## 3 Experimental Verification
### Advection Equation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **Model** & **MAE** & **rMAE** & **rRMSE** \\ \hline \hline Test1 & PIDEepONet & 0.0218 & 0.118 & 0.091 \\  & MulSPIDeepONet & **0.0083** & **0.004** & **0.039** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test results for solving linear advection equations. MulSPIDeepONet is superior to the baseline model in all indicators. Figure 5: Comparison of PIDEepONet, MulSPIDeepONet, and analytical solution based on Figure 6. (**a**,**d**) are the real solutions \(s(t,x)\) of the random sampling test variable function \(u(x)\), (**b**,**e**) are the prediction results of PIDEepONet and MulSPIDeepONet, respectively. (**c**,**f**) are the absolute errors between the predicted results and the numerical solutions given by PIDEepONet and MulSPIDeepONet, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **Model** & **MAE** & **rMAE** & **rRMSE** \\ \hline \hline Test2 & PIDEepONet & 0.0337 & 0.117 & 0.111 \\  & MulSPIDeepONet & **0.0108** & **0.038** & **0.033** \\ \hline Test3 & PIDEepONet & 0.0307 & 0.177 & 0.167 \\  & MulSPIDeepONet & **0.0070** & **0.040** & **0.039** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test results for solving linear advection equations. MulSPIDeepONet is superior to the baseline model in all indicators. Figure 7: Comparison of PIDeepONet, MulSPIDeepONet, and analytical solution based on Figure 9. (**a,d**) are the real solutions \(s(t,x)\) of the random sampling test variable function \(u(x)\), (**b,e**) are the prediction results of PIDeepONet and MulSPIDeepONet, respectively. (**c,f**) are the absolute errors between the predicted results and the numerical solutions of PIDeepONet and MulSPIDeepONet, respectively. Figure 6: Input sample of \(u(x)\) for Figure 5.  
Figure 8: Comparison of PIDeepONet, MulSPIDeepONet, and analytical solution based on Figure 10. **(a,d)** are the real solutions \(s(t,x)\) of the random sampling test variable function \(u(x)\), **(b,e)** are the prediction results of PIDeepONet and MulSPIDeepONet, respectively. (**c,f**) are the absolute errors between the predicted results and the numerical solutions of PIDeepONet and MulSPIDeepONet, respectively. Figure 9: Input sample of \(u(x)\) for Figure 7.  
The MulSPIDeepONet architecture represents an improvement over the PIDeepONet method. First, by sharing the independent variable information in the variable function and the solution function, and then applying feature fusion between these two, an indirect mapping of the solution function is efficiently achieved. This method simplifies the task of the network in capturing extremely complex and highly nonlinear relationships, which not only simplifies the learning process of the network, but also improves the ability to identify and analyze complex physical phenomena. In dealing with large gradient changes, as are often encountered in physical problems, the experimental results show that the proposed method effectively captures and reveals the complex physical processes behind the phenomenon. Through comparisons and analysis of the solution results, we find that the MulSPIDeepONet architecture proposed in this paper has an obvious effect in reducing the error in large-gradient regions. Compared with the baseline model, the error is reduced by a factor of three, which reflects the superior generalization performance of MulSPIDeepONet compared with the baseline model. This improvement provides a new perspective and solution for dealing with complex physical problems in regions with large gradient changes. In general, the MulSPIDeepONet architecture has unique advantages and higher-level analysis capabilities in the face of complex physical scenes with large gradient changes. Thus, it provides a new approach for deep learning in the field of physical problems.