# Research on 3D Geological Modeling Method Based on Deep Neural Networks for Drilling Data
## 3 Methods  
### Deep Neural Network  
A deep neural network is a type of neural network with multiple hidden layers, achieved by extending the binary linear perceptron model through the addition of hidden layers, expanding the output layer, and incorporating various activation functions. These hidden layers can comprise a single neuron or multiple neurons, substantially enhancing the neural network's modeling capabilities while simultaneously increasing its complexity. Likewise, the output layer can accommodate multiple neurons, enabling the model to flexibly represent diverse information, rendering it well suited for applications in classification, regression, dimensionality reduction, clustering, and various other domains. Activation functions provide a wide array of options, each with its own adaptability, advantages, and limitations [36; 37; 38; 39; 40; 41]. The selection of different activation functions can further enhance the neural network's expressive capabilities. Taking into account the representation of borehole data and the characteristics of 3D geological modeling, the commonly employed Sigmoid function in neural networks, originally designed for binary classification, may not be the most suitable choice. In contrast, Softmax is well suited for multi-category classification, and ReLU effectively mitigates the gradient vanishing issue. Both of these activation functions are more appropriate for the task of 3D geological modeling.  
A deep neural network comprises hidden layers, with each neuron in layer \"i\" connected to every neuron in the subsequent layer, \"i + 1\". Although deep neural networks may appear complex, they fundamentally integrate linear relationships and activation functions at specific locations. The linear coefficients, often referred to as weights, within a deep neural network establish connections between different layers of the network. Each neuron is associated with a corresponding weight as it connects to the subsequent layer. These weights determine the strength of the connections between neurons in the preceding layer and those in the succeeding layer. They are utilized to linearly transform the input data by performing a weighted sum with the input values. Throughout the training process, the neural network autonomously learns and adjusts these weights to minimize the loss function. In deep neural networks, biases are constant offsets assigned to each neuron within the network. They function as additional parameters for each neuron, working independently of the input data. Biases enable neurons to activate even in the absence of input, and they provide a means to adjust a neuron's activation threshold by modifying the bias. These biases introduce a form of translational invariance within the network, thereby enhancing the network's flexibility and expressive capabilities.  
During the forward propagation of deep neural networks, individual neurons are situated within distinct layers. Neurons within each layer receive signals from neurons in the preceding layer and generate signals for subsequent transmission to the following layer. The network operates without feedback, and the propagation of signals follows a unidirectional path from the input layer to the output layer. Information within neural networks is propagated through the continuous iteration of Equations (1) and (2).  
\[\mathbf{Z(l)}=\mathrm{W(l)a(l-1)+b(l)} \tag{1}\]  
\[\mathbf{a(l)=f_{l}(Z(l))} \tag{2}\]  
In this equation, \(\mathbf{Z(l)}\) denotes the net input or net activity value of the neuron in layer \"l\", \(\mathrm{W(l)}\) represents the weight matrix connecting layer \"l - 1\" to layer \"l\", \(\mathbf{a(l)}\) denotes the output or activity value of the neuron in layer \"l\", \(\mathbf{b(l)}\) signifies the bias from layer \"l - 1\" to layer \"l\", and \(\mathrm{f_{l}(^{*})}\) denotes the activation function of the neuron in layer \"l\".