## 4 Result and Simulations  
The comprehensive simulations for the numerical outcomes using GNN-GAASM to solve the second kind of LE-NSM are presented in this section. **Problem I:** Consider the second kind of LE-NSM involving exponential functions is written as:  
\[\left\{ \begin{array}{l}u^{\prime\prime}(t)+\frac{0.5}{t}u^{\prime}(t)+(e^{2 u(t)}-0.5e^{u(t)})=0,\ \ t\in(0,1),\\ u(0)=\ln(2),\ \ u(1)=0.\end{array} \right. \tag{10}\]  
For the above equation, the MF is given as follow:  
\[E=\frac{1}{N}\sum\limits_{k=1}^{N}{\left( \begin{array}{l}t_{k}\hat{\mu}^{ \prime\prime}(t_{k})+0.5\hat{\mu}^{\prime}(t_{k})+\\ t_{k}(e^{2u(t_{k})}-0.5e^{u(t_{k})})\end{array} \right)^{2}}+\frac{1}{2}\Big{(} \left(\hat{u}_{0}-\ln(2)\right)^{2}+\left(\hat{u}_{N}\right)^{2}\Big{)}. \tag{11}\]  
The exact solution is \(\ln\Big{(}\frac{2}{(t^{2}+1)}\Big{)}\). **Problem II:** Consider the second kind of LE-NSM is written as:  
\[\left\{ \begin{array}{l}u^{\prime\prime}(t)+\frac{7}{2}u^{\prime}(t)+u^{5}(t )=0,\ \ t\in(0,1),\\ u(1)=0.75,\ \ u^{\prime}(0)=0.\end{array} \right. \tag{12}\]  
\begin{table}
\begin{tabular}{l} \hline \hline \multicolumn{1}{c}{**Process of ASM Starts**} \\ \multicolumn{1}{c}{**Inputs:** Starting point: \(\boldsymbol{W_{\text{GA-Best}}}\)} \\ \multicolumn{1}{c}{**Output:** Best GAASM weights are signified as \(\boldsymbol{W_{\text{GAASM}}}\)} \\ \multicolumn{1}{c}{**Initialize:** Take \(\boldsymbol{W_{\text{GA-Best}}}\), assignments, Bounded constraints,} \\ \multicolumn{1}{c}{generations and other values of the deceleration.} \\ \multicolumn{1}{c}{**Terminate:** The process stops, when any of the below criteria meets [Fit = \(e_{FILT}\) = \(10^{-17}\)], [Iterations = 500], [Max Evals Fun = 272,000], [TolX = TolCon= TolFun = \(10^{-18}\)] \\ \multicolumn{1}{c}{**While [Terminate]**} \\ \multicolumn{1}{c}{**Fitness assessment:** Assess the Fit, \(\boldsymbol{W}\), using Equations (5)–(7).} \\ \multicolumn{1}{c}{**Modifications:** For the SQP scheme, Invoke [fmincon]. Adjust\({}^{\prime}\)  
For the above equation, the MF is given as follow:  
\[ \begin{array}{l}E=\frac{1}{N}\sum\limits_{k=1}^{N}\big{(}t_{k}\hat{u}^{\prime }(t_{k})+2\hat{u}^{\prime}(t_{k})+t_{k}\hat{u}^{5}(t_{k})\big{)}^{2}\\ \quad\quad\quad+\frac{1}{2}\Big{(}(\hat{u}_{N}-0.75)^{2}+(\hat{u}_{0}^{\prime })^{2}\Big{)}.\end{array}  \tag{13}\]  
The exact solution is \(\sqrt{\frac{3}{l^{2}+3}}\). **Problem III:** Consider the second kind of LE-NSM having an exponential function is written as:  
\[\left\{ \begin{array}{l}u^{\prime\prime}(t)+\frac{1}{t}u^{\prime}(t)+e^{u(t) }=0,\;\;t\in(0,1),\\ u^{\prime}(0)=0,\;\;u(1)=0.\end{array} \right. \tag{14}\]  
For the above equation, the MF is given as follow:  
\[E=\frac{1}{N}\sum\limits_{k=1}^{N}\Big{(}t_{k}\hat{u}^{\prime\prime}(t_{k})+ \hat{u}^{\prime}(t_{k})+t_{k}e^{\hat{u}(t_{k})}\Big{)}^{2}+\frac{1}{2}\Big{(} \big{(}\hat{u}_{0}^{\prime}\big{)}^{2}+(\hat{u}_{N})^{2}\Big{)}. \tag{15}\]  
The exact solution is \(2\ln\Big{(}\frac{4-2\sqrt{2}}{1+(3-2\sqrt{2})l^{2}}\Big{)}\). To optimize the second kind of LE-NSM based on all problems by functional the GAASM system using the activation GF for independent hundred executions to find the system parameter variables. The set of best weight validates the estimated numerical outcomes for 10 neurons. The mathematical form of the obtained results is given as:  
\[