## 3 Experimental Verification
### Advection Equation  
We first study the ability of PIDeepONet and MulSPIDeepONet to process advection equations. The physical process of such PDEs in fast flowing media (such as a gas or liquid) is somewhat complex, so the solutions to numerical simulations become complicated. In particular, sharp gradients or non-physical oscillations of the solution may occur [30]. The traditional reduced-order model also faces significant challenges in obtaining a solution [31]. Therefore, this paper considers a linear advection partial differential system with simple hyperbolic variable coefficients:  
\[\frac{\partial s}{\partial t}+u(x)\frac{\partial s}{\partial x}=0,(x,t)\in(0, 1)\times(0,1). \tag{18}\]  
The initial boundary conditions are as follows:  
\[\text{s}(x,0)=\sin(\pi\text{x}),\] \[\text{s}(0,t)=\sin(\frac{\pi}{2}\text{t}),\]where \(u(x)\) is a variable function. To make the input function \(u(x)\) strictly positive, we let \(\mathrm{u}(x)=\
u(x)-\min_{x}\
u(x)+1\), where \(\
u(x)\) is sampled from a Gaussian random field. The data used to train the MulSPIDeepONet model are obtained by sampling the random function \(\
u(x)\) generated by the Gaussian random field [25]. The generalization performance of the model is better verified by evaluating the prediction accuracy using the variable function \(u(x)\), which is not included in the training process. The training goal is to learn the solution operator that maps the variable coefficient (variable function) \(u(x)\) to the solution \(s(x,t)\). The model learning rate is set as 0.001, and the Adam optimizer is used. The training set includes 1000 groups of training data. The specific network size settings are listed in Table 1, and the network settings determine the depth of each network and the number of neurons in each layer. The results for the residual loss (Loss-Res) and boundary loss (Loss-Bcs) during the training process are shown in Figure 4.  
It can be seen from Figure 5 that PIDeepONet trained by physics-informed constraints is in good agreement with the reference PDE solution derived under the conditions specified in Figure 6, but the accuracy needs to be improved when dealing with PDE systems with rigid, turbulent, or chaotic dynamics [25]. For example, in this test case, the PIDeepONet model produces obvious errors in areas with large gradient changes (such as the area in the red triangle in Figure 5), and the average absolute error is 0.0218. Using the proposed method, the average absolute error of the test results is 0.0083 in the area where the gradient changes significantly. More comparative results are given in Table 2, and additional experimental results are shown in Figures 7 and 8, where Figures 7 and 8 correspond to the solutions under the functions presented in Figures 9 and 10, respectively. In Figure 7, the average absolute error between the prediction results of PIDeepONet and the analytical solution in the region with large gradient changes is 0.0337, while the average absolute error between the prediction results of MulSPIDeepONet and the analytical solution in this region is 0.0108. In Figure 8, the average absolute error between the prediction results of PIDeepONet and the analytical solution in the region with large gradient changes is 0.0307, while the average absolute error between MulSPIDeepONet and the analytical solution in this area is 0.0070. More comparative indicators are presented in Table 3.  
Figure 4: Solving a advection equation: Boundary and residual losses of a physics-informed DeepONet and multi-step physics-informed DeepONet. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Name** & **Subnetwork Categories** & **Network Setting** \\ \hline \hline PIDEepONet & Trunk Net & [2,100,100,100,100,100,100] \\  & Branch Net & [100,100,100,100,100,100,100,100] \\ \hline \hline  & Trunk Net & [2,100,100,100,100] \\ MulSPIDeepONet & Branch Net & [100,100,100,100,100] \\  & Transition SubNet & [1,100,100,100,100,100] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Network settings.