## 3 Results
### One-Dimensional Burgers Equation  
The 1D Burgers equation is typically used to study phenomena in the fields of fluid dynamics, nonlinear fluctuation theory, and phase transition dynamics. This equation describes the nonlinear fluctuation behavior in a nonviscous, incompressible fluid. Although its mathematical form is relatively simple, its analytical solution is often difficult to obtain due to the inclusion of nonlinear terms, causing researchers to focus more on numerical simulations, approximation methods, and mathematical analysis. The numerical experimental case set up for the 1D Burgers equation is as follows:  
\[ \begin{gathered} u_{t}+uu_{x}-(0.01/\pi)u_{xx}=0,\\ u(x,0)=-\sin{(\pi x)},\\ u(\pm 1,t)=0\end{gathered}  \tag{16}\]  
Equation (16) denotes the 1D Burgers equation with the coefficient of viscosity \(v=0.01/\pi\), and both boundary and initial condition settings are included. Equation (16) can be obtained via shifting the terms of Equation (2). The range is set to \(x\in[-1,\ 1]\), \(t\in[0,\ 1]\). \[u_{h}(x,t)=-sin(\pi x)+(x+1)(x-1)\cdot t\cdot u(x,t) \tag{17}\]  
The hard-constraint function of the 1D Burgers equation is shown in Equation (17). The hard constraint is implemented in the same way as in the case of the AC equation, imposed according to the initial and boundary conditions. \(u(x,t)\) is the output of the approximator neural networks. \(u_{h}(x,t)\) is output of the neural network part of PINNs with hard constraints imposed using the penalty method and the ALM. The input of the neural network is \((x,t)\), and the output is \(u(x,t)\). HWPINN and SWPINN use the same neural network parameter settings. The neural network part was set to 8 layers and 32 neurons in the hidden layers, the wide-body network part was set to 8 layers and 32 neurons. For HPINN and SPINN, the parameters in the neural network were set to 8 layers of 64 neurons in the hidden layers, and the maximum number of iterations was set to 10,000 iterations. The density of the collection points was set to \(N_{C}=51\), and the density of the collection points on the boundary was similarly set to \(N_{IC/BC}=51\). These collection points contained only spatial coordinate information \(x\) and discrete temporal information \(t\). Figure 7 illustrates the training loss curves for HWPINN, HPINN, SPINN, and SWPINN. It can be inferred that the PINN models (HWPINN and SWPINN) that incorporate wide-body structures performed more smoothly in terms of the training loss curves. It is worth mentioning that the residual function was discarded when hard constraints were used, resulting in a loss function value that does not match the traditional soft-constrained \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **MSE (\%)** & **RMSE (\%)** & **R2 (\%)** & \
\begin{tabular}{c} **Relative L\({}_{2}\)** \\ **Error (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Computational** \\ **Time (s)** \\ \end{tabular} \\ \hline HPINN & 0.36\% & 6.01\% & 99.21\% & 8.49\% & 1641 \\ HWPINN & 0.25\% & 5.03\% & 99.45\% & 7.10\% & 1654 \\ SPINN & 14.18\% & 37.66\% & 68.86\% & 53.10\% & 1752 \\ SWPINN & 14.00\% & 36.43\% & 69.23\% & 52.78\% & 1791 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error evaluation and computational time of the HWPINN, HPINN, SPINN, and SWPINN models for the prediction of 1D AC equations. PINNs (SPINN and SWPINN) in terms of scale, and the results presented in Figure 7 are only used as a reference for approximating the smoothness. The accuracy of the prediction results was compared in terms of percentage error. Figure 8 shows the FDM solution of the 1D Burgers equation set up in the numerical experiments described in this subsection. Figure 9a-d display the results of the 1D Burgers equation predicted using HPINN, HWPINN, SPINN, and SWPINN; a comparison revealed that HWPINN, HPINN, and SWPINN all perform relatively well.