## 3 Experimental Verification
### Eikonal Equation  
This example aims to highlight the ability of PIDeepONet and MuSPIDeepONet to handle different input functions in two dimensions. We consider the following two-dimensional eikonal equation:  
\[\left\|\
abla\mathrm{s}(\mathrm{x})\right\|_{2}=1, \tag{24}\] \[\mathrm{s}(\mathrm{x})=0,\mathrm{x}\in\partial\Omega,\]  
where \(\mathrm{x}=(x,y)\in R^{2}\) denotes two-dimensional space coordinates, and \(\Omega\) is an open domain with a piecewise smooth boundary \(\partial\Omega\). The solution to Equation (24) is a signed function measuring the distance from the point in \(\Omega\) to the nearest point on the boundary \(\partial\Omega\). For example,  
\[\mathrm{s}(\mathrm{x})= \left\{ \begin{array}{l}\mathrm{d}(\mathrm{x},\partial\Omega), \;\mathrm{x}\in\Omega,\\ -\mathrm{d}(\mathrm{x},\partial\Omega),\;\mathrm{x}\in\Omega^{C},\end{array} \right. \tag{25}\]  
where \(\mathrm{d}(\cdot\,,\cdot)\) is defined as the distance function. The purpose of this example is to explore an effective method for mapping the closed curve \(F\) to its corresponding signed distance function while satisfying the eikonal Equation [25]. In this paper, circular boundaries with different radii centered on the origin are considered to fully explore the characteristics of the mapping process. The training data are a random selection of 1000 circular boundaries of different radii taken from a uniform distribution, and the test data are the circular boundaries of different radii that are not involved in the training process. In this example, the learning rate is set as 0.001, and the Adam optimizer is used. The specific network size is listed in Table 6. The results for the residual loss (Loss-Res) and boundary loss (Loss-Res) during the training process are shown in Figure 16. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Network Name** & **Subnetwork Categories** & **Network Setting** \\ \hline \multirow{2}{*}{PIDeepONet} & Trunk Net & [2,50,50,50,50,50] \\  & Branch Net & [200,50,50,50,50,50,50] \\ \hline \multirow{3}{*}{MulSPIDeepONet} & Trunk Net & [2,50,50,50,50,50] \\  & Branch Net & [200,50,50,50,50,50] \\ \cline{1-1}  & Transition SubNet & [1,50,50,50] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Network settings. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a\) & **Model** & **MAE** & **rMAE** & **rRMSE** \\ \hline \multirow{2}{*}{\(a=4.4\)} & PIDeepONet & 0.6567 & 0.2432 & 0.3177 \\  & MuSPIDeepONet & **0.1704** & **0.0631** & **0.1583** \\ \hline \multirow{2}{*}{\(a=5.0\)} & PIDeepONet & 0.7726 & 0.2575 & 0.3308 \\  & MuSPIDeepONet & **0.1918** & **0.0639** & **0.1650** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Test results for solving one-dimensional parametric Euler equation. MulSPIDeepONet is superior to the baseline model in all indicators. The test results are displayed in Figure 17, indicating that PIDeepONet achieves good consistency between the exact value and predicted value of the signed distance function. The L2 norm error of the test result given by the proposed method is 0.001904, whereas that of the baseline model is 0.002975. The error is significantly reduced by using MulSPIDeepONet. Additional test results and error metrics are presented in Figure 18 and Table 7, illustrating that the model proposed in this study outperforms the baseline model. Figure 16: Solving a Eikonal equation: boundary and residual losses of a physics-informed DeepONet and multi-step physics-informed DeepONet. Figure 17: PIDeepONet and MulSPIDeepONet compared with analytical solutions. (**a,d**) are the real solutions \(s(x)\) of numerical simulations for any arbitrary test variable function (circular boundary centered on the origin, with different radii), (**b,e**) are the prediction results of the baseline model (PIDeepONet) and proposed method (MulSPIDeepONet), respectively, and (**c,f**) are the absolute errors between the prediction results and the numerical solution.