# A Deep Learning-Based Approach for the Identification of a Multi-Parameter BWBN Model
## 3 Deep Learning Identification Model
### The Network Parameters of the Deep Learning Model  
The back-propagation (BP) neural network is a multi-layer feed-forward neural network (FNN), which is trained according to the error backpropagation algorithm. Feed forward refers to the data flow direction and reverse refers to the error flow direction. The BP neural network is composed of two modules: a forward-propagation network of information and a back-propagation network of error. The signal propagates forward and the error propagates backward. The process of the BP neural network is mainly divided into two stages. The first stage is the forward propagation of the signal, from the input layer to the hidden layer and finally to the output layer. The second stage is the error back-propagation, from the output layer to the hidden layer, and finally to the input layer, adjusting the weight and bias from the hidden layer to the output layer, the weight and bias from the input layer to the hidden layer in turn. Commonly used activation functions that are utilized include linear rectification function (ReLU), Sigmoid function and tanh function. Among them, the nonlinear activation functions are sigmoid, ReLU, and tanh. In this paper, we choose the ReLU function for the nonlinear identification problem. The BP model reversely adjusts the value of neuron weight by the loss function. The commonly used loss functions are mean square error (MSE), the sum of the square error (SSE), and mean absolute error (MAE). Among them, MSE is selected, which is more suitable for continuous value regression. The process of numerical iteration to approach the optimal solution needs an optimization algorithm. The common optimization algorithms include the gradient descent method, the Newton method, momentum, the Nesterov Momentum, the Adagrad, the Adam, and the Rmsprop [69; 70]. The Rmsprop is used to solve the problem of excessive swing in the optimization process and is suitable for parameter identification of the BWBN model. The learning rate of this paper is 0.9.  
The learning data come from the experimental data of the PEER database. In total there are 416 sets of data which are divided into FS, FF and SF failure modes. In order to study the generality of the present method, only some failure data are deducted, the Failure data are some test distortion data. The remaining 386 groups of data are used as learning data. The training process is that the supervised label of each group of experimental data, namely BWBN model parameters, is established by using the traditional NSGA-II algorithm. Its evaluation index is given by Equation (25). Then, the regression characteristic solution of the feasible solution region is found. The regression analysis adopts the BP deep learning network model. Its evaluation index is provided by Equations (23) and (24). The appropriate parameter solution can be found through the training of neural networks. The optimal hyperparameters of the network model can be found. The number of hidden layers and neurons of the deep learning model are shown in the following Table 1. It can be seen that the more layers and neurons there are, the better. The prediction accuracy is not positively correlated with the model complexity. Simply increasing the model complexity will produce over-fitting phenomenon, which can add a dropout layer to the model. Therefore, for different engineering problems, the complexity of the model needs to be determined separately, for example, the stress problems of RC beam and RC column need to be determined separately.