## 2 Network Architecture and Training
### Incorporation of Physical Priors  
To enhance model performance, physical prior information is integrated into the loss function of the network. The multi-scale feature fusion module and Laplace constraint are complementary: the former captures multi-scale spatial features of geological bodies, while the latter ensures that features at different scales conform to physical laws via a gravitational field harmonic constraint, thereby preventing non-physical artifacts. During the model training process, the Laplace equation--which describes the gravitational potential field in geophysics--is introduced as a physical prior before the loss function (Laplace term). In regions distant from density anomalies, the gravitational potential function satisfies the Laplace equation (Equation (4)). \[\
abla^{2}u=0 \tag{4}\]  
where \(\
abla^{2}\) denotes the two-dimensional Laplace operator, specifically (Equation (5)). \[\
abla^{2}u=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y ^{2}}+\frac{\partial^{2}u}{\partial z^{2}} \tag{5}\]  
The introduction of the Laplace constraint can better preserve the low-frequency components in gravity data. For the analysis of its mathematical principle, first, the Laplacian operator is represented in the frequency domain for energy analysis. From the Fourier transform of the function \(u(x)\in L^{2}(\mathbb{R}^{n})\) (where \(n\) is the spatial dimension), the transform of the Laplacian operator into the frequency domain is given in Equation (6), where \(|\omega|=\sqrt{\omega_{1}^{2}+\ldots+\omega_{n}^{2}}\),  
\[\mathcal{F}\Big{[}\
abla^{2}u\Big{]}(\omega)=-|\omega|^{2}\cdot\mathcal{F}[u] (\omega) \tag{6}\]  
According to [PERSON]'s theorem, the total energy of a function in the spatial domain is equal to the total energy of its Fourier transform into the frequency domain multiplied by a normalization factor. The frequency-domain energy density is defined as \(\mathrm{E}(\omega)=\left|\mathcal{F}[\mathfrak{u}](\omega)\right|^{2}\), while the total energy can be decomposed into low-frequency \(|\omega|\leq\Omega\) and high-frequency \(|\omega|\leq\Omega\) components. \[\parallel u\parallel_{2}^{2}=\frac{1}{\left(2\pi\right)^{n}}\left(\int_{| \omega|\leq\Omega}E(\omega)d\omega+\int_{|\omega|>\Omega}E(\omega)d\omega\right) \tag{7}\]  
Assume the network output is \(\hat{u}\), after adding the Laplace constraint term to the loss function, \(\mathcal{L}_{\mathrm{Lap}}=\parallel\
abla^{2}\hat{\mathfrak{u}}\parallel_{2} ^{2}\), from Equations (6) and (7), the frequency-domain representation of the constraint term can be expressed as Equation (8), where \(\mathrm{E}_{\hat{\mathfrak{u}}}(\omega)=\left|\mathcal{F}[\mathfrak{u}]( \omega)\right|^{2}\) is the spectral energy density of \(\hat{u}\):  
\[\parallel\
abla^{2}\hat{u}\parallel_{2}^{2}=\frac{1}{\left(2\pi\right)^{n}} \int_{\mathbb{R}^{n}}\left|\mathcal{F}\Big{[}\
abla^{2}u\Big{]}(\omega)\right| ^{2}d\omega=\frac{1}{\left(2\pi\right)^{n}}\int_{\mathbb{R}^{n}}\left|\omega \right|^{4}\mathrm{E}_{\hat{\mathfrak{u}}}(\omega)d\omega\Bigg{)} \tag{8}\]  
We determine its bound \(\mathrm{C}\) according to the real scenario and decompose the integral into high-frequency and low-frequency parts in Equation (9). \[\int_{\mathbb{R}^{n}}\left|\omega\right|^{4}\cdot E_{\hat{u}}(\omega)d\omega= \int_{|\omega|\leq\Omega}\left|\omega\right|^{4}\cdot E_{\hat{u}}(\omega)d \omega+\int_{|\omega|>\Omega}\left|\omega\right|^{4}\cdot E_{\hat{u}}(\omega)d\omega \tag{9}\]To analyze the restriction on high-frequency energy, we focus on the integral of the high-frequency part in Equation (10). \[C\geq\frac{1}{(2\pi)^{\mathrm{n}}}\int_{|\omega|>\Omega}|\omega|^{4}\cdot E_{ \mathrm{d}}(\omega)d\omega \tag{10}\]  
In the high-frequency region, the frequency modulus satisfies \(|\omega|\geq\Omega\). Thus, \(|\omega|^{4}\geq\Omega^{4}\). Substituting this into the high-frequency integral inequality gives Equation (11).