## 2 Materials and Methods
### Model Architecture and Training  
The proposed SHM application now utilizes the physics-driven, synthetic and pristine training data for the identification of damages in strain sensor data acquired from the real structure. The underlying concept is to learn the behavior of each strain value based on the behavior of all other strain values. In unseen data, each strain value is isolated and the remaining strain values are used as input to predict the value of the isolated element. The resulting prediction error is finally used as an anomaly score, characterizing the health state at the corresponding sensor location. The outlined task is broken down into three main process steps, namely training, strain data transformation and damage identification. To learn the relationship between strain sensor values, the developed overall model consists of an ensemble of regression models, which is governed by the layout of the sensor grid providing the data. Each strain value measured by the sensor grid is considered as a feature \(j\), where \(J\) is the total number features. In the training stage, the synthetic training data is first separated into feature data \(\mathbf{X}_{j}^{R}\in\mathbb{R}^{L\times J-1}\) (strain values which are used for the prediction) and a target label \(\mathbf{y}_{j}^{R}\in\mathbb{R}^{L}\) (target strain which is to be predicted), where \(L\) denotes the number of samples and \(R\) denotes the reference configuration of the structure, i.e., the pristine state. This data is then further split into a training and a benchmark set. A regression model \(f_{j}\) is subsequently fitted with the training data, resulting in the trained model \(f_{j}^{R}\). With this, the benchmark set is passed to \(f_{j}^{R}\), which provides predictions \(\mathbf{\hat{y}}_{j}^{R}\). The residual \(\delta_{j}^{R}\in\mathbb{R}^{L}\) is then calculated element-wise using the absolute error  
\[\delta_{j}^{R}=\Big{|}\mathbf{y}_{j}^{R}-\mathbf{\hat{y}}_{j}^{R}\Big{|}. \tag{4}\]  
This serves as a benchmark for the following damage identification. The residuals of all features together define the benchmark residual set \(\mathbf{\Delta}^{R}\in\mathbb{R}^{L\times J}\). Now, when passing unseen strain data of the real structure to the trained model, the prediction results of the fitted regression model \(f_{j}^{R}\) are again evaluated against the corresponding true strain value. For this, the sample, which is to be classified, is once more separated into feature data \(\mathbf{x}_{j}\in\mathbb{R}^{J-1}\) and target label \(y_{j}\). The feature data is then directly fed as input to \(f_{j}^{R}\), which produces a prediction \(\hat{y}_{j}\). The residual \(\delta_{j}\) is calculated as in (4) and the vector of all \(J\) residual values of one sample is subsequently denoted as \(\delta\in\mathbb{R}^{J}\). When the fitted regression models \(f_{j}^{R}\) are provided with strain values from a pristine structure, they are able to reproduce each strain \(j\). In contrast, when they receive strain values from a damaged structure, there will be a larger difference between true and predicted value, consequently indicating an anomaly. The above describes the transformation stage, in which sensor grid strain data from the feature space is transformed into a residual space, where variations due to different load cases are compensated and differences due to anomalies, such as damages, are emphasized. The described process is the same for all \(J\) features and is therefore summarized in a generic Strain Prediction Unit (SPU). Hence, each feature is transformed by its own SPU. The architecture and corresponding process flow of a single unit is visualized in Figure 6.  
Theoretically, any regression model can be chosen for \(f_{j}\), e.g., an ANN as chosen by [PERSON] et al. [19] or linear regression, as used by [PERSON] et al. [20]. For this SHM application, a Histogram-based Gradient Boosting Regression Tree (GBRT), provided by the _scikit-learn 1.4.1_[25] package, was implemented in _Python 3.11_.