## 2 Materials and Methods
### Computational Methodology
#### 2.2.2 Artificial Neural Network Modeling  
ANN was used as a computational tool [36] for the development of a mathematical model that correlates the environmental and operational parameters of the solar dryer to the drying velocity. The main reason for its choice is that it is one of the multivariate regression techniques with the greatest adaptability in complex thermal processes such as those exposed in this work [37]. In short, an ANN is a set of mathematical subfunctions called neurons, which are distributed in layers and interconnected with each other through constant values (weights). The ANN architecture used in this work consisted of an input layer, a single hidden layer, and an output layer. The numbers of neurons in the input and output layers were defined by the experimental variables in Table 3, while the hidden layer was made up of sigmoid neurons whose number was difficult to specify and required an iterative process for identification. In this work, two ANN architectures were evaluated with two different activation functions in the hidden layer, a hyperbolic tangent function and a logistic sigmoid, whose mathematical expressions with respect to their network architectures are presented in Equations (3) and (4).  
\[v_{d,\ ANN}=\sum_{j=1}^{s}\left[\frac{2\times W_{o}(1,j)}{1+\exp\left(-2\left( \sum_{k=1}^{K}(W_{i}(j,k)\cdot In(k))+b_{1}(j)\right)\right)}+1\right]+b_{2}, \tag{3}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Experimental Variables** & **Working Range** & **Unit** \\ \hline Inputs & & \\ Solar radiation & 143–1222 & W\(\cdot\)m\({}^{-2}\) \\ Ambient temperature & 27–44 & \({}^{\circ}\)C \\ Relative humidity & 24.6–95.7 & \({}^{\circ}\)\% \\ Wind velocity & 0–2.6 & m/s \\ Input temperature dryer & 31.5–49.7 & \({}^{\circ}\)C \\ Output temperature dryer & 17–48 & \({}^{\circ}\)C \\ Voltage of fan & 6, 9, 12 & V \\ Output & & \\ Drying velocity & 0–6.4 & g/min \\ \hline \hline \end{tabular}
\end{table}
Table 3: Operation range of input and output experimental variables.  
\[v_{d,\ ANN}=\sum_{j=1}^{s}\left[\frac{W_{o}(1,j)}{1+\exp\left(-\left(\sum_{k=1}^{K} (W_{i}(j,k)\cdot In(k))+b_{1}(j)\right)\right)}\right]+b_{2}, \tag{4}\]  
where \(W_{i}\) and \(W_{o}\) correspond to the connection weights of the neurons between the input-hidden layer and the hidden-output layer, respectively. The elements \(b_{1}\) and \(b_{2}\) represent factors of adjustments of the neural network associated with the neurons of the hidden layer and output layer, respectively. \(S\) and \(K\) represent the total number of neurons in the hidden layer and the input layer, respectively. Lastly, \(In\) represents the \(k\)-th input variable of the mathematical model.  
For the development of the ANN model, the normalized input parameters contained in the database were used (Table 3). The data were randomly divided into subsets of training (80%), validation (10%), and testing (10%). By using a backpropagation algorithm to train the ANN through an iterative trial and error process, different numbers of neurons in the hidden layer were evaluated to find the optimal values of \(W_{i}\), \(W_{o}\), \(b_{1}\) and \(b_{2}\) for Equations (3) and (4) that minimized the error between the experimental data (\(x_{out,exp}\)) and those estimated by the ANN (\(x_{out,sim}\)). In this work, the backpropagation algorithm applied was Levenberg-Marquardt. This is a modification of the gradient descendent algorithm that performs numerical minimization processing to solve nonlinear least squares problems from an estimated initial parameter vector. The main advantage of using this algorithm is its fast convergence time, which is why it is widely used in ANN training [38].  
To quantify the performance of the neuronal model, the following statistical criteria were used: root-mean-square error (\(RMSE\)), linear regression coefficient (\(R\)), coefficient of determination (\(R^{2}\)), and mean absolute percentage error (\(MAPE\)), synthesized in Table 4.