## 2 Materials and Methods
### Model Architecture  
Figure 1 illustrates the proposed deep learning model framework for simulating nonlinear fiber pulse propagation. The dataset is uniformly sampled at 1024 points for both input and output signals, with real and imaginary parts fed into the branch network. The input signal, determined by different fiber parameters and peak power, is normalized and processed in the trunk network. BranchNet utilizes convolutional layers followed by residual blocks to extract hierarchical features from the input pulse, effectively capturing the complex dynamics of the signal. After feature extraction, an adaptive average pooling layer reduces the feature dimensions, and a fully connected layer maps these features into a compact representation. Specifically, BranchNet is designed with a consistent network width that is proportional to the input size, ensuring that the feature extraction process is both deep and wide enough to handle the complexity of the pulse dynamics. Figure 1: DeepONet model framework for simulating nonlinear dynamics of fiber pulse propagation. TrunkNet incorporates a self-attention mechanism to process the normalized fiber parameters, enhancing the model's ability to capture interactions between different parameters. TrunkNet's width is aligned with that of BranchNet, with both networks maintaining a proportional relationship between their layers to ensure balanced learning and feature integration. The resulting features are further refined through fully connected layers, progressively capturing the intricate relationships between the input parameters. Finally, the outputs from both BranchNet and TrunkNet are multiplied and passed through a multilayer fully connected network, which generates the final output pulse. This final network layer is also designed with a width proportional to the preceding layers, ensuring that the combined features are adequately processed. This architecture leverages the strengths of both convolutional feature extraction and attention mechanisms, along with carefully calibrated network widths, to achieve high accuracy in simulating fiber pulse dynamics, offering a significant improvement over traditional methods. To ensure model accuracy, we employed a mean squared error (MSE) loss function, defined as  
\[\text{MSE}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2},\]  
where \(N\) is the number of samples, \(y_{i}\) represents the actual values, and \(\hat{y}_{i}\) represents the predicted values. The model was trained using the RMSprop optimizer, with a cosine annealing scheduler to adjust the learning rate dynamically. An early stopping mechanism was also implemented to prevent overfitting. The dataset was split into training and validation sets in a 9:1 ratio, and model generalization was verified on an unseen dataset. This model's unique architecture, particularly its use of dynamically adjustable network components and efficient training strategies, demonstrates significant improvements in both accuracy and computational efficiency over conventional methods. The structure of the input-output data for the model is outlined as follows:  
**Trunk Network**  
The trunk network encodes the location of the output function. The input to the trunk network is the position \(T\), corresponding to the output location and its associated parameters, effectively representing the high-dimensional location information of the equation. The output of the trunk network is a set of feature representations related to the input location. Testing by inputting only the variable among the model parameters can effectively improve its accuracy and training speed. * **Input**: \(T=\{T_{1},T_{2},\dots,T_{m}\}\)
* **Output**: \(\text{Trunk}(T)=\{\text{Trunk}(T_{1}),\text{Trunk}(T_{2}),\dots,\text{Trunk}(T_{ m})\}\)  
**Branch Network**  
The branch network processes the values of the input function at fixed sensor points. The input to the branch network is the function value at the sensor points \(A(T_{i})\).