## 3 Methodology
### Cascade forward Neural Networks (CFNNs)  
CFNN is a specialized form of the MLP with a unique network structure. Generally, CFNN includes an input layer, one or more hidden layers, and an output layer, mirroring the architecture of the MLP model. However, unlike the MLP, which employs a feedforward network structure to transmit data from the input to output layers via hidden layers, CFNN introduces additional direct connections between the input and output layers [40]. This difference results in CFNN having a denser interconnection of weights and biases compared to the traditional MLP. The CFNN architecture in this study is depicted in Figure 2.  
The CFNN model constructs its cascade architecture within the network by incorporating newly added neurons and their connections. The additional connections between the input layer and subsequent layers potentially enhance the learning rate of the network by adjusting the weights of the newly formed neurons exclusively [40]. The total inputs of the \(j\)th hidden node from the \(i\)th input node are described as Equation (5):  
\[n_{inputs,j}=\sum_{i=1}^{N}w_{ij}u_{i}\quad 1\leq j\leq N \tag{4}\]  
where \(N\) represents the count of input variables; \(w_{ij}\) signifies the weight connecting the \(i\)th input node to the \(j\)th hidden node; and \(u_{i}\) is the \(i\)th input. The output of the \(j\)th hidden node is determined as Equation (6). \[O_{hidden,j}=f(n_{\text{input},j}) \tag{5}\]  
Figure 2: CFNN architecture in this study. Ultimately, the output of the CFNN model is calculated as Equation (7). \[Y_{n}=g(\sum_{j=1}^{M}w_{jn}^{2}O_{j}+\sum_{i=1}^{N}w_{i,n}^{3}u_{i})\quad\text{ for }1\leq n\leq N \tag{6}\]  
where \(M\) is the number of hidden nodes. To improve the CFNN, the architecture and training parameters of the model were optimized using the GWO, specifying the number of neurons, learning rate, batch size, and epochs for two networks. The parameter bounds were set to [(0.0001, 0.01), (8, 128), (8, 128), (8, 128)] with 50 iterations and 20 wolves. After tuning, the model commences with an input layer that matches the dimensionality of the feature space (10 neurons), followed by the first hidden layer comprising 26 neurons. Subsequently, two additional hidden layers are introduced in a cascaded fashion, with 128 and 21 neurons, respectively. Each of these hidden layers is connected to the input layer and all the preceding hidden layers. The activation function employed for the hidden layers is the Rectified Linear Unit (ReLU), which introduces non-linearity into the model and enables it to learn complex patterns present in the data. The output layer consists of a single neuron, aiming to predict the compressive strength of the geopolymer concrete. During the training process, the Adam optimizer is utilized. The MSE serves as the loss function, optimizing the model to minimize the squared differences between the predicted and actual compressive strength values. To measure the performance of the predictive models, six types of indicators are utilized, namely, \(R^{2}\), MAE, RMSE, MAPE, RSR, and WMAPE. The chosen evaluation metrics offer specific benefits in comparing the models. \(R^{2}\) elucidates how well each model captures the variance in the target variable, crucial for assessing predictive capability and overall model fit. MAE and RMSE provide tangible measures of prediction accuracy, enabling direct comparison of error magnitudes and helping identify which model yields predictions closer to actual values. RSR normalizes errors by dataset variability, facilitating assessment of model fit relative to inherent data characteristics, which is particularly valuable when dealing with datasets of varying scales or levels of variability.