## 3 Results
### Validation with Real Gravity Data  
To further evaluate the proposed algorithm, we utilized global satellite gravity data for validation (available at [[https://www.godac.jamstec.go.jp/darwin_cruise/view/base?lang=en](https://www.godac.jamstec.go.jp/darwin_cruise/view/base?lang=en)]([https://www.godac.jamstec.go.jp/darwin_cruise/view/base?lang=en](https://www.godac.jamstec.go.jp/darwin_cruise/view/base?lang=en)) (accessed on 13 June 2025)). The global gravity data were divided into grids of 3\({}^{\circ}\)\(\times\) 3\({}^{\circ}\), and noise with levels in the range of [0.05, 0.08, 0.10, 0.13, 0.15, 0.18, 0.20, 0.23, 0.25, 0.28, 0.30] was added to augment the training samples. The dataset was split into 80% for training and 20% for testing, and the model was trained for 300 epochs. To compare the effect of introducing physical constraints, ablation experiments were conducted. Figure 15 shows the comparison between the model without physical constraints and the model with physical constraints incorporated into the neural network training scheme. Figure 15 consists of four subplots, displaying the trends of the loss function, mean squared error (MSE), and structural similarity index (SSIM) during training. The two plots on the left show the training and validation loss curves as a function of the number of iterations for both models. It can be observed that the training loss of both models decreases rapidly in the early stages, but the model with physical constraints achieves a lower and more stable training loss. The validation loss of the physically constrained model exhibits less fluctuation, indicating more robust performance on the validation set. The two plots on the right show the changes in MSE and SSIM on the training and validation sets during training. Although the model without physical constraints also achieves a rapid decrease in MSE initially, its validation error fluctuates more than that of the physically constrained model, especially after prolonged training. The SSIM curves indicate that the model with physical constraints achieves higher validation stability, suggesting that physical constraints enhance the model's ability to preserve visual quality. The low fluctuation and high SSIM values of the physically constrained model on the validation set indicate that introducing physical constraints during training helps improve the model's generalization performance and perceptual quality. Figure 16 compares the PSNR values of the physically constrained model and the baseline model without physical constraints under different noise levels. The horizontal axis represents the level of added Gaussian noise, ranging from 3% to 11%, while the vertical axis shows the PSNR value, which is used to assess the reconstruction quality of the models. It can be observed that the PSNR of the physically constrained model is significantly higher than that of the baseline model without physical constraints and remains stable at around 26 across all noise levels. This demonstrates that physical constraints help improve the robustness and reconstruction accuracy of the model under noisy conditions. In contrast,  
Figure 14: Comparison of PSNR values for different denoising methods. the PSNR of the baseline model without physical constraints is significantly lower, around 24.2, and does not show notable improvement as the noise level increases. This highlights the effectiveness and necessity of physical constraints in mitigating the impact of noise. The denoising results for real gravity data are shown in Figure 17, where three regions with distinct geological features were selected for comparative analysis. In this study, a U-Net network incorporating physical prior information was used for gravity data denoising. As shown in the figure, the improved U-Net network achieves remarkable denoising performance on real gravity data.