## 5 Deep Neural Network Framework  
An ANN is a type of mathematical model that resembles the biological characteristics of the human brain. It has numerous artificial neurons in input, output, and hidden layers, which are fully connected. In this study, the DNN algorithm, which has multiple hidden layers (minimum two or above), was employed. The DNN is a type of ANN with several hidden layers between the input and the output layers, whereas the ANN model has one hidden layer. It contains multiple neurons (or nodes) arranged in the input, hidden, and output layers. All the neurons at each layer are fully connected and the information flows in one direction from the input layer to the output layer through the hidden layers. Recently, neural networks have been widely used to model complex nonlinear relationships that cannot be solved deterministically using traditional mathematical models. In the general structure of an ANN or DNN, the neurons in the layers are linked through weights and bias, as illustrated in Figure 8a,b and in Equation (11). The accuracy of ANN or DNN models depends on the accuracy of the weights and bias, which are updated  
Figure 7: Comparisons between bracing moments of design provision and FEA: (**a**) cross-section types; (**b**) W-sections; and (**c**) built-up sections. by a backpropagation process. The input vector of the neuron is \(x=[x_{1},\ x_{2},x_{3},\cdots,x_{n}]\) and the weighted sum of the input (\(z\)) is expressed as in Equation (11). \[z=\sum_{i=1}^{n}w_{i}x_{i}+b_{i}. \tag{11}\]  
where \(z\) is a weighted sum of the input; \(w_{i}\) is the weight, which is equal to \([w_{1},\ w_{2},\cdots,w_{n}]\), and \(b_{i}\) is the bias. The weighted sum of the input (\(z\)) is converted into y in Equation (12), in which \(y\) is an activation value obtained through an activation function. \[y=\sigma(z) \tag{12}\]  
The activation function is the essential element during training for deep neural networks. In this study, ReLU functions were used, as shown in Equation (13). In the ReLU activation function,  
\[\sigma(z)=\max(0,z) \tag{13}\]  
In the neural network algorithms, the weights of each link between neurons at each layer are initially assumed with arbitrary values, and then they are adjusted during the training process, which is called backpropagation. The results of the training are evaluated by loss functions, such as MSE or other error estimation functions, by evaluating actual and predicted values. These evaluation processes are so-called epochs. To implement the training process using the DNN algorithm and provide accurate prediction models, Python and TensorFlow were used in all process steps. In this study, DNN models are developed for predicting bracing moments and forces in torsional bracing steel I-girder systems, using a database of 16,576 samples, as shown in Table 4. The database is divided into two parts, which comprise 80% for training and 20% for testing. Furthermore, the training sets include 25% for validation to evaluate the accuracy of the prediction models. The best prediction models can be determined by a parametric study of the hyperparameters. The hyperparameters are parameters used in DNN algorithms affecting  
Figure 8: Neural network structures: (**a**) a typical artificial neural network and (**b**) an example of the deep neural network structure used in this study. the accuracy of the prediction models. Table 7 shows the range of the hyperparameters used in the DNN algorithm. In this study, we use 16 input neurons, and one and two neurons in the output layer for \(F_{br}\) and \(M_{br}\) predictions, respectively. To evaluate the accuracy of the prediction models, this study included a parametric study on the number of hidden layers and the number of neurons at each hidden layer. The number of neurons at each hidden layer ranges from 50 to 200 with three to five hidden layers. The MSE is used as a loss function to evaluate the accuracy of the weights in the neural network.