## 2 Artificial Neural Network
### Network Modeling  
In general, the modeling procedure can be defined as the procedure of delineating a physical phenomenon using mathematical functions [29; 30]. It is critical to optimize the network's design to provide a lightweight and high-quality product simultaneously [31]. Considering a direct solution is not available for estimating the number of neurons for each layer and the number of hidden layers. These numbers can be estimated using a trial-and-error process. To offer the best possible architecture for the ANN model, numerous structures using different neurons and hidden layer numbers have been analyzed. The findings on the network's performance indicated that a single hidden layer network including 19 neurons provides the maximum accuracy to predict the PFRC's compressive strength for 28 days. In order to maintain the initial database's relationships, the input features are normalized with a linear approach in a range between 0 and 1 [19]. Training the neural network was fulfilled by adopting the Levenberg-Marquardt (LM) algorithm, since this method has suitable convergence, more precision, and is time-efficient [32; 33]. The LM algorithm randomly separated the datasets into three parts: 15% was reserved for validation, 70% was used to train the network, and the remaining percentage was used to test the performance of network. TANSIG (Equation (1)) and PURELIN (Equation (2)) were adopted as the activation functions for the hidden and output layers, respectively. The learning process of the network terminates if the desirable performance can be achieved. \[y=tansig(x)=\frac{2}{(1+e^{-2x})}-1 \tag{1}\]  
\[y=purlin(x)=x \tag{2}\]  
Figure 2: The histograms for features of the network (**a**) Binder (kg/m\({}^{3}\)), (**b**) Water (kg/m\({}^{3}\)), (**c**) Coarse aggregate (kg/m\({}^{3}\)), (**d**) Fine aggregate (kg/m\({}^{3}\)), (**e**) PET fiber (kg/m\({}^{3}\)), (**f**) Fiber tensile strength (MPa), (**g**) Fiber modulus of elasticity (MPa), (**h**) Fiber length (mm), (**i**) Fiber width (mm), (**j**) Fiber unit weight (kg/m\({}^{3}\)), (**k**) Fiber shape, (**l**) Compressive strength (MPa). The concrete's mechanical characteristics, including PET fibers, can be evaluated after training the neural network. In this regard, the convoluted relationships between the input data and their impact on the reported output can be evaluated. The networks' performances to determine the concrete's 28-day compressive strength are presented in Figure 2(a). The best validation performance was reported as 0.0042 at the 13 th epoch for PFRC's compressive strength. The estimation quality is the function of R as the coefficient of determination in the network for all the datasets presented in Figure 2(b), which illustrates the correlation between the empirical date (target) and the ANN output. The overall response is based on the R value almost equal to 1, verifying that the trained network produced the optimal outcomes. The R value, including all analyzed datasets, is 0.9806, which is reasonable. Moreover, the comparison of the target and output of the proposed network is illustrated in Figure 4. It was observed that a desirable match is achieved between the empirical results and outcomes. This correlation highlights the capability of the proposed network to determine the PFRC's compressive strength. Having the PFRC's compressive strength may help engineers and stakeholders better understand the response of concrete-containing PET fibers, and lays the groundwork for further adoption of this clean and environmentally friendly concrete. The statistical characteristics of the reported errors, such as root mean square error (_RMSE_), correlation coefficient (_R_), mean square error (_MSE_), and mean absolute percentage error (_MAPE_), were determined for evaluating the concrete's compressive strength by the ANN model. These errors can be calculated by Equation (3). Table 2 offers a comparison regarding these error metrics including all the analyzed datasets incorporated in the network.