## 3 Results
### Two-Dimensional Wave Equation  
In this subsection of numerical experiments, we extended the PDE case from 1D to 2D, with the aim of validating the performance of the proposed HWPINN model in 2D problems. Having extensibility across dimensions is one of the advantages provided by the PINN architecture in combination with strict physical laws. The wave equation finds diverse applications in oceanography, including simulating waves, tides, and tsunamis; analyzing marine structures' and ships' responses to waves and studying underwater acoustics, providing essential insights for marine engineering, safety, and environmental monitoring [41]. The 1D wave equation is a mathematical model describing fluctuations propagating along a straight line, and the 2D wave equation is its generalization in 2 dimensions, introducing a variable \(y\). For the definition of a 2D wave equation, see Equation (3). The experimental set up for the 2D wave equation is as follows:  
\[ \begin{split} u_{tt}=c^{2}\big{(}u_{xx}+u_{yy}\big{)},\\ u(x,y,t)|_{t=0}=e^{-[(x-3)^{2}+(y-3)^{2}]}\\ u_{t}|_{t=0}=0.\end{split}  \tag{18}\]  
where \((x,y,t)\) is the input, \(u(x,y,t)\) is the output of the neural network and the velocity of wave \(c=1\). The range is set to \(x\in[0,6],y\in[0,6],t\in[0,1]\). The physical definition of this can be understood as applying a peak initial velocity to the membrane \(\Omega=[0,6]\times[0,6]\) in a 2D region at moment \(t=0\). The membrane vibrates back and forth in time, and, at the same time, the resulting wave propagates in all directions. The initialization of the 2D wave equation is shown in Figure 11. The 2D wave field is formed in a two-dimensional plane. The solution to the 2D wave equation is a function of the wave field. The implementation of the hard-constraint function is defined by the following:  
\[u_{h}(x,t)=e^{-[(x-3)^{2}+(y-3)^{2}]}+\frac{1}{36}\cdot t\cdot u(x,t) \tag{19}\]  
where \(u_{h}(x,t)\) is the output constrained by the hard-constraint function. \(u(x,t)\) represents the output of the approximator neural networks. \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 & 0.34\% & 5.91\% & 99.07\% & 9.59\% & 2144 \\ HWPINN & 0.23\% & 5.01\% & 99.45\% & 6.10\% & 2145 \\ SPINN & 19.98\% & 44.64\% & 47.86\% & 72.67\% & 2245 \\ SWPINN & 1.22\% & 11.07\% & 96.74\% & 18.03\% & 2290 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Error evaluation and computational time of the HWPINN, HPINN, SPINN, and SWPINN models for the prediction of 1D Burgers equation. As for the network parameters, HWPINN and SWPINN used 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 \(N_{\mathrm{C}}=31\), and the density of the collection points on the boundary was similarly set to \(N_{\mathrm{C}/\mathrm{BC}}=31\). These collection points contained only spatial coordinate information \((x,y)\) and discrete temporal information \(t\). As shown in Figure 12, the convergence intervals of the training loss curves of HWPINN, HPINN, SPINN, and SWPINN are very close to each other in the case of the 2D wave equation, and it can be noticed that the training curves of HWPINN and HPINN are much smoother. It can be concluded that the network structure of HWPINN is more numerically stable than those of SPINN, SWPINN, and HPINN.