## 2 Method
### DeepONet and PIDEepONet Architectures  
Inspired by the nonlinear operator approximation theorem, [PERSON] et al. [21] recently proposed the DeepONet architecture. In this architecture, they named the network structures related to the input function \(u\) and the output variable \(y\) as the branch and trunk networks, respectively. Therefore, the general nonlinear operator approximate expression can be written as follows:  
\[\left|\mathrm{G}(u)(y)-\sum_{k=1}^{N}\underbrace{\left[\sum_{i=1}^{M}c_{i}^{k} g\left(\sum_{j=1}^{m_{k}}\varepsilon_{i,j}^{k}u(x_{j})+\theta_{i}^{k}\right) \right]}_{\text{BranchNet}}g\underbrace{\left(\sum_{l=1}^{n}w_{k,l}y_{l}+ \zeta_{k}\right)}_{\text{Trunk Net}}\right|<\delta. \tag{3}\]  
The subexpressions related to the trunk and branch networks are explicitly divided into stacked and unstacked components according to whether the branch network has multiple subnetworks. Essentially, the unstacked type merges multiple subbranches of the network into one branch (as demonstrated in Figure 1).  
[PERSON] et al. also proposed a generalized approximation theorem for operators based on the unstacked DeepONet architecture [21], which is expressed as follows:  
\[\left|\mathrm{G}(u)(y)-\left\langle\underbrace{g\Big{(}u(x_{1}),u(x_{2}),\cdots,u (x_{m})\Big{)}}_{\text{Branch Net}},\underbrace{\mathrm{f}(\mathrm{y})}_{\text{ Trunk Net}}\right\rangle\right|<\delta. \tag{4}\]  
[PERSON] et al. [25] were inspired by PINNs and the DeepONet deep neural operator. They combined the advantages of DeepONet with those of PINNs to produce PIDeepONet. This architecture embeds the explicit formula of the PDE system into the network loss function as an additional physical constraint through automatic differentiation technology, thereby greatly reducing the need for training data. The network architecture of PIDeepONet is shown in Figure 2. The biggest difference between PIDeepONet and DeepONet is that the network output uses automatic differentiation to formulate an appropriate regularization mechanism, so that the target output function satisfies any given differential constraint. Therefore, the loss function of PIDeepONet is written as follows:  
\[L(\theta)=L_{operator}(\theta)+L_{physics}(\theta), \tag{5}\]  
where  
\[L_{physics}(\theta)=\frac{1}{NQ}\sum_{i=1}^{N}\sum_{j=1}^{Q}\Big{|}\mathcal{ N}(u^{i},G_{\theta}(u^{i})(y^{i}_{u,j}))\Big{|}^{2}, \tag{6}\]  
and  
\[L_{operator}(\theta)=\frac{1}{NP}\sum_{i=1}^{N}\sum_{j=1}^{P}\big{|}G_{\theta} (u^{i})(y^{i}_{u,j})-G(u^{i})(y^{i}_{u,j})\Big{|}^{2}, \tag{7}\]  
in which \(\theta\) represents the set of all trainable weights and bias parameters in the trunk and branch networks, \(\left\{u^{i}\right\}_{i=1}^{N}\) represents the \(N\) independent input functions for sampling each \(u^{i}\), and \(\left\{y^{i}_{u,j}\right\}_{j=1}^{P}\) are \(P\) positions determined by data observations, initial conditions, or boundary conditions. Additionally, \(\left\{y^{i}_{u,j}\right\}_{j=1}^{Q}\) is a set of collocation points that can be randomly sampled in the domain. Therefore, \(L_{operator}(\theta)\) is used to fit the available solution measurements, while \(L_{physics}(\theta)\) imposes the basic PDE constraints.  
We propose the MulSPIDeepONet architecture, which is based on the general nonlinear operator approximation theorem and incorporates the PIDeepONet structure.  
Figure 1: Architecture diagram of unstacked DeepONet.