## 2 Methodologies  
### Deep Belief Network (DBN)  
The DBN has the special network structure represented by the multiple-layer restricted Boltzmann machines (RBMs) [28], as shown in Figure 1. For its excellent data analysis ability, DBN has become one of the most commonly used deep learning methods, and it is widely used in big data prediction, data mining, recognition and classification [28; 29]. For DBN, the unsupervised pre-training method is applied to learn the input features by the multiple-layer RBMs. Multiple-layer RBMs will increase the upper bound of the log-likelihood, which can greatly enhance the data mining ability and improve the prediction accuracy. In comparison with ANN, in DBN, the initial weights are learned from the structure of the input data, which is closer to the global optimum. The learned weights are used as the initial values of other networks with the same structure, and thus, the drawbacks of initialization parameters falling into the local optima and a long training time can be avoided. Moreover, for its complex network structure, the DBN can treat big data problems. Nowadays, DBNs have already been used to solve the big data problems in geotechnical engineering [30]. Therefore, in this study, the DBN is applied to the performance prediction of a utility tunnel in a soft foundation during the operation period. For one RBM in the DBN (Figure 2), there are only two layers, which are the visible layer (\(\mathbf{v}\)) and hidden layer (\(\mathbf{h}\)). Therefore, the visible layer of the first RBM is the first layer of the DBN, for which the original data are inputted. The hidden layer of the first RBM is the second layer of the DBN and is also the visible layer of the second RBM. The DBN including multiple hidden layers can be constructed. For multiple RBMs in the DBN, the DBN has a strong learning ability which can extract deep features of the complex data [31]. As a typical deep learning method, the training of a DBN is complex and generally includes two parts, unsupervised learning and supervised learning, corresponding to two stages, namely, forward pre-training and reverse fine-tuning. The training process of the DBN is shown in Figure 3.  
Figure 1: Structure of the DBN. Figure 3: Training process of the DBN. Figure 2: Structure of the RBM. The detailed steps are as follows:  
(1) Parameter initialization  
First, for the first RBM, its model parameters \(\mathbf{\theta}=(\mathbf{W},\mathbf{a},\mathbf{b})\) should be initialized randomly, which include the connection weights \(\mathbf{W}=(w_{ij})\in(0,1)\) between the visible layer \(\mathbf{v}_{i}\) and the hidden layer \(\mathbf{h}_{j}\) and the biases of visible and hidden layers \((\mathbf{a}=(a_{1},a_{2},\cdots,a_{l})^{T}\) and \(\mathbf{b}=(b_{1},b_{2},\cdots,b_{j})^{T})\). The other initial parameters should be determined by the control variable method [29], including the number of RBM layers \(K\), the numbers of neurons in hidden layers \(n_{k}\) (\(k=1\), \(K\)), the learning rate \(\eta\) and the maximum iteration numbers of pre-training and fine-tuning (\(T_{1}\) and \(T_{2}\)). (2) Forward pre-training  
In this stage, the network is trained by the unsupervised learning method. Therefore, the input data have no labels, and the model parameters of DBN are updated by the greedy layer-by-layer learning algorithm [31]. That is to say, the first RBM is trained by the original input data, and thus, its model parameters can be obtained. Then, the model parameters of the first RBM remain unchanged, and its output data are used to train the second RBM until the last RBM is trained.