## 2 Method
### MultiSPIDeePONet Architecture  
We now describe the MulSPIDeePONet structure, which considers the indirect mapping between the input function and the output function. Through physical constraints, this architecture achieves a better ability to solve arbitrary PDEs with relatively little training data. MulSPIDeepONet is based on the advanced network architecture of DeepONet, which consists of a branch network, trunk network, and independent transition subnetwork. An independent transition subnetwork is established to learn the transition operator, thus enhancing the information utilization between the variable function and the independent variable of the solution function. This subnetwork also improves the feature recognition of the network with regard to the independent variable. Our model uses the DeepONet architecture [21] and considers the form of general parametric PDEs. MulSPIDeepONet can be expressed as follows:  
\[\left|F(u)(y)(G)-\sum_{s=1}^{u}\underbrace{g\left(\sum_{l=1}^{n}w_{s,l}y_{l}+ \xi_{s}\right)}_{\text{Trunk Net}}\underbrace{g\left(\sum_{q=1}^{d}p_{q,s}G(u,y )_{q}+\xi_{q}\right)}_{\text{Transition SubNet}}\right|<\delta, \tag{8}\]  
where \(G(u,y)\) is the output of DeepONet:  
\[G(u,y)=\sum_{k=1}^{N}\underbrace{\left[\sum_{i=1}^{M}c_{i}^{k}g\left(\sum_{j=1 }^{m_{k}}\varepsilon_{i,j}^{k}u(x_{j})+\theta_{i}^{k}\right)\right]}_{\text{ BranchNet}}g\underbrace{\left(\sum_{l=1}^{n}w_{k,l}y_{l}+\zeta_{k} \right)}_{\text{Trunk Net}}, \tag{9}\]  
in which \(c_{i}^{k}\), \(\varepsilon_{i,j}^{k}\), \(\theta_{i}^{k}\), \(w_{k,l}\), \(\zeta_{k}\), \(p_{q,s}\), and \(\zeta_{q}\) are the weights and biases of the corresponding network, which are constants. \(N\), \(M\), and \(U\) are network hyperparameters. Therefore, MulSPIDeepONet can be expressed as follows:  
\[\left|F(u)(y)(G)-\left\langle FNN\left(\left\langle\underbrace{g\left(u(x_{1}),u(x_{2}),\ldots,u(x_{m})\right)}_{\text{branch}},\underbrace{f(y)}_{\text{ trunk}}\right\rangle\right)\underbrace{f(y)}_{\text{trunk}}\right\rangle\right|<\delta, \tag{10}\]  
where \(F(u)(y)(G)\) is the final network output, and \(FNN\) represents the transition subnetwork. The main purposes are to simplify the mapping of the network to the solution  
Figure 2: PIDeepONet architecture. The asterisk (*) represents the set of trainable parameters in the network. function and reduce the difficulty of capturing the complex and highly nonlinear relationship. The MulSPIDeepONet network architecture is depicted in Figure 3.  
MulSPIDeepONet adopts an unstacked DeepONet structure. First, the outputs from the trunk and branch networks are merged through the dot product operation to obtain the transition operator, which is used as the intermediate variable for mapping the solution operator and forms the input to the transition subnetwork. Through the transition subnetwork, the characteristics of the branch and trunk networks are fused to enhance the utilization of hidden information between the two networks, thereby establishing an indirect mapping relationship with the solution function. The accuracy of the PINN model is closely related to the physical constraints between the independent variables of the solution function. The key is to enhance the feature fusion of the solution function and the independent variables, as this has a great impact on improving the accuracy of the network model. To this end, the proposed architecture connects the input function and the trunk network with the transition subnetwork, and forms an association through the dot product operation. Finally, the outputs of the transition subnetwork and the trunk network are combined through the dot product operation to obtain the final network output. This improves the physics-informed fusion of the independent variables of the solution function and expands the recognition ability of the physical laws. In this way, MulSPIDeepONet captures regional information with complex physical changes more accurately.