## 3 Results and Discussion
### One-Dimensional Wave Equation  
The wave equation (Equation (13)):  
\[\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\
abla^{2}u, \tag{13}\]  
models the oscillations of a one-dimensional string (\(u=u(x,t)\)), the oscillations of a two-dimensional thin membrane (\(u=u(x,y,t)\)), or the pressure oscillations of an acoustic wave in air (\(u=u(x,y,z,t)\)). The constant \(c\) denotes the velocity of wave propagation for the oscillations and is also known as the wave velocity in certain literature studies. Although typically discussed in just one spatial dimension (\(x\)) due to time (\(t\)) being the only independent variable, it is important to mention that the variable we are studying (\(u\)) can represent movement in another direction, like up and down (\(y\)). For example, this occurs when a string is not only moving horizontally (\(x\)) but also vertically (\(y\)), as seen on a flat surface. The unknown function \(u\) depends on space \(x\) and time \(t\), and can be represented as an equation  
\[u=u(x,t).\]  
We also need initial conditions and boundary conditions to solve the function. In the experiments, we use the conditions given in Equation (14):  
\[ \begin{cases}u_{tt}=c^{2}u_{xx}\\ u(x,0)=\sin(x)\\ u_{t}(x,0)=\sin(x)\\ u_{b}(0,t)=u_{b}(\pi,t)=0\end{cases} \quad\quad\text{for }0\leq t\leq 2\pi, \quad 0\leq x\leq\pi \tag{14}\]  
We solve the case where \(c=1\). Specifically, we address the equation with homogeneous Dirichlet conditions, \(c=1\), and compare the results with the analytical solution. As we increase \(c\) from \(1\) to \(2\), we observe that the solution takes much longer to converge. To address this, we employ transfer learning. We first train the model for \(c=1\) and then use this knowledge to approximate the solution for \(c=2\). To do so, we approximate the underlying solution with a feedforward dense neural network with tunable parameters \(\theta\):  
\[\hat{u}_{\theta}(x,t)\approx u(x,t).\]  
This approach allows us to efficiently model and compare solutions under different conditions, providing a deeper understanding of the system's behaviour. The loss function is referred to as Equation (5). The interior residual is given by Equation (15):  
\[r_{\text{int},\theta}(x,t):=\hat{u}_{\theta,H}(x,t)-c^{2}(\hat{u}_{\theta,xx} (x,t)),\quad\forall\;t\in[0,T],\;x\in[0,\pi]. \tag{15}\] \begin{table}
\begin{tabular}{l l l l} \hline \multicolumn{3}{c}{**L2 Relative Error (\%) of Adam, LBFGS, and Transfer Learning**} \\ \hline  & **Adam** & **LBFGS** & **Transfer Learning (Adam)** \\ \hline
20 & 0.95 & 0.92 & - \\
30 & 1.24 & 0.88 & - \\
40 & 0.80 & 0.95 & 0.037 \\
50 & 3.15 & 0.95 & 0.881 \\
60 & 71.54 & 0.95 & 0.218 \\ \hline \end{tabular}
\end{table} 
Table 1: Relative L2 error (%). The spatial boundary residual or boundary conditions are given by Equation (16):  
\[r_{\text{sb},\theta}(0,t) :=\hat{a}_{\theta}(0,t)-u_{b}(0,t),\] \[r_{\text{sb},\theta}(\pi,t) :=\hat{u}_{\theta}(\pi,t)-u_{b}(\pi,t), \tag{16}\] \[\forall t \in(0,T].\]  
The temporal boundary residual is given by Equation (17):  
\[r_{\text{tb},\theta}(x) :=\hat{a}_{\theta}(x,0)-u(x,0),\quad\forall x\in[0,\pi], \tag{17}\] \[r_{\text{tb},\theta}(x) :=\hat{a}_{\theta,t}(x,0)-u_{t}(x,0),\quad\forall x\in[0,\pi].\]  
With the training input points corresponding to low-discrepancy Sobol sequences, the loss terms are given in Equations (18)-(20):  
\[L_{F}(\theta) =\frac{1}{N_{\text{int}}}\sum_{i=1}^{N_{\text{int}}}r_{\text{int},\theta}^{2}(x_{i},t_{i}) \tag{18}\] \[L_{B}(\theta) =\frac{1}{N_{\text{sb}}}\sum_{i=1}^{N_{\text{sb}}}r_{\text{sb}, \theta}^{2}(t_{i},0)+\frac{1}{N_{\text{sb}}}\sum_{i=1}^{N_{\text{sb}}}r_{ \text{sb},\theta}^{2}(t_{i},\pi),\] (19) \[L_{I}(\theta) =\frac{1}{N_{\text{tb}}}\sum_{i=1}^{N_{\text{tb}}}r_{\text{tb}, \theta}^{2}(x_{i}) \tag{20}\]  
where \(N_{int}\) is the number of collocation points or the PDE points; \(N_{sb}\) represents each spatial boundary or boundary condition points; and \(N_{tb}\) is either the temporal boundary points or the initial condition points.