## 4 Proposed Methods
### Zone Selection-Based ANNs  
ANNs are made up of multilayered structures. Multi-layer perceptron (MLP) is the approach that artificial neural networks most frequently utilize to approximate functions. Figure 5 depicts the MLP's structural layout. MLP consists of input layers, hidden layers, and output layers, as shown in Figure 5. A typical MLP usually has multiple nodes connecting it. The network structure must first be established before building an artificial neural network. The computation of the solution only requires one hidden layer if the input nodes can be linearly separated. Finding the optimal solution is difficult in situations with eight input nodes, like the one described in this study. In this study, we built an artificial neural network training model using stochastic gradient descent (SGD) among backpropagation algorithms. SGD is a variant of gradient descent that uses minibatches, or just one data point, to calculate the error and update the weights. By dividing the data to be calculated into small batches, memory is efficiently used. Backpropagation algorithms like these show significant differences in learning effectiveness and predictive power depending on the number of hidden layers and nodes. The maximum number of learning iterations was set at 1000 to enable the determination of the ideal artificial neural network topology. The number of hidden layers was set to three, the output layer node to two, and the input layer node to eight. There were 16 nodes in the first hidden layer, 8 nodes in the second, and 4 nodes in the third hidden layer. The logarithmic sigmoid function was employed as the activation function in the hidden layers. The input layer was composed of virtual region labeling data that included node signals from the 16 virtual regions and the four fixed nodes. Results are output through the three hidden layers and two output layers. The output value was selected in four rows and four columns \((i,j)\) and was represented by zone selection in 16 cases. Because the weight changes in neural network learning were taken randomly, the results differed, and the number of attempts varied each time. Thus, the generation seed was fixed, and the optimal method and loss function were selected for learning. The model obtained the optimum MSE (highest) value using the Adam optimization method and a sequential network comprising five dense layers. The loss function for polynomial classification was defined as binary cross-entropy, while the performance  
Figure 4: Defined border line and zone number. evaluation was influenced by accuracy. The fitting method was used to carry out the learning process as with other neural networks, and callback functions were employed to prevent overfitting. To ensure the lowest loss feasible, the training process was stopped if the validation loss increased three times in a row. The learning model's findings were verified using the average squared error (MSE, mean squared error) and the coefficient of determination. The mean squared error (MSE) and coefficient of determination \(R^{2}\) were used to evaluate the performance of the learning model [33]. MSE is a statistic for evaluating the learning effectiveness of ANNs. A more reliable model is one that has a lower MSE value. R2 estimates the percentage of the dependent variable's variability that can be predicted from the independent variables. The value \(R^{2}\) = 1 shows that all sample observations lie on the estimated regression line, and the closer the value is to 1, the better the fit. To evaluate the learning outcomes, 60, 20, and 20% of the total learning data were randomly assigned for training, validation, and testing, respectively. The SGD optimization model and a sequential network with five dense layers, as previously reported [34], were used to produce the optimum MSE (best value). While accuracy affects performance evaluation, binary cross-entropy is the definition of the loss function for polynomial classification.