# Prediction of the Load-Bearing Behavior of SPSW with Rectangular Opening by RBF Network
## 3 Neural Network Modeling  
Nowadays, artificial neural networks (ANNs) have been used in several engineering problems for modeling or function approximation applications [41; 42]. ANN is used in this paper to model the effects of the rectangular openings in the SPSWs with different plate thicknesses. The RBF is selected for modeling as the neural network.  
RBF network is a feedforward network that has one hidden layer and its activation functions are radial basis functions [43]. RBF output is a linear combination of the inputs radial basis functions and neuron parameters. The RBF network structure is similar to the MLP network, but there is one hidden layer in the RBF network. The neurons in the hidden layer of the RBF network are called the RBF centers [44]. The hidden layer outputs are summed in the output layer, as shown in Equation (1).  
\[y=\sum_{i=1}^{n}w_{i}qi, \tag{1}\]  
where, \(w_{i}\) are the weight values. The neuron outputs in the hidden layer, \(qi\), can be written as Equation (2):  
\[qi=\exp\left(-\frac{(x_{i}-c_{i})^{2}}{\sigma_{i}^{2}}\right). \tag{2}\]  
In Equation (2), \(X\) is input vector, \(C\) is the basis function and \(\sigma_{i}\) is the spread of the Gaussian function. Radial basis functions are frequently used to create neural networks for regression-type problems [45].  
As mentioned earlier, the rectangular opening effects should be studied in the behavior of the SPSW. In the proposed model, rectangular opening surface to overall SPSW plate surface ratio (A\({}_{o}\)/A\({}_{s}\)) and thickness of the SPSW plate, H (mm), are assumed as the model input parameters. The model output variables are normalized shear strength, P, normalized energy absorption, E and the normalized stiffness, K. The structure of the proposed ANN model is illustrated in Figure 6.  
After defining the model parameters, the RBF model specifications should be selected. The neurons number in the output and input layers are considered equal to the number of output and input parameters, respectively. Several RBF models with different specifications were defined to find the best network. The proposed RBF network schematic is shown in Figure 6 and the network specifications are given in Table 4.  
Figure 6: Structure of the proposed radial basis function (RBF) model.  
The performance of the network evaluated using mean absolute and mean squared errors (MAE and MSE) in this paper, which are defined as follows.  
\[MAE=\frac{\sum_{i=1}^{N}|Y_{Ri}-Y_{Pi}|}{N}. \tag{3}\]  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}\left(Y_{Ri}-Y_{Pi}\right)^{2}, \tag{4}\]  
where, in these equations, \(Y\) is the real value and \(N\) is total data number. The \(R\) and \(P\) indices represent the real and predicted values. 110 shear walls with different rectangular opening sizes and different infill plate thicknesses were modeled and analyzed in this paper, which forms data set for the proposed model. We used 77 data for training the network model and the 33 data are chosen for the testing process of the proposed model. Data are normalized before training to increase the precision of the model.