PSI - Issue 77

Douaa Benhaddouche et al. / Procedia Structural Integrity 77 (2026) 152–160 Author name / Structural Integrity Procedia 00 (2026) 000 – 000

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structure to extract relevant features of the structural state and forecast the future structural responses. The prediction errors are employed then to identify damages and to estimate the global structural state. The remainder of this paper is organized as follows: Section 2 provides an overview of the proposed method. Section 3 presents the experimental setup and results that validate the effectiveness of the proposed method, along with a discussion of the main findings. And Section 4 concludes the paper and outlines directions for future research. 2. Methodology In recent years, Graph Neural Networks (GNNs) have gained significant attention in anomaly detection (Deng and Hooi, 2021; Ma et al., 2021) due to their capacity to model complex and irregular data structures. This makes them highly applicable in domains such as sensor networks, cyber-security, and SHM, where the detection of anomalous entities or interactions between them can prevent system failures. Graph Convolutional Networks (GCNs) are the most widely known and studied GNN (Wu et al., 2022). They extend the concept of convolution from grid data to graph data, enabling efficient feature extraction by aggregating information from each node’s local neighborhood. When GCNs are combined with other architectures such as autoencoders or recurrent neural networks (RNNs) like LSTMs they can be used for advanced tasks such as time-series forecasting and signal reconstruction on graphs. The methodology proposed in this paper is a forecasting-based approach, integrating LSTM units to predict future structural response data from features extracted by the GCN. In this method, damage identification is performed using prediction errors, while the global structural state is estimated with the Kolmogorov – Smirnov statistic (KS statistics). More details about each stage of the method are provided in the following subsections. 2.1 Constructing graph from accelerometer network Sensor graph structure is established using adjacency matrix of sensors. This adjacency matrix rely on similarities between the sensor records. In bridge monitoring, vibrations caused by loads propagate along stay cables at finite speeds determined by the cable’s tension and mass per unit length, leading to delays of up to a second between sensors far apart. Along the deck and towers, faster wave speeds result in only millisecond-scale lags. Additionally, delay can be introduced by sensor filtering, wireless transmission latency, and unsynchronized clocks. These physical and measurement system effects distort the similarity between sensor, thereby motivating the use of techniques that handle this delay. Dynamic Time Warping (DTW) is is a widely used method in time series analysis, designed to identify the optimal alignment between two timeseries, by exploring temporal distortions between them (Sakoe and Chiba, 1978). Unlike Euclidean distance, which requires sequences to be of equal length and aligned point-to-point, this method allows for flexible alignment by "warping" the time axis. It slides one sequence over the other to minimize the cumulative distance between corresponding points. Let us consider two time series and ′ of respective lengths n and m . Here, all elements and ′ are assumed to lie in the same-dimensional space and the exact timestamps at which observations occur are disregarded. The DTW distance between and ′ is given by: ( , ′ )= min ∈ ( , ′ ) ( ∑ ( , ′ ) ( , )∈ ) 1 (1) where ( , ′ ) is the set of all possible alignment paths between and ′ and is a sequence of index pairs (i, j ) telling us which points in are matched with which points in ′ . ( , ′ ) is a local distance between point and point ′ (often Euclidean distance if they are scalars). The ∑ ( , ′ ) ( , )∈ is the total accumulated cost along a given alignment path. To construct the adjacency matrix, a threshold is set to preserve dependencies between sensors based on their similarity, ensuring that each sensor is connected to at least two others with similar response signals. 2.2 Spatio-temporal feature extraction and modeling In bridge monitoring, vibration responses at different locations are shaped by geometry, boundary conditions, and stiffness – mass distribution, which together define the spatial signature of the structure. This means that the