## 1 Introduction
The application of transfer learning under specific conditions can alleviate this limitation [19; 20]; although, this often leads to strong targeting of new target domains and a loss of generalization reasoning capability. Thus, PINNs cannot produce inferences in real time under different boundaries, initial conditions, and loads. If a cross-scenario prediction method with generalized reasoning capabilities could be established, the time cost would be almost negligible compared with traditional numerical solvers (such as CFD or solid mechanics simulators). To establish the generalization reasoning capability of neural networks, [PERSON] et al. [21] proposed the deep operator network (DeepONet). This network is based on the universal approximation theorem of operators [21; 22], which has been theoretically proven. DeepONet predefines the input space with input generalization ability before training, such that the trained model can generalize predictions on the input space and achieve the generalization reasoning capability. Additionally, DeepONet is unaffected by the curse of dimensionality in the input space [23], which has great application value in engineering problems. [PERSON] et al. [24] employed a machine learning framework with deep neural operators (DeepONet) to map microstructures to mechanical responses in metamaterials, achieving prediction accuracies within 5-10% from sparse data. This approach marks a significant advance in the inverse design of materials with complex, nonlinear properties. The cost of DeepONet is that it requires numerous expensive experiments or high-fidelity simulation datasets to express the input space, making it difficult for DeepONet to solve PDEs containing variable functions. Moreover, fitting the output function in DeepONet is not guaranteed to satisfy the deep PDEs and only provides a rough approximation of the target solution operator. Thus, DeepONet lacks interpretability at the physical level. DeepONet and PINNs are complementary approaches. Therefore, [PERSON] et al. [25] combined DeepONet with a PINN to give PIDeepONet, which makes use of NNs and solution function spaces to directly establish mapping relationships. This requires the network to capture very complex and highly nonlinear relationships [26]. Hence, the network learning process can be difficult and false model convergence may occur, meaning that accurate prediction results are not guaranteed. This article establishes a new method for directly solving arbitrary PDEs. The proposed method does not rely on data and has strong predictive generalization capabilities and high accuracy. Inspired by the latent variable model [27], our approach breaks the direct mapping relationship between PIDeepONet and the solution function space and establishes an intermediate variable model. The model uses latent variables to encode the information bottleneck between the inputs and output and incorporates a multi-step deep network structure. Essentially, our proposed multi-step physics-informed deep operator neural network (MulSPIDeepONet) for solving PDEs combines the advantages of PINNs and DeepONet. This paper describes the novel NN structure of MulSPIDeepONet, which simplifies the mapping process between variable functions and solution functions to achieve enhanced interpretability and generalization capabilities. The architecture of MulSPIDeepONet can be described as follows. First, the independent variables of the variable function and the solution function are represented by two subnetworks, and we obtain the transition operator of the solution function through the dot product operation. Second, the transition operator of the solution function provides the input to the transition subnetwork. Third, to enhance the information fusion between the network output and the trunk network, we take the dot product of the transition subnetwork and the trunk network to obtain the final solution function. Finally, physics-informed constraints are applied as additional penalty terms in the loss function.