## 3 Background
### Graph Convolutional Neural Networks
In order to achieve this, a loss function is utilized that necessitates iterative-based optimality. This function is applied to a training set containing pairs of true \(x_{t,(i)}\) and current iterate \(x_{k,(i)}\) for \(i\) ranging from 1 to \(N\). \[\text{Loss}(\theta_{k})=\frac{1}{N}\sum_{i=1}^{N}\left(\Lambda_{\theta_{k}}([x _{k},\phi x_{k}],\mathcal{A})_{(i)}-x_{t,(i)}\right)^{2} \tag{6}\]  
To evaluate the duplication of a given text, the assessment involves analyzing a quantity \(x\) on a finite element method (FEM) mesh consisting of \(M\) elements and updating \(\delta x\). To ensure plagiarism detection, we must consider two vital elements in this evaluation: an adjacency matrix and a feature matrix. The feature matrix, denoted as \(H\in\mathbb{R}^{M\times f}\), primarily comprises rows that correspond to the nodes of the graph, while the columns embed the features defined across these nodes. On the other hand, the adjacency matrix, denoted as \(\mathcal{A}\in\mathbb{R}^{M\times M}\), is a sparsely populated matrix that depicts the interconnectedness of graph nodes. Only the non-zero entries \(\mathcal{A}_{ij}=1\) indicate the connectivity between graph nodes \(i\) and \(j\). In the given context, every component of the network functions as a node in the graph, and two components are deemed linked if they possess at least one shared mesh node. Alternatively, employing the FEM nodes as the graph nodes would be the most suitable selection if the resolution is established at that level. The problem domain is divided into a discrete FEM, where each element represents a specific region. These elements consist of nodes, discrete points within the region. To extract relevant features from the FEM solution, we consider various factors such as node values, element properties, boundary conditions, and material properties. These features are then  
Figure 2: Graph convolution structure based on [PERSON]’s method. The initial two iterations of GCNM are depicted in the **top left**. An individual block within the network is illustrated in the **top right**, where \(\Lambda_{\theta}\) represents a NN with parameters \(\theta\). Both GCNM and GResNet utilize the same block structure. In the GCNM, the input \(H^{(0)}\) is created by concatenating \(x_{k}\) and \(\phi x_{k}\), while the output \(H^{(0)}\) corresponds to \(x(k+1)\). In GResNet, \(H^{(0)}\) represents \(x_{1}\) in the first block, and in the subsequent blocks, it represents the sum of the input and output from the previous block. The **bottom** part showcases GResNet, which includes the first iteration of a traditional Newton method as input in the first block, followed by a total of \(k_{\text{max}}\) blocks. used as input for the GNN. The architecture of the network is designed by specifying the number of layers, the type of activation function, and how the layers are interconnected. The input layer of the GNN corresponds to the extracted features from the FEM solution, while the output layer corresponds to the desired output or prediction. To train the GNN, we generate a dataset by selecting a set of FEM problems, solving them using an FEM solver, and recording the input-output pairs. The input consists of the extracted features, while the output represents the desired solution or prediction. The training process involves feeding the input-output pairs to the network, calculating the loss or error between the predicted output and the desired output, and optimizing network parameters using techniques such as back-propagation and gradient descent. Once the network is trained, its performance is evaluated on a separate validation or test dataset. Finally, the trained network can be used to predict the output or solution of a new FEM problem by providing the extracted features as input to the network.