## 2 Construction of Finite Element and Neural Network Models
### Neural Network Model  
The artificial neural network is a parallel, distributed processing structure composed of processing units interconnected with undirected signal channels called connections. These processing units have local memory and can perform local operations that depend solely on the current value of all input signals that arrive at the processing unit via the input link as well as the value stored in the processing unit's local memory. Each processing unit has a single output connection, and the output signal can be any desired mathematical model. There are many kinds of neural networks, such as radial basis neural networks, Hopfield networks, competitive neural networks, and back-propagation networks [19; 20; 21].  
The neural network is a typical multi-layer feedforward (usually three-layer) artificial neural network. Structurally, it consists of input, hidden, and output layers with nodes in each layer. Adjacent layer nodes are connected by weights, but the nodes in each layer are independent of each other. Theoretically, it has been proven that a neural network with a single hidden layer can approximate any nonlinear function with arbitrary precision [22; 23; 24]. In practical applications, the neural network with a single hidden layer can also meet the needs of calculation well. The structure diagram of the neural network is shown in Figure 1:  
In the figure, \(x_{n}\) is the input variable; \(i\), \(j\), and \(k\) are the number of nodes in the input layer, hidden layer, and output layer, respectively; \(w_{ij}\) and \(w_{jk}\) are the connection weights between layers; and \(y_{ij}\) and \(y_{jk}\) are the output value of the hidden layer and the output value of the output layer, respectively. The flow of the operation process is shown in Figure 2:  
Figure 1: Structure diagram of neural network.  
The learning process of the neural network algorithm consists of forward propagation and back-propagation. In the forward propagation process, the input signal of the learning sample is first sent to the input layer. Then it is transmitted to the output layer through the hidden layer, and the corresponding predicted value is output after the calculation of the output layer. When the error between the predicted value and the real value does not meet the preset target accuracy requirements, the network turns to back-propagation. The network feedbacks the error information from the output layer to the input layer by layer, and adjust the weights and thresholds between the layers. Through repeated loop iterations, the error between the output value of the network and the expected output value of the sample is gradually reduced until the set number of loops or accuracy requirements are met. At this point, the learning process of the network ends, and the optimized weights and thresholds are obtained. Then, based on the internal relationship, the input information of the unknown sample is extracted, and the mapping of the unknown sample can be obtained.  
In this paper, the iron core material was an isotropic linear material. Additionally, the final magnetic field cloud map B was related to the coil current, the number of coil turns and the magnetization model. As shown in Table 1, three input quantities were used for the research.  
Figure 2: Operation process of neural network.  
The study considered two magnetization models. The relative permeability was used for the linear material, and the _B-H_ curve was used for nonlinear material. The _B-H_ curve for nonlinear material is shown in Figure 3.  
The specific structure of the neural network model in this study is shown in Figure 4. The number of hidden layer nodes was determined by the empirical formula: \(j=\sqrt{i+k}+a\), where a is a constant between 1 and 10. Through experimental comparison, we determined that there were nine hidden layers.