## 2 Network Architecture and Training
### Incorporation of Physical Priors
\[\mathrm{C}\geq\frac{1}{(2\pi)^{\mathrm{n}}}\int_{|\omega|>\Omega}|\omega|^{4} \cdot\mathrm{E}_{\mathrm{d}}(\omega)\mathrm{d}\omega\geq\frac{1}{(2\pi)^{ \mathrm{n}}}\cdot\Omega^{4}\cdot\int_{|\omega|>\Omega}\mathrm{E}_{\mathrm{d}} (\omega)\mathrm{d}\omega \tag{11}\]  
Rearranging the above equation yields Equation (12). \[\int_{|\omega|>\Omega}\mathrm{E}_{\mathrm{u}}(\omega)\mathrm{d}\omega\leq \frac{\mathrm{C}\cdot(2\pi)^{\mathrm{n}}}{\Omega^{4}} \tag{12}\]  
From Equation (12), when the high-frequency cut-off frequency increases (\(\Omega\) increases), the upper bound \(\frac{\mathrm{C}\cdot(2\pi)^{\mathrm{n}}}{\Omega^{4}}\) will decay at the rate of \(\Omega^{4}\). This shows that the Laplace constraint is very strict on the restriction of high-frequency energy. The higher the frequency, the less allowed energy to better preserve the low-frequency components. After each forward pass, in addition to the conventional reconstruction loss, the discrete Laplace operator is evaluated on the network output (Equation (13)). \[\
abla^{2}u_{ij}=\frac{u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}}{h^{2}} \tag{13}\]  
where \(u_{i,j}\) represents the function value at a grid point (i, j), and \(h\) is the grid spacing. The numerator approximates the second-order derivative by summing the four neighboring values and subtracting four times the center value. The weight of the physical constraint is dynamically adjusted according to the training performance. When the physical loss decreases slowly, the weight increases to strengthen the constraint. When the MSE loss rises significantly, the weight is reduced to mitigate the impact of the physical constraint on data fitting, enabling the model to more intelligently balance data fitting and physical constraints. The specific training process is shown in \[\
abla^{2}u=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y ^{2}}+\frac{\partial^{2}u}{\partial z^{2}} \tag{5}\]0 The left side of Figure 4 presents the variation in loss function during training, and the right side presents the variation in physical constraint weight during training. In the early stage of training, the physical constraint weight is small, and the model prioritizes learning the basic patterns of the data during this stage. In the middle and late stages of training, the physical constraint weight dynamically fluctuates and increases. Through the adjustment logic driven by periodicity and performance, the model's predictions consider both data fitting and physical rules. By introducing the Laplace prior as a soft constraint, the output is enforced to satisfy the harmonic characteristics of real physical fields--even with limited training data--thus preventing the occurrence of non-physical artifacts. Regardless of the type or magnitude of the input noise, as long as the gravitational field is harmonic, the network output naturally approaches the physical \"true solution\", demonstrating robust cross-distribution generalization. The Laplace term encourages global smoothness; however, with U-Net's skip connections and multi-scale mechanism, the model can still capture geological body boundaries, achieving a balance between physical rationality and detail preservation.