proposes a hierarchical Bayesian FEMU framework considering the effects of operational loads, including temperature and traffic loads. A linear temperature-elastic modulus relationship and a vehicle load estimation method based on weigh-in-motion (WIM) data are established to quantitatively consider the operational loads' effects on structural dynamic properties. In addition, the identified natural frequencies and measured expansion joint displacements are incorporated into the objective function to update structural static and dynamic properties simultaneously. A two-step Markov Chain Monte Carlo (MCMC) sampling method and a response surface surrogate model are proposed to accelerate the updating process. The proposed method is validated on a long-span arch bridge using two-year monitoring data. The updated model is then used to predict the structural static and dynamic responses by taking into account of operational loads, parameter uncertainties, and modelling errors, and the measured responses generally fall within the predicted 95% confidence interval. Finally, a structural state indicator, which has considered the effects of operational loads on structural properties, is defined to assess the structural condition and successfully detect the pavement replacement process on the bridge.

The expanding structural health monitoring (SHM) systems on bridge structures have provided an abundance of multi-source data for finite element model updating (FEMU). The SHM systems on bridges usually include surveillance cameras, vibration sensors (e.g., accelerometers, strain gauges, and displacement sensors), and sometimes a weight-in-motion (WIM) system. Currently, the majority of FEMU studies focus on identified modal parameters derived from vibration data, neglecting the incorporation of video and WIM data in the updating process, which impedes a thorough quantification of uncertainty associated with the structural parameters of interest. Therefore, this paper proposes a hierarchical Bayesian FEMU framework to comprehensively integrate a variety of information sources, including videos, WIM, and vibration data. The data features comprise the static deflections of the bridge under traffic load and modal parameters identified from acceleration measurements. The measured static deflections are extracted from raw displacement data using the locally weighted regression and smoothing scatterplots method. Computer vision-based technology is employed to pinpoint the location of vehicle load on the bridge, which is then integrated into a FEM to predict vehicle-load-induced static deflection. A two-stage Markov Chain Monte Carlo sampling approach is proposed to evaluate the high-dimensional posterior distribution efficiently. The effectiveness of the proposed method is demonstrated on a laboratory three-span bridge model. The results show that the hierarchical Bayesian FEMU can provide accurate estimation and uncertainty quantification on structural stiffness and mass parameters. The updated model accurately predicts both static deflection and modal parameters, exhibiting model-predicted variability in close alignment with the identified values for observed and unobserved responses. Remarkably, this holds true even for unseen loading conditions which are not included in the updating process. These observations validate the capability of the proposed method for multi-source data fusion and uncertainty quantification of real-world bridge structures under operational conditions.