## 3 Neural Network-Based Q-Analogies of the Numerical Scheme
In Tables 1-11, we evaluate the numerical results produced with our parallel numerical scheme for various values of q. In Table 1, we report on the error and approximate solution up to two decimal places whereas the approximated root is computed up to 64-digit floating point arithmetic using the stopping criterion \(\Lambda_{i}^{[*]}=\left| \begin{matrix}\dot{\vec{x}}_{i}+1\\ \dot{\vec{x}}_{i}\end{matrix} -\dot{\vec{x}}_{i}^{[\vec{x}]}\right|\). The maximum error, number of iterations, and elapsed CPU time for the Q-MM\({}^{\vec{x}_{1}}\) method for various values of q are shown in Tables 2-4. On the other hand, Tables 5-9 displays the numerical results of Q-MM\({}^{\sigma 2}\) in terms of approximated values, residual errors, maximum errors, number of iterations, and elapsed CPU time for finding all roots of (53) using different values of q. The rate of convergence of Q-MM\({}^{\sigma 1}\)-Q-MM\({}^{\sigma 2}\) increases significantly as the q values increase from 0.2 to 1.1 on random initial values, demonstrating global convergence behavior. Figure 3a-e illustrates the neural network implementation and its results using Algorithm 1. The error histogram curve of the neural network demonstrates the consistency of the proposed scheme; the transition statistics curve reflects the effective convergence rate of the neural network; the fitness curve shows accuracy and stability; the regression curve illustrates the linear relationship between the expected and actual outcomes; and the mean square error demonstrates how well the target solution and expected outcomes matched. Figure 3 displays the neural network's outputs in the following ways: (a) the error histogram; (b) the transition statistics curve; (c) the mean square error; (d) the regression curve; and (e) the fitness curve. As illustrated in Figure 3a-e and Table 9, neural network simulations for this engineering application produce reliable and consistent results. Figure 4a,b depicts the residual error and approximate order of convergence, whereas Figure 5a-f depicts the local computational order of convergence of the neural network-based parallel root finding scheme. The following values are chosen as initial guesses:

\[ \begin{array}{rcl}\dot{\tilde{x}_{1}}&=&20.1,&\dot{\tilde{x}_{2}}=50.8,&\dot{ \tilde{x}_{3}}=125.5.\end{array} \]

As shown in Table 10, the accuracy of the proposed numerical schemes improves when the outputs of the neural networks are used as initial guess values in Q-MM\({}^{\sigma_{1}}\), TAM\({}^{\zeta_{*}}\), BSM\({}^{\zeta_{3}}\), NAM\({}^{\zeta_{3}}\), NDM\({}^{\zeta_{3}}\), NAM\({}^{\zeta_{5}}\), Q-MM\({}^{\sigma_{2}}\). Table 11 presents the overall convergence behavior of neural network outputs based on q-analogies. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Method** & \(\mathbf{e_{1}^{(3)}}\) & \(\mathbf{e_{2}^{(3)}}\) & \(\mathbf{e_{3}^{(3)}}\) & \(\mathbf{\rho_{i}^{(\sigma-1)}}\) \\ \hline \([\)Q-MM\({}^{\sigma_{1



In Table 11, Max-Err\({}_{\Lambda_{1}^{[*]}}\) represents the maximum error; Max-it\({}_{\Lambda_{1}^{[*]}}\) represents the maximum number of iterations; Max-CPU\({}_{\Lambda_{1}^{[*]}}\) represents the maximum elapsed CPU time; Max-CO\({}_{\Lambda_{1}^{[*]}}\) represents the maximum order of convergence achieved; and \(\rho_{i}^{[\sigma-1]}\) is the local order of convergence of q-analogies based neural network parallel numerical scheme on this application using \(\Lambda_{i}^{[*]}=\left| \begin{matrix}[\sigma+1]\\ \tilde{x}_{i}\end{matrix} - \begin{matrix}[\sigma]\\ \tilde{x}_{i}\end{matrix} \right|\) as stopping criterion. Figure 3: (**a–e**) Error histogram of neural network for engineering application 1 (**b**) Transition statistics curve of neural network for engineering application 1 (**c**) Mean square error curve of neural network for engineering application 1 (**d**) Regression curve of neural network for engineering application 1 (**e**) Fitness curve of neural network for engineering application 1. (**a**) The neural network’s error histogram.