## 3 Background
### Graph Convolutional Neural Networks  
In recent years, the field of GNNs [34] has experienced rapid growth and development. The primary aim of GNNs is to utilize graphs to transfer learning information and compute the properties of nodes and edges in the diagram. The fundamental principle entails creating a computational graph composed of nodes and expanding it to encompass all feasible nodes. A specific aggregation technique is utilized to map the structural data within the graph, allowing the model to grasp the relationships between nodes and the characteristics of edges [35]. This approach empowers the model to provide a more comprehensive portrayal of the node associations and interactions within the graph. The advancement of GNNs holds considerable potential for applications in diverse fields [36]. The GCNM, as opposed to conventional neural networks (CNN) that exclusively function in Euclidean spaces, is a deep neural network that effectively handles input that adheres to a graph structure [21]. The GCNM has recently garnered significant attention owing to its versatility in various industries [37]. To perform edge prediction, graph classification, and node classification, the GCNM offers a technique for extracting features from graph data. Graph convolution signifies a feature propagation methodology in semi-supervised learning, surpassing a mere label propagation technique. [PERSON] et al. integrate a GN and finite element technique to expedite finite element simulations [38]. The graph is constructed using the discretized geometry derived from a finite element pre-processor, and the GNN is deployed to address the boundary value problem in the discretized domain. The finite element grid is utilized in this study to approach the optimal solution through the utilization of the iterative Newton-type optimization method. Usually, the image of \(x\) is obtained through the solution of an inverse problem from impedance data acquired from measurements. Still, the combination of the current iteration \(x_{k}\) and its updated \(\phi x_{k}\) is changed. The classical method obtains the next iteration as \(x_{k+1}=x_{k}+\phi x_{k}\), where \(\phi x_{k}\) is calculated by a particular optimization technique, such as the \(LM\) method or the GN method. Then, the trained network block \(\Lambda_{\theta_{k}}\) and the adjacency matrix \(\mathcal{A}\) are used for computing the next iteration, where \(\phi x\) is the conventional update, such as GN, LM, etc. \[x_{k+1}=\Lambda_{\theta_{k}}([x_{k},\phi x_{k}],\mathcal{A}) \tag{5}\]  
The graph convolution block takes as input the concatenated terms \(x_{k}\) and \(\phi x_{k}\). As seen in Figure 2, the output of the GCN block delivers \(x_{k}+1\) for the following iteration. In this study, the topology of the network block \(\Lambda_{\theta}\) remains constant between iterations; however, each block has a unique set of trainable parameters \(\theta_{k}\). We use the LM algorithm in the experiment for iterative solutions without explicit data priors since we want the network to be able to learn features from the training set. As with previous model-based methods [39, 40], there are two alternatives for training the network: end-to-end training of the complete system consisting of \(k_{\text{max}}\) blocks, or sequential training of each block. Nevertheless, for our scenario, conducting end-to-end training is not feasible because of two primary factors. Initially, performing back-propagation through the updates \(\phi x_{k}\) would be necessary to update the network parameters. To carry out plagiarism checking, adjustments can be made to the text to avoid consecutive identical words. Firstly, it is not possible to compute these updates by assessing the model equations using a FEM solver. Secondly, the evaluation of the model equations consumes a considerable amount of time and leads to extensive training periods. Hence, our approach is to adopt a sequential training method where each block is trained separately.