## 2 Network Architecture and Training
### Model Training and Optimization  
To enhance the denoising performance of the model, multiple optimization strategies were employed during the training and optimization process. These include data augmentation for the input data, loss function optimization, and the use of the Adam optimizer combined with an exponential learning rate decay strategy to improve training stability and convergence speed. To prevent model overfitting, L2 regularization, and spectral normalization techniques were applied during training. The complete training process of the U-Net model is illustrated in Figure 6; the flow chart in Figure 6 will be explained in more detail in the subsequent paragraphs. During model training, the mean squared error (MSE) was adopted as the basic loss function (Equation (15)), which captures pixel-level differences. \[\mathcal{L}_{MSE}=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2} \tag{15}\]  
To further enhance the physical plausibility and robustness of the model, a Laplace constraint was introduced by computing the gradient field differences using the discrete Laplace operator. This approach maintains the physical properties and smoothness of the data while optimizing the model. The Adam optimizer, an adaptive learning rate optimization algorithm, was selected for training. [PERSON] dynamically adjusts the learning rate based on the first and second moments of the gradients, thereby improving the convergence speed and stability of the model. Additionally, implicit regularization was achieved using batch normalization (BatchNormNorm). By normalizing the inputs, BatchNorm not only stabilizes and accelerates the training process but also provides a certain degree  
Figure 5: Spatial distribution of randomly generated subsurface objects. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Prism ID**} & **Density** & **Length** & **Width (km)** & **Burial** & **Depth** & **Center X** & **Center Y** \\  & **(kg/m\({}^{3}\))** & **(km)** & **Depth (km)** & **Exent (km)** & **(km)** & **(km)** \\ \hline
1 & 2437.96 & 3.8 & 1.83 & 2.7 & 2.81 & 4.18 & 5.14 \\
2 & 2314.86 & 4.09 & 2.19 & 4.16 & 1.12 & 7.39 & 15.97 \\
3 & 4510.36 & 2.93 & 4.81 & 4.03 & 1.57 & 13.3 & 5.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data of randomly generated subsurface objects. of regularization, helping to prevent overfitting. For hyperparameter tuning, a dynamic learning rate with an exponential decay strategy was employed (Equation (16)):  
\[\eta_{t}=\eta_{0}\cdot\gamma^{t}\quad(\gamma=0.95) \tag{16}\]  
where \(\eta_{t}\) is the learning rate at the (t)-th iteration, \(\eta_{0}\) is the initial learning rate, \(\gamma\) is the decay factor, and \(t\) is the current iteration number. In deep learning training, a larger learning rate is required in the early stages to quickly traverse the \"flat regions\" of the loss function and approach the optimal solution. In the later stages, the learning rate should be reduced to avoid overshooting the optimum or oscillating near the optimal solution. The exponential decay strategy allows for a larger learning rate at the beginning to accelerate convergence and gradually decreases the learning rate as training progresses to improve model stability. To further improve model performance and generalization, a hybrid regularization mechanism was constructed by combining L2 regularization with the Laplace constraint. L2 regularization introduces a weight decay term into the optimization objective, constraining the parameter space and suppressing the risk of overfitting. The Laplace constraint further optimizes the feature learning process from the perspective of physical plausibility, enhancing the model's adaptability to real-world scenarios. By introducing an adjustable coefficient \(\lambda\), the smoothness strength can be flexibly controlled, achieving a fine balance between model complexity and generalization ability. Figure 6: Full training process of the U-Net model.