Deep Learning-Based Gravity Inversion Integrating Physical Equations and Multiple Constraints

Abstract

Three-dimensional gravity inversion technology involves inferring the underground density structure based on observed gravity anomaly data. In addition to gravity inversion based on physics-driven methods, deep learning, as a purely data-driven technique, is increasingly gaining attention in geophysical inversion problems. However, purely data-driven methods rely on the implicit relationships within the data during the inversion process, which results in a lack of clear physical significance. This study proposes a three-dimensional gravity inversion method that integrates physical equations with deep learning. Based on the U-Net architecture, the gravity forward equation is incorporated as a physical constraint term, and a composite loss function—comprising three-dimensional mean squared error, a depth-weighting function, and three-dimensional intersection-over-union loss—is constructed to enhance inversion accuracy. Numerical experiments indicate that this method outperforms traditional algorithms in terms of density recovery accuracy and boundary clarity. When applied to gravity anomaly data from the Tangshan earthquake region in China, this method successfully inverted the three-dimensional subsurface density structure, revealing a high-density anomaly beneath the seismic source area, which provides important evidence for understanding the regional earthquake generation mechanism.

1. Introduction

Gravity exploration, as one of the most classical and widely applied methods in geophysical exploration, plays an important role in many fields, including resource exploration and regional subsurface structural interpretation [1]. Gravity observation data are a direct reflection of the internal density structure of the Earth. When there are concealed anomalies with densities inconsistent with the surrounding rocks in the subsurface, local gravity anomalies and gravity gradient anomaly responses will be induced [2,3]. However, the complex characteristics of the subsurface cannot be extracted based on geological interpretation of the observed data [4]. Therefore, to obtain the density structure of subsurface anomalies, it is usually necessary to isolate the gravity anomaly regions corresponding to the target objects and invert them based on their numerical values to estimate the structural information of the subsurface anomalies in the target area.

Gravity inversion is one of the key methods for studying subsurface density structures. It enables the estimation of both the geometry and density of underground bodies through the analysis of gravity anomaly data. Existing inversion methods can be divided into two main categories: physics-driven methods and data-driven methods. Physics-driven inversion methods are based on inversion theory. They start from an initial model and iteratively calculate physical equations to reduce the discrepancy between the model’s theoretical values and the actual data. This helps obtain a subsurface density model that matches the observed data [5,6]. The process essentially involves minimizing the objective function. Common algorithms include least squares fitting, gradient descent, Newton’s method, and conjugate gradient methods [7]. However, inversion problems are often underdetermined and have multiple solutions. Therefore, constraints need to be added to the objective function, and regularization methods are used to improve the inversion results. Bell, Tikhonov, and Arsenin (1978) introduced a regularization method that incorporated a constraint term, which effectively enhanced the well-posedness of the inversion problem [8]. This provided a theoretical basis for ill-posed problems in geophysical inversion. Li and Oldenburg (1998) further developed a 3D gravity data inversion method [9]. They used spatial derivatives as physical constraints to reduce the ambiguity of the inversion. They also introduced a depth-weighted function to address the issue of inversion results focusing on shallow layers due to the decay of the kernel function with depth. This allowed for the better recovery of deep subsurface density structures. When it comes to selecting regularization parameters, traditional methods are computationally intensive for large-scale 3D inversion problems [10]. To address this, Chen et al. (2024) proposed an Adaptive Conjugate Gradient Least Squares Regularization (ACGLSR) method [11]. This method combines the conjugate gradient algorithm with adaptive regularization parameters, allowing for dynamic adjustment during the iteration process. It not only improves computational efficiency but also ensures the stability and accuracy of the inversion results. It can also effectively recover the depth of density interfaces and the dip angles of geological bodies. In addition, to address the high computational demands and storage requirements of physics-driven 3D inversion, researchers have proposed solutions such as wavelet compression functions and adaptive data sampling strategies [12,13]. These approaches help optimize the use of computational resources.

However, physics-driven inversion methods face dual constraints in practical applications. On the one hand, their inversion accuracy is highly dependent on the rationality of the initial model. On the other hand, the settings of inversion parameters are often influenced by subjective experience [14]. It is worth noting that with the increasing scale of geophysical data acquisition and the rapid development of high-performance computing technologies, data-driven inversion methods are triggering a paradigm shift in research. These methods, utilizing deep neural networks and other machine learning techniques, can autonomously explore the implicit nonlinear mapping relationships between observed data and subsurface parameters [15]. Compared with traditional methods, they completely eliminate the dependence on initial models, achieving truly data-driven inversion. Moreover, deep learning models employ feature-adaptive parameter optimization mechanisms, automatically determining inversion parameters through data-driven learning processes. This effectively reduces the impact of human intervention on inversion results [16,17].

In recent years, deep learning techniques have demonstrated significant advantages in the field of geophysical inversion, especially achieving a series of breakthroughs in gravity data interpretation [18]. Deep learning methods are commonly used for identifying and extracting image features. Since gravity data and image data have similar structures and can both be represented in a gridded format, researchers have begun to explore new forms of data-driven gravity inversion based on the feature extraction capabilities of convolutional neural networks [19]. Guo, Zhu, and Lu (2012) developed a 3D gravity inversion method based on BP neural networks with a large number of parameters, successfully verifying the effectiveness and feasibility of deep learning methods in gravity inversion [20]. Subsequently, Yang et al. (2021) proposed a gravity inversion method based on CNNs [21]. This method relies more on data-driven training. Unlike traditional methods, it does not depend on prior assumptions during the prediction phase after model training. This demonstrates the superiority of deep learning in handling complex geological structures and noisy data. However, these methods use relatively regular structures when training models, and the density contrasts are all positive values. This limits their performance in inverting composite models and multiple density anomalies. To more accurately simulate the complex shapes and arbitrary locations of subsurface anomalies, Huang et al. (2021) proposed a model-building method based on random walks [22]. They used a U-Net network for gravity data interface inversion and adopted the Dice function as the objective function to develop a 3D sparse inversion method, which showed significant advantages in handling complex geological structures. Moreover, various variants of U-Net have become increasingly popular in gravity inversion. Chen et al. (2024) proposed an improved U-Net network by introducing attention mechanisms, which significantly enhanced the vertical resolution of gravity inversion [23]. Xu et al. (2023) proposed a fast reconstruction method for underground density models based on Res-U-Net [24]. By using residual connections, this method effectively addressed the degradation problem of deep networks, accelerated network convergence, and improved model generalization. Lv, Zhang, and Liu (2023) further proposed a multi-task U-Net3+ network structure that can simultaneously locate anomalies and reconstruct density contrasts, thereby greatly improving inversion efficiency and accuracy [25]. Wang et al. (2024) proposed a multi-scale MS-U-Nets inversion method [26]. The network uses squeeze-and-excitation (S-E) and strip pooling (S-P) modules to gradually reconstruct the underground density structure at different resolutions, further enhancing the accuracy and efficiency of gravity inversion.

Deep learning methods have demonstrated remarkable nonlinear fitting capabilities in geophysical inversion. However, several limitations remain. On the one hand, conventional deep learning models often involve a large number of parameters and complex network architectures, which can easily lead to overfitting and consequently reduce the model’s generalization ability in practical geological scenarios. On the other hand, purely data-driven approaches tend to overlook underlying physical constraints, potentially resulting in inversion outcomes that are not fully consistent with real geological conditions. Although Physics-Informed Neural Networks (PINNs) have successfully integrated data-driven modeling with physical equations in fields such as fluid dynamics and heat transfer [27], their application to geophysical inversion—particularly gravity inversion—remains in an exploratory stage [28,29,30]. In this study, we introduce physical equations into the deep learning algorithm and add multi-faceted constraints to enhance the interpretability of the model and improve inversion accuracy. To more effectively constrain the boundary information of the model, we incorporate the 3D Intersection over Union (3D_IoU) loss. Given that the traditional Mean Squared Error (MSE) loss is typically used to describe deviations in one-dimensional data, it may not be fully suitable for subsurface density structures. These structures are three-dimensional spatial data. Therefore, we adopt a 3D MSE loss to evaluate the deviations in the density distribution of the inversion model. Additionally, to address the skin effect in gravity inversion, we introduce a depth-weighted function. This improvement has been validated through experiments on single and composite models, demonstrating its effectiveness. Finally, we apply this method to gravity anomaly data from the Tangshan earthquake region of China, successfully obtaining a new 3D subsurface density structure, which provides new evidence for the upwelling of magma beneath the surface during the Tangshan earthquake.

2. Background

2.1. Physics-Driven Methods

Physics-driven methods infer subsurface structures through physical equations, where the relationship between surface gravity measurements and subsurface anomalies is governed by the following equation [31]:

$$\left[\begin{array}{c}{\mathrm{d}}_1\\ {\mathrm{d}}_2\\ {\mathrm{d}}_3\\ .\\ .\\ .\\ {\mathrm{d}}_{\mathrm{N}}\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{g}}_{\mathrm{1,1}}& \dots & {\mathrm{g}}_{1,\mathrm{M}}\\ ⋮& \ddots & ⋮\\ {\mathrm{g}}_{\mathrm{N},1}& \dots & {\mathrm{g}}_{\mathrm{N},\mathrm{M}}\end{array}\right]\left[\begin{array}{c}{\mathrm{m}}_1\\ {\mathrm{m}}_2\\ {\mathrm{m}}_3\\ .\\ .\\ .\\ {\mathrm{m}}_{\mathrm{M}}\end{array}\right]$$

(1)

It can be simplified as:

$$\mathrm{d}=\mathrm{G}\mathrm{m}$$

(2)

[d₁, d₂, d₃,..., d_N]^T is the observed gravity data vector, where N is the number of surface observation points. [m₁, m₂, m₃,..., m_M]^T is the subsurface density model, with M representing the number of density units in the model. G is an N × M forward operator, where g_i,j quantifies the gravitational contribution of the j-th subsurface density body to the i-th surface observation point.

In gravity inversion with physics-driven methods, the establishment of the initial model m₀ is a critical step. Using the gravity forward equation, the predicted gravity values on the surface can be computed based on the initial model m0. Subsequently, the predicted gravity values are compared with the actual observed gravity values, and the difference between them is typically defined as the objective function. Through iterative calculations, the initial model m₀ is gradually corrected to minimize the discrepancy between the predicted and observed gravity values. When this difference reaches a pre-defined threshold, the corrected subsurface density model is output as the final gravity inversion result [5,6]. However, in practical applications, the number of observation points N is typically smaller than the number of model parameters M, leading to non-uniqueness and instability in the solution of the physical equation. To address this issue, new constraints must be introduced to limit the solution space. Tikhonov regularization, a widely used 3D gravity inversion method, effectively handles the ill-posedness of the inversion problem. By introducing a regularization term that constrains model complexity, this method significantly improves the stability and reliability of the inversion results [32,33]. The objective function can be expressed as:

$${\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}}={\sigma }_1\left(\mathrm{d}\right)+{\sigma }_2\left(\mathrm{m}\right)$$

(3)

σ₁(d) is the data fitting term, used to measure the difference between the model predictions and the observed values. The specific relationship is expressed as:

$${\sigma }_1\left(\mathrm{d}\right)={‖{\mathrm{d}}_{\mathrm{o}\mathrm{b}\mathrm{s}}-\mathrm{G}\mathrm{m}‖}^2$$

(4)

where dobs represents the gravity values observed on the surface, and GM represents the theoretical gravity values calculated based on the subsurface density model.

σ₂(m) is the regularization term, used to constrain the model complexity and incorporate prior knowledge into the model. The specific relationship is expressed as:

$${\sigma }_2\left(\mathrm{m}\right)=\lambda {‖\mathrm{L}\left(\mathrm{m}-{\mathrm{m}}_0\right)‖}^2$$

(5)

where λ is the regularization parameter, which controls the balance between the data fitting term and the regularization term; L is the regularization matrix, used to impose constraints on the model; m is the predicted subsurface structure model, and m0 is the initial model.

In gravity inversion with physics-driven methods, the selection of the initial model is crucial. The quality of the initial model not only directly affects the convergence speed of the inversion process but also significantly determines the accuracy and reliability of the final inversion results. An inappropriate initial model may cause the inversion process to converge to a local minimum or fail to converge to a reasonable geological model, thereby compromising the credibility of the inversion results [4,9]. Furthermore, although regularization methods address the ill-posedness of the inversion problem to some extent, they introduce additional challenges. The inclusion of regularization terms requires the introduction of prior information. While this helps to constrain the solution space, it also limits the flexibility of the model. As a result, the model finds it difficult to adapt to complex and variable geological conditions. Additionally, the selection of regularization parameters often relies on subjective experience or trial-and-error approaches, lacking systematization and generalizability. This dependency makes regularization methods difficult to quickly adapt to different datasets and application scenarios, necessitating parameter adjustments and thereby increasing the complexity and uncertainty of the inversion process [33].

2.2. Data-Driven Methods

Unlike physics-driven methods, data-driven approaches have demonstrated unique advantages in gravity inversion. By utilizing large-scale input–output datasets, these methods are capable of autonomously learning the underlying complex patterns and relationships within the data, without relying on manually defined prior information. This characteristic substantially reduces the potential influence of subjective factors, thereby enhancing the objectivity and robustness of the inversion results. Furthermore, with the continuous advancement of computational power, data-driven approaches can fully exploit modern parallel computing architectures, enabling more efficient and scalable inversion processes. This computational efficiency not only accelerates the overall workflow but also facilitates the processing of significantly larger datasets, improving the applicability of these methods under complex geological conditions [18].

At the core of data-driven approaches lies the construction of large-scale training datasets for neural network modeling. Through this process, the network learns the complex nonlinear mapping between input data and output models in an automated manner. In the context of gravity inversion, this mapping can be interpreted as the transformation from surface gravity anomaly data to subsurface density distribution. By training on extensive datasets, the neural network extracts the intrinsic relationship between gravity anomalies and subsurface anomalous bodies and encodes this relationship in a parameterized form. Once trained, the network is capable of accurately inferring the subsurface density distribution from new gravity anomaly inputs using the learned parameters. This mapping relationship can be expressed as [34]:

$$\mathrm{m}=\mathrm{N}\mathrm{e}\mathrm{t}\left(\mathrm{d},\theta \right)$$

(6)

where m denotes the subsurface density model, d represents the surface gravity anomaly data, and θ denotes the learned parameters of the network.

However, data-driven methods also present certain limitations in practical applications. These approaches primarily rely on the implicit prior information contained within the data and typically lack explicit physical constraints or theoretical support. As a result, their performance in terms of model interpretability and general applicability may be inferior to that of physics-driven methods. The latter are grounded in well-established physical principles and make use of prior geological knowledge to constrain the inversion process, thereby producing results that are more physically interpretable. In contrast, data-driven methods emphasize automatic pattern learning from data, which, while effective in handling complex nonlinear relationships, often struggle to establish a direct link between the learned patterns and underlying physical processes. Furthermore, the strong dependence on the quality and diversity of training datasets makes data-driven methods vulnerable to performance degradation when applied to scenarios significantly different from those represented in the training data. In such cases, physics-based methods, supported by their theoretical foundations, often exhibit better adaptability to varied geological settings [35].

3. Combining Physical-Driven and Data-Driven Inversion Methods

In the study of subsurface density structures using gravity inversion, data-driven approaches demonstrate strong data-handling capability and adaptability by autonomously learning from large-scale datasets to construct suitable subsurface models. In contrast, physics-driven methods are highly sensitive to the quality of the initial model—its reasonableness directly determines the accuracy and reliability of the final inversion results. To fully leverage the advantages of both approaches and mitigate the limitations of each when used independently, this study proposes a Multi-Constraint inversion method that integrates both physics-driven and data-driven frameworks.

We adopt deep learning algorithms as the core tool on the data-driven side and incorporate physical equations along with additional constraints into the model architecture. The proposed workflow consists of two main stages, as shown in Figure 1. First, the model is trained on a large dataset of surface gravity anomalies. The deep learning algorithm autonomously extracts key features from the data and establishes a mapping relationship between inputs and subsurface models. This stage fully leverages the powerful learning capability of data-driven methods to efficiently mine valuable information from complex datasets. Then, physical equations and other constraints are integrated into the training process to iteratively refine and optimize the model. This hybrid strategy ensures the physical consistency of the inversion results while continuously improving model accuracy and reliability by fitting observed data. The incorporation of physical constraints provides a robust theoretical foundation for the inversion process and addresses the lack of physical interpretability that is commonly encountered in purely data-driven approaches. By combining the efficient learning capacity of data-driven methods with the theoretical rigor of physics-based modeling, the proposed approach aims to construct subsurface models that are not only physically plausible but also highly interpretable. This integrated framework offers an accurate and theoretically grounded solution for gravity inversion under complex geological conditions.

4. Methodology

4.1. Network Architecture

To efficiently extract subsurface density information from raw gravity data, we employ a fully convolutional network (FCN) based on end-to-end learning. Originally proposed by Long, Shelhamer, and Darrell (2015), the FCN architecture replaces the fully connected layers in traditional convolutional neural networks (CNNs) with convolutional layers, thereby enabling direct end-to-end learning from input to output [36]. This design not only simplifies the overall network structure but also significantly reduces the number of hyperparameters, greatly improving computational efficiency. At the same time, FCN avoids the information loss caused by the one-dimensional processing of data in the fully connected layers of traditional CNNs, thereby better preserving the spatial structure information of the input data. In our implementation, two-dimensional gravity anomaly maps obtained from surface observations are used as input, while the output consists of three-dimensional subsurface density models. This approach allows the FCN to directly learn the complex mapping from surface gravity anomalies to subsurface density distributions without relying on intermediate feature extraction stages.

In this study, we employ a specialized form of the fully convolutional network (FCN), namely the U-Net architecture, as the foundational model. The U-Net is widely used in image processing due to its powerful feature extraction capabilities and efficient pixel-level segmentation performance [37]. As shown in Figure 2, the U-Net architecture consists of an encoder and a decoder, which are interconnected through skip connections to facilitate the effective transmission and fusion of feature information [38]. During the decoding phase, we utilize two 3 × 3 convolutional operations to extract feature information from the input data. To minimize the loss of detailed information, we avoid traditional pooling operations and instead apply convolution operations with a stride of 2 for downsampling. This design not only enables efficient feature extraction from the data but also prevents information loss that may arise during pooling, thereby preserving the fine details and structure of the input data. In the encoding phase, to restore the feature maps to their original size, upsampling is required. However, conventional upsampling methods, such as nearest-neighbor interpolation, bilinear interpolation, and bicubic interpolation, rely on fixed mathematical formulas for upsampling and cannot automatically learn the optimal upsampling method from the data [39,40,41]. These methods often perform poorly when dealing with complex image structures, making them inadequate for high-precision inversion tasks. Therefore, in this study, we employ transposed convolution for upsampling. Transposed convolution learns the weights of the convolutional kernels to perform upsampling, and it can be trained end-to-end as part of the network, with weights updated through backpropagation to optimize the performance of the entire network [42]. This learning-based upsampling approach automatically adjusts the convolutional kernel weights based on the features of the input data, thereby better adapting to complex image structures and significantly improving the accuracy and effectiveness of the upsampling process.

4.2. Loss Function

In the field of gravity inversion, to more accurately capture the complexity of subsurface geological structures and enhance the reliability of inversion results, we propose a custom loss function L′. This loss function consists of three key components, aimed at comprehensively optimizing the inversion process.

$${\mathrm{L}}^{\prime }={\lambda }_1\ast 3\mathrm{D}\_\mathrm{I}\mathrm{o}\mathrm{u}\left({\mathrm{m}}^{\prime }-\mathrm{m}\right)+{\lambda }_2\ast \mathrm{D}\mathrm{e}\mathrm{e}\mathrm{p}\_\mathrm{M}\mathrm{S}\mathrm{E}\left({\mathrm{m}}^{\prime }-\mathrm{m}\right)+{\lambda }_3\ast \mathrm{M}\mathrm{S}\mathrm{E}\left({\mathrm{d}}_{\mathrm{o}\mathrm{b}\mathrm{s}}-\mathrm{G}\mathrm{M}\right)$$

(7)

Here, m’ denotes the predicted subsurface model, m represents the true or labeled subsurface model, λ is the weighted factor, d is the surface observation data, and G is the kernel function used for forward modeling. The first component of the loss function is designed to measure the similarity between the predicted model and the actual model. This metric intuitively reflects the overall performance of the model shape inversion and ensures that the inverted model is structurally consistent with the actual geological conditions, thereby capturing the basic outline of the underground structure. The specific formula is as follows:

$$3\mathrm{D}\_\mathrm{I}\mathrm{o}\mathrm{u}\left({\mathrm{m}}^{\prime }-\mathrm{m}\right)=1-{\displaystyle \frac{{\mathrm{m}}^{\prime }\cap \mathrm{m}}{{\mathrm{m}}^{\prime }+\mathrm{m}-{\mathrm{m}}^{\prime }\cap \mathrm{m}}}$$

(8)

The second part utilizes a three-dimensional depth-weighted mean squared error (Deep_MSE). Since the resulting subsurface density model is three-dimensional, the traditional MSE loss function has limitations, as it lacks the ability to account for continuous spatial variations. The three-dimensional MSE is more accurate in describing and simulating the complexity of the subsurface structure, providing richer information and more precise error metrics when handling complex spatial data. Additionally, due to the significant attenuation of gravity data in the vertical direction, a depth weighting is introduced to enhance the deep subsurface information [43]. This weighting is applied on top of the three-dimensional mean squared error, with the specific formula given as follows:

$$\mathrm{W}\left(\mathrm{H}\right)={\left(\mathrm{z}+{\mathrm{z}}_0\right)}^{\beta }$$

(9)

$$\mathrm{D}\mathrm{e}\mathrm{e}\mathrm{p}\_\mathrm{M}\mathrm{S}\mathrm{E}\left({\mathrm{m}}^{\prime }-\mathrm{m}\right)={\displaystyle \frac{1}{\mathrm{N}}}\ast {\displaystyle \frac{1}{\mathrm{L}\ast \mathrm{W}\ast \mathrm{H}}}\sum _{\mathrm{x}=0}^{\mathrm{L}-1}\sum _{\mathrm{y}=0}^{\mathrm{W}-1}\sum _{\mathrm{z}=0}^{\mathrm{H}-1}{\left({\mathrm{m}}^{\prime }-\mathrm{m}\right)}^2\ast \mathrm{W}\left(\mathrm{H}\right)$$

(10)

where Z₀ controls the baseline value of the shallow layer weight, N represents the batch size, and L, W, and H correspond to the spatial dimensions of length, width, and height, respectively.

The third part introduces the data fitting loss function, with the focus on the error between the actual observed surface gravity anomaly value dobs and the data calculated based on the prediction model Gm’. The calculation in this part is based on forward modeling with physical constraints. The forward calculation of the predicted model is performed with the precomputed kernel function G to generate theoretical gravity anomalies corresponding to the observations. Then, the predicted data is compared with the actual observed data, and the MSE loss function is used to quantify the prediction accuracy of the model. The specific formula is as follows:

$$\mathrm{M}\mathrm{S}\mathrm{E}\left({\mathrm{d}}_{\mathrm{o}\mathrm{b}\mathrm{s}}-\mathrm{G}{\mathrm{m}}^{\prime }\right)={\displaystyle \frac{1}{\mathrm{N}}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{d}}_{\mathrm{i}}-\mathrm{G}\left({\mathrm{m}}_{\mathrm{i}}^{\prime }\right)\right)}^2$$

(11)

where M represents the predicted underground model, dobs is the surface observation data, and G is the kernel function used for forward modeling.

The final loss function L’, by comprehensively considering three key factors—model shape similarity, spatial density distribution differences, and gravity anomaly data fitting consistency—achieves a comprehensive optimization of the inversion model. This multidimensional optimization strategy not only effectively enhances the overall performance of the inversion model but also ensures the reliability and scientific validity of the inversion results.

5. Experiments

5.1. Dataset

In geophysical research, accurately modeling the study area is a key step in understanding the characteristics of geophysical field sources. Li et al. (2023) pointed out that parameterizing geophysical field sources using multiple models is an important approach to achieving this goal [44]. To accurately describe the random distribution characteristics of subsurface density, this study employs random-walk-generated stochastic models to simulate underground structures. Specifically, the subsurface space of the study area is discretized into a three-dimensional grid of 64 × 64 × 16 cells, as shown in Figure 3. The position and side length of each cubic cell are known, and each cell is assigned a relative density value. The total number of cubes directly determines the number of parameters to be optimized during the inversion process. This discretization method not only simplifies complex underground structures but also provides a clear parameterization framework for the inversion. In this study, gravity observation stations are located at the center of each grid cell on the top surface of the model, with measurements taken from fixed ground observation points. This ensures a uniform distribution of observation data and provides high-quality input for the inversion process. Subsurface density models are often represented using regular geometric bodies, each with a specific density value. Common regular density models include spheres, cylinders, and rectangular prisms. Although these models are of significant theoretical value, in practical applications, the complexity of underground structures often makes it difficult to accurately describe them using a single, regular geometric body. Therefore, for the discretized subsurface density inversion problem, this paper adopts the random walk method to generate various simulated source models. We assume that the potential subsurface density model consists of a combination of random structures of different sizes and densities. This approach enables flexible adaptation to complex geological conditions.

In this study, we constructed a dataset comprising 43,000 pairs of distinct 3D density models and their corresponding 2D gravity anomaly data, referred to as the Random_GravInv dataset. Four different categories were designed: (1) single random model structure, (2) double random model structure, (3) triple random model structure, and (4) quadruple random model structure. Each random model was generated using the random walk method with a step length in the range [30,40], simulating various unknown complex subsurface structures. Since subsurface density variations are typically small, the density values of the models were set within the range of [−0.10, 0.10] g/cm³ to reflect these subtle changes. The three-dimensional representation and detailed parameters of the density models are shown in Figure 4. To train and validate the network’s performance, the Random_GravInv dataset was divided into two subsets: a training set containing 40,000 samples for the network learning process, and a validation set containing 3000 samples for evaluating the network’s capability.

In addition, to further demonstrate the performance and generalization capability of the network, we also constructed a set of simple regular models that differ from the aforementioned models, referred to as the Rule_GravInv dataset, comprising 2000 pairs in total. This dataset also includes four categories: (1) single cube structure, (2) stepped structure, (3) upper–lower cube structure, and (4) horizontal cube structure. The 3D representations and detailed parameters of these density models are shown in Figure 5.

5.2. Evaluation Metrics

To quantitatively analyze the model inversion results, we introduce two key metrics: Model Fitting Error (E_m) and Data Fitting Error (E_d), to provide a comprehensive evaluation of the inversion accuracy. Specifically, the Model Fitting Error (E_m) quantifies the discrepancy in density between the inverted model and the true model, while the Data Fitting Error (E_d) measures the deviation between the predicted gravity anomalies corresponding to the inversion result and the actual observed gravity anomalies. The formulas are as follows:

$${\mathrm{E}}_{\mathrm{m}}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{‖{\mathrm{m}}_{\mathrm{i}}^{\prime }-{\mathrm{m}}_{\mathrm{i}}‖}_2}{{‖{\mathrm{m}}_{\mathrm{i}}‖}_2}}$$

(12)

$${\mathrm{E}}_{\mathrm{d}}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\displaystyle \frac{{‖{\mathrm{d}}_{\mathrm{i}}^{\prime }-{\mathrm{d}}_{\mathrm{i}}‖}_2}{{‖{\mathrm{d}}_{\mathrm{i}}‖}_2}}$$

(13)

where m is the true model, m′ is the inverted model, n is the number of samples, d is the theoretical gravity anomaly, and d′ is the gravity anomaly calculated from the inverted model.

5.3. Inversion Results

By inputting the 2D gravity data into the trained networks, we obtained the three-dimensional subsurface inversion structures. Figure 6 shows the loss curves for the conventional U-Net and the proposed Deeply weighted Multi-Constraint U-Net. In the figure, the blue solid lines represent the training loss variation, while the orange dashed lines correspond to the validation loss variation. The experimental results indicate that, whether using the traditional MSE loss function or the proposed Multi-Constraint loss function, both the training and validation loss curves exhibit similar convergence behaviors. Specifically, the loss values decrease rapidly during the early stages of training; as the number of epochs increases, the rate of decline slows and eventually stabilizes. Moreover, the training and validation loss curves converge synchronously without noticeable divergence, indicating that neither underfitting nor overfitting occurred throughout the entire training process for both methods.

5.3.1. Random Model Inversion

To provide a more intuitive comparison of the inversion performance between the Deeply weighted Multi-Constraint U-Net and the conventional U-Net, four models were selected from the random model test set for evaluation and analysis. By gradually increasing the number of density bodies to be inverted, we systematically examined how the two networks perform under different levels of structural complexity. Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the inversion results for the four validated models.

The comparative results indicate that the Deeply weighted Multi-Constraint U-Net delivers significantly better inversion performance than the conventional U-Net. While conventional U-Net can reasonably predict the positions of the models, it tends to produce blurred boundaries and exhibits considerable discrepancies in density values—showing relatively accurate density predictions in shallower layers but suffering from severe attenuation in deeper layers. In contrast, the Deeply weighted Multi-Constraint U-Net more clearly reconstructs the shapes of the models and provides more accurate predictions of deep-layer density values.

Furthermore, we performed forward gravity calculations based on the predicted models. As shown in Figure 8, Figure 10, Figure 12 and Figure 14, the gravity anomalies computed from both networks’ predicted models exhibit shapes that are largely consistent with the input gravity anomalies. However, analysis of the gravity deviations reveals that the predicted models from the Deeply weighted Multi-Constraint U-Net yield smaller gravity deviations than those from the conventional U-Net. This demonstrates that the Deeply weighted Multi-Constraint U-Net outperforms the conventional U-Net in both model prediction accuracy and gravity data fitting quality.

However, it is inevitable that the inversion results for shallow layers outperform those for deeper layers, indicating insufficient information in the gravity data—an inherent limitation of gravity inversion. To improve deep-layer inversion performance, the loss function of the proposed Deeply weighted Multi-Constraint U-Net incorporates a depth-weighting function, enabling the network to place greater emphasis on information from deeper layers during training. We further investigate the role of the depth-weighting function in gravity inversion. The subsurface domain is stratified, and the errors between the predicted and true density values are calculated for each layer, as shown in Figure 15. It can be observed that all three network models exhibit a general trend of increasing inversion error with depth. The Multi-Constraint network without depth weighting clearly outperforms the conventional U-Net, while incorporating depth weighting leads to further improvement in deep-layer inversion accuracy.

In addition, we conducted a statistical analysis of the inversion errors for both the Deeply weighted Multi-Constraint U-Net and the conventional U-Net. As shown in Figure 16 and

Table 1. Statistical Summary of Inversion Errors for Models I to IV.

Model	Em	Ed
Model Ⅰ	0.6613	0.1448
Model Ⅰ(Deeply weighted Multiple constraints)	0.3470	0.0517
Model Ⅱ	0.5674	0.0569
Model Ⅱ(Deeply weighted Multiple constraints)	0.3849	0.0515
Model Ⅲ	0.5996	0.0695
Model Ⅲ(Deeply weighted Multiple constraints)	0.4349	0.0594
Model Ⅳ	0.6670	0.1031
Model Ⅳ(Deeply weighted Multiple constraints)	0.5549	0.0681

, the proposed method exhibits a clear advantage in error control, achieving significant improvements in the accuracy and reliability of the inversion results. This demonstrates that the introduction of multiple constraint terms can effectively enhance the performance of deep learning-based inversion methods, enabling the inversion outcomes under complex geological conditions to more closely approximate the actual subsurface structures.

5.3.2. Rule Model Inversion

To evaluate the generalization capability of the Deeply weighted Multi-Constraint U-Net on unseen models, we selected four types of rule-based models, distinct from those in the training set, as an independent test set. Their corresponding gravity anomaly data were fed into the trained network, and the prediction results are shown in Figure 17, Figure 18, Figure 19 and Figure 20.

Overall, the Deeply weighted Multi-Constraint U-Net was able to predict shapes and positions across different model types that closely match the true models, indicating that the method maintains a certain degree of adaptability when handling previously unseen structures. However, the inversion performance was relatively poorer for the more complex inclined dyke model. This may be attributed to the lack of similar samples in the training set and the unique geometry of the model. The inclined shape causes the gravity response to be more spatially dispersed, increasing the difficulty of feature extraction and inversion for the network. Additionally, the projected shape of the inclined structure on the observation plane changes with depth, introducing asymmetric features that challenge the network’s ability to leverage learned patterns from other models for accurate fitting.

6. Practical Applications

To validate the application effectiveness of the proposed method on real data, we employed the Multi-Constraint deep learning approach to perform subsurface density inversion based on surface gravity anomalies from the M_s7.8 Tangshan earthquake. On 28 July 1976, a major earthquake with a magnitude of M_s7.8 struck the Tangshan region. The epicenter was located in Yuehe Township, Kaiping District, Tangshan, at geographic coordinates 39.6° N latitude and 118.2° E longitude, with a focal depth of approximately 11 km. The source mechanism of this earthquake revealed a complex rupture process characterized by prominent strike-slip features. The occurrence and preparation of earthquakes are closely linked to tectonic movements within the crust, which induce crustal deformation and density variations, thereby causing gravity anomalies [45]. Therefore, an in-depth study of the subsurface density structure in the seismic zone is of great significance for understanding the earthquake preparation mechanisms. The gravity anomaly characteristics of the Tangshan seismic zone are prominent, as shown in Figure 21. Observational data indicate a distinct positive gravity anomaly in the Tangshan area, which may be closely related to subsurface material migration activities. The Tangshan seismic zone lies on the eastern side of the Taiyuan–Yanqing Bouguer gravity anomaly gradient zone (i.e., the middle segment of the gravity step zone in eastern China) and the Moho discontinuity steep variation zone. Studies have shown that the crustal density structure in this area exhibits a distribution pattern of higher density in the south and lower density in the north, with scattered shallow high-density bodies gradually converging with increasing depth. This variation is possibly associated with magmatic activities in the deep upper mantle [46].

This study constructed a deep learning Deeply weighted Multi-Constraint U-Net model based on synthetic datasets. Gravity data of the study area were sampled using a regular grid, discretizing the region into a 64 × 64 × 16 three-dimensional grid space, with each cell uniformly sized at 3.125 km × 3.125 km × 3.125 km to achieve high-precision 3-D subsurface density inversion. Based on the inversion results, different depth-level horizontal profiles and three-dimensional structural diagrams were systematically generated to reveal the large-scale density distribution patterns, as shown in Figure 22, Figure 23 and Figure 24. In order to deeply analyze the causes of the positive gravity anomaly in the Tangshan earthquake zone, two vertical cross-sectional diagrams (Diagram A) along the fault direction and a cross-sectional diagram perpendicular to the fault (Diagram B) were drawn to explore the relationship between the fault structure and the density anomaly. The vertical section results are presented in Figure 25, revealing subsurface structural features near the earthquake epicenter. A prominent positive density body can be clearly observed, extending downward from approximately 6 km depth to about 40 km depth, seemingly cutting through the Moho discontinuity. This positive density anomaly is likely associated with subsurface magma, indicating possible magmatic activity in the region. Since magma typically has a higher density than surrounding rocks, its upward migration may cause the observed positive gravity anomaly. Moreover, during the earthquake, these high-density magmatic materials exhibited significant upward surges along the fault lines, intensifying the vertical heterogeneity of the density structure. This phenomenon not only reflects the active deep material processes in the study area but also provides important insights for explaining the seismic triggering mechanisms and deep dynamic processes in the region.

7. Conclusions

This study proposes a Multi-Constraint deep learning gravity inversion method (Deeply weighted Multi-Constraint U-Net) that integrates data-driven and physics-driven approaches. By optimizing the U-Net architecture and employing a composite Multi-Constraint loss function, the method significantly improves the accuracy and reliability of gravity inversion.

(1)

Network Architecture Design: Several enhancements were made to the traditional U-Net framework. First, conventional pooling operations in the downsampling path were replaced with convolutional operations to better preserve critical fine details. Second, transpose convolution was introduced in the upsampling path, overcoming the limitations of traditional interpolation methods and enabling dynamic learning during feature reconstruction. These improvements allow the network to learn more complex data mappings.

(2)

Loss Function Construction: A composite Multi-Constraint loss function was developed, including 3D Intersection-over-Union (IoU) loss to strengthen geometric constraints; Spatial 3D Mean Squared Error (MSE) loss, an extension of traditional MSE, to better describe spatial continuity in the subsurface density field; depth-weighting functions to mitigate the skin effect commonly encountered in gravity inversion; and gravity field equation constraints to ensure physical plausibility. This multi-scale, multi-physics constraint framework provides rigorous mathematical and physical guarantees for the inversion results.

(3)

Dataset Construction: A highly diversified training dataset was created using the random walk method, effectively simulating complex underground anomaly structures.

Numerical experiments demonstrate that the proposed Deeply weighted Multi-Constraint U-Net outperforms the conventional U-Net in inversion performance. Systematic comparative analysis reveals inherent shortcomings of the traditional U-Net, such as blurred boundaries, underestimated densities, and signal attenuation with increasing depth. These limitations become more pronounced as model complexity grows. In contrast, the Deeply weighted Multi-Constraint U-Net exhibits superior inversion capabilities, accurately capturing density variations, burial depths, and boundary features of anomalies. Forward modeling further confirms that although gravity anomalies generated by both networks resemble reference models in overall morphology, the deviations produced by the Deeply weighted Multi-Constraint U-Net are significantly smaller.

Applied to gravity anomaly data from the Tangshan seismic zone in China, the method yielded 3D subsurface density images. Near the earthquake epicenter, a prominent positive density anomaly extending beneath the Moho discontinuity was identified. Combined with regional geological analysis, this anomaly is interpreted as representing deep magmatic material. This finding suggests that during seismic events, such high-density materials may vertically migrate along fault zones, resulting in significant density contrasts. The results not only reveal dynamic features of deep material activity within the study area but also provide geophysical evidence crucial for understanding the earthquake triggering mechanisms in this region.