## 2 Network Architecture and Training  
### Network Architecture Overview  
The Convolutional Neural Network (CNN) is inspired by the biological visual system, aiming to replicate the way biological vision processes information by emulating the mechanisms of \"local receptive fields\" and \"parameter sharing\", allowing for the efficient extraction of key features from images [27]. Over the past few years, CNNs have achieved remarkable progress in fields such as image recognition, object detection, image generation, and many other domains. While traditional neural networks are limited in their ability to control input dimensions and lack translational invariance, CNNs introduce convolutional operations that enable neurons to perceive local inputs within their receptive fields. By reusing the same convolution kernels at different spatial locations, CNNs effectively reduce the number of parameters required during computation [28,29]. The encoder-decoder network framework within CNNs provides an end-to-end learning algorithm, which is particularly well suited for image-denoising tasks [30]. [PERSON], [PERSON], and [PERSON] were the first to apply the encoder-decoder architecture to image denoising [31]. Through skipping connections, features extracted by the encoder are directly forwarded to the decoder, thereby better preserving and recovering the detailed information in the images. The network's training process can be regarded as learning a mapping from noisy data to clean data, which can be formulated in Equation (1). \[\theta=\text{argmin}_{t}\frac{1}{N}\sum_{i=1}^{N}\parallel\widehat{\mathbf{d}} _{\text{out}}^{(i)}-\text{Net}\Big{(}\mathbf{d}_{\text{in}}^{(i)};t\Big{)} \parallel^{2} \tag{1}\]  
Here, \(d_{out}\) represents the desired clean data, \(d_{in}\) denotes the input noisy data, and \(\theta\) symbolizes the trainable parameters of the network, which are determined through the training process. Specifically, the network architecture proposed in this work is based on improved U-Net architecture, as illustrated in Figure 1. The network accepts single-channel images of size 200 \(\times\) 200 as input, with the output maintaining the same spatial dimensions and channel configuration as the input. Multi-scale feature fusion modules and attention mechanisms are integrated into the network to enhance the feature learning and extraction capabilities. The U-Net network designed in this study contains 5-layer encoders. Among them, there is 1 DoubleConv layer, which is responsible for mapping the number of input channels (in_channels) to the number of feature channels (features). Layers 2-5 are down-sampling blocks that double the number of channels. The decoder part has five layers, and the number of channels is halved during each up-sampling until the final output. Residual connections are retained in the network design to help the model improve training stability. The \"physical constraint\" in this study is added to the training process as a penalty term in the loss function. Here, assuming the predicted output of the model and the true label, the physical constraint loss function can be defined in Equation (2). \[L_{\textit{phys}}=\alpha\cdot\parallel\Delta\hat{y}-\Delta y\parallel_{2}^{2} \tag{2}\]  
Among them, \(\alpha\) is the weight coefficient of the physical constraint loss, which is used to control the degree of importance of the physical constraint in the total loss (Equation (3)). \[L_{\textit{total}}=L_{\textit{mse}}+L_{\textit{phys}}=\parallel\hat{y}-y \parallel_{2}^{2}+\alpha\cdot\parallel\Delta\hat{y}-\Delta y\parallel_{2}^{2} \tag{3}\]  
The attention mechanism can be incorporated as a module within the U-Net network to compensate for the limitations of conventional convolutional operations [32,33]. By aggregating features across multiple resolution levels, the attention mechanism enhances the model's sensitivity to multi-scale spatial structures such as geological anomalies.