## 2 Research Method
### Deep Learning  
The MLP is one of the basic types of deep learning architecture. Its architecture follows the neural network system principle, learns, and predicts data. The MLP learns in the perceptron; the weight is changed after each data processing. The weight is adjusted using an algorithm, and the deviation in the training process is reduced to minimize the error in the amount of output and prediction results. The main advantage is the ability to solve complex problems rapidly. The MLP is a feedforward neural network composed of the multilayer structure of linear and nonlinear activation functions. Each layer is composed of basic elements of neurons, and each neuron is fully connected to all others in the previous and subsequent layers. Neurons in the network have a bias value \(b\) and an activation function \(f\). The connection between neurons in different layers is defined by connection weight, \(w_{i}\in R,\ i\in\{1,2,\ldots,n\}\). These parameters are updated during the training of MLP. The general activation functions are ReLU, TANH, sigmoid, and softmax. The output value \(y\) of a neuron is defined as:  
\[y=f\left(\sum_{i=1}^{n}w_{i}x_{i}+b\right) \tag{1}\]  
where \(x_{i}\) is the output value of neuron \(i\) of the previous layer, \(i\in\{1,2,\ldots,n\}\). The MLP network generally includes one input layer, one output layer, and one or more dense layers, detailed below. 1. Input layer: the number of neurons of the input layer is determined by the point \(k\) of input data. 2. Hidden layer: fully connected layer between the input layer and output layer. 3. Output layer: the neuron of the output layer is determined by the number of classes or the output of the approximation function. In the training process of MLP, the weight \(w_{i}\) and offset parameter \(b\) are updated in every iteration, and the update target is a minimum loss function. After network training, it can be used for classification or function approximation. The architecture is shown in Figure 2 [48; 49; 50; 51; 52]. In this study, we used Adaptive Moment Estimation (Adam) optimization to train MLP and minimize the Mean Squared Error (MSE). Adam optimization was applied to calculate the adaptive training rate of the parameters [53]. This method, in addition to storing the descending mean of the square of the past of the gradient or \(v_{t}\) and the mean of the descending of the past of the gradient or \(m_{t}\), is kept as momentum. This is why the momentum can be seen as a ball sliding on a sloping surface with no friction and, therefore, can be placed at the minimum error level [54]. Both parameters of the mean descending average of the square of the gradient and the average of the descending of the past of the gradient can be calculated as follows:  
\[m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t}\]  
\[v_{t}=\beta_{2}v_{t-1}+(1-\beta_{2})g_{t}^{2}\]  
where \(g_{t}\) is the gradients at subsequent time steps, \(m_{t}\) and \(v_{t}\) are estimates of the first moment (the mean) and the second moment (the uncentered variance) of the gradients, respectively (hence the name of the method). As \(m_{t}\) and \(v_{t}\) are initialized as vectors of zeros, the authors of Adam observe that they are biased towards zero, especially during the initial time steps, especially when the decay rates are low (i.e., \(\beta_{1}\) and \(\beta_{2}\) are close to 1). The prediction of unknown data has always been a complex problem, and it is also an important application in neural networks. The high-level network architecture of deep learning can more accurately predict and describe the distribution and changes of unknown data. The ERT profile is an inverted trapezoid; two sides are blank and short of resistivity data. This study used machine learning, data prediction, and computational ability to reduce the error between the actual value and prediction value of MLP for the blank in the field test ERT profile by MLP. The result was obtained after reiteration to convergence, forming a complete rectangular section.