FNO deep learning model for gravity inversion, training method and gravity inversion method

Abstract

The invention discloses an FNO deep learning model for gravity inversion, a training method and a gravity inversion method, and relates to the technical field of gravity inversion in geophysical exploration, the model comprises a feature improvement layer, at least two spatial adaptive operator layers and a feature projection layer arranged behind the last spatial adaptive operator layer which are connected in sequence, and at least one cross-layer residual jump connection; each spatial adaptive operator layer comprises a global Fourier branch, a local convolution branch, an adaptive fusion module, a spatial attention module and an activation module. The method overcomes the defects that a traditional inversion method is high in dependence on an initial model and low in calculation efficiency during gravity inversion and a traditional deep learning model is low in generalization ability and difficult to capture global features, and gravity data intelligent inversion with high precision, high efficiency and high generalization ability is achieved.

Description

Technical Field

[0001] This invention relates to the field of gravity inversion technology in geophysical exploration, specifically to an FNO deep learning model, training method, and gravity inversion method for gravity inversion. Background Technology

[0002] Gravity exploration is a geophysical method that infers the distribution of subsurface density by measuring changes in the Earth's surface gravity field. It has wide applications in mineral exploration, oil and gas resource assessment, and geological structure research. Gravity inversion is the core component of gravity exploration, and its goal is to quantitatively estimate the density structure of the subsurface medium based on observed gravity anomaly data.

[0003] Traditional gravity inversion methods are mainly divided into two categories: linear inversion and nonlinear inversion. Linear inversion methods (such as least squares and conjugate gradient methods) are computationally efficient, but their results heavily depend on the selection of the initial model and the construction of regularization constraints, making them prone to getting trapped in local optima and difficult to handle strongly nonlinear problems. Nonlinear inversion methods (such as genetic algorithms, simulated annealing, and particle swarm optimization) possess global search capabilities, overcoming the dependence on the initial model to some extent. However, their computational cost is extremely high, and their iterative convergence speed is slow, making them difficult to apply to large-scale three-dimensional inversion problems.

[0004] In recent years, deep learning technology has provided new insights into gravity inversion. Researchers have utilized architectures such as convolutional neural networks and U-Net to learn end-to-end mappings from gravity data to density models, achieving faster inversion speeds than traditional methods. However, these grid-based deep learning models have inherent limitations: they learn mappings on specific discrete grids, and their generalization ability is limited by the grid resolution of the training data; their receptive field is limited, mainly expanding slowly by stacking convolutional layers, making it difficult to effectively capture crucial global features in the gravity field, resulting in insufficient accuracy in inverting subsurface structures and complex geological boundaries.

[0005] The Fourier Neural Operator (FNO), as an emerging neural operator framework, focuses on learning mappings between infinite-dimensional function spaces. FNO efficiently processes global information using the Fast Fourier Transform by parameterizing the integral kernel in the frequency domain, naturally possessing a global receptive field. However, while standard FNO also possesses a global receptive field, its fixed frequency domain processing method fails to fully consider the physical fact that the sensitivity of the gravity field to sources at different depths is different, resulting in limited inversion accuracy for shallow fine structures and deep anomalies. Therefore, developing an intelligent inversion model or method that can capture global dependencies, focus on local details, and embed physical laws has become an urgent need in the field of gravity exploration. Summary of the Invention

[0006] To address the aforementioned shortcomings in existing technologies, the FNO deep learning model, training method, and gravity inversion method provided by this invention solve the problem that existing FNO deep learning models, when performing gravity inversion, fail to fully consider the physical fact that the sensitivity of the gravity field to sources at different depths underground is different, resulting in limited inversion accuracy for shallow fine structures and deep anomalies.

[0007] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: A deep learning model for gravity inversion using the Fiber-Nutrition (FNO) method is provided, comprising sequentially connected feature lifting layers, at least two spatial adaptive operator layers, a feature projection layer placed after the last spatial adaptive operator layer, and at least one cross-layer residual skip connection; each spatial adaptive operator layer includes a global Fourier branch, a local convolutional branch, an adaptive fusion module, a spatial attention module, and an activation module; wherein: The feature enhancement layer is used to concatenate the input scalar gravity field data with the coordinate grid representing the spatial location and project it into a high-dimensional feature space to obtain the high-dimensional features corresponding to the gravity field data. The global Fourier branch uses the fast Fourier transform to perform a linear transformation on the high-dimensional features corresponding to the gravity field data in the frequency domain to capture low-frequency trends and outputs the global Fourier features corresponding to the gravity field data. The local convolution branch is used to perform convolution operations on the high-dimensional features corresponding to the gravity field data to capture high-frequency trends and output the local convolution features corresponding to the gravity field data. The adaptive fusion module is used to fuse the global Fourier features and local convolutional features corresponding to the gravity field data, and output the fused features corresponding to the gravity field data; the fusion weights are adaptively adjusted according to the gravity field depth and spatial location. The spatial attention module is used to generate attention weights corresponding to the fused features through the spatial attention mechanism, multiply the fused features by the corresponding attention weights, and output the features for density anomaly identification. The activation module is used to activate the output of the spatial attention module; The feature projection layer predicts the three-dimensional density model corresponding to the gravity field data by mapping the output of the activation module back to the target output dimension, thus completing gravity inversion. A cross-layer residual jump connection is used to fuse the input of the k-th spatial adaptive operator layer and the output of the p-th spatial adaptive operator layer, and the fusion result is used as the input of the next layer structure of the p-th spatial adaptive operator layer; pk≥1.

[0008] The beneficial effects of this invention are as follows: 1. This model deeply integrates the physical characteristics of gravity field inversion (especially the law of sensitivity decay with depth: fusion weights) into the network design, and dynamically adjusts the feature extraction method through a spatial adaptive mechanism to achieve high-precision inversion of the entire scale from shallow details to deep anomalies. 2. This model captures high-frequency details and low-frequency trends through two branches, and dynamically fuses the features of local and global branches through "fusion weights." The fusion weights are learnable and related to depth information, enabling the network to automatically learn to "focus" on local details in shallow layers and "pay attention" to global trends in deeper layers. A spatial attention mechanism is also introduced to emphasize regions with significant gravity anomalies or rich features, while suppressing the feature contributions of noisy or unimportant background regions. 3. This model uses cross-layer residual skip connections, allowing the input signal to skip one or more layers and be directly added to the outputs of these layers. This reduces common gradient vanishing and model degradation problems, ensuring effective gradient propagation within the model and enhancing feature reuse capabilities. This allows the model to maintain strong representational power while ensuring training stability, ultimately supporting high-precision mapping learning from two-dimensional gravity field observation data to three-dimensional spatial density distribution. This design is particularly suitable for multi-scale physical problems like gravity inversion, which require simultaneous modeling of shallow details and deep structures. It is one of the key technical guarantees for achieving high-precision inversion while maintaining the training stability of deep networks. 4. This model assigns different fusion weights to each depth channel. For example, the local convolution branch has a higher weight for shallow channels, while the global Fourier branch has a higher weight for deep channels. This allows the model to adaptively handle the physical characteristics of gravity inversion problems.

[0009] Furthermore, within the global Fourier branch, specific methods for capturing low-frequency trends by using the Fast Fourier Transform to linearly transform the high-dimensional features corresponding to the gravity field data in the frequency domain include: The high-dimensional features corresponding to the gravity field data are transformed to the frequency domain by fast Fourier transform. The transformed frequency domain feature data is then multiplied point-by-point with a set of learnable complex weight tensors. The result of the point-by-point multiplication is then transformed back to the spatial domain by inverse Fourier transform to obtain the global Fourier features corresponding to the gravity field data. Among them, some feature data are 32 low-frequency Fourier modes in the X direction and 17 low-frequency Fourier modes in the Y direction.

[0010] Furthermore, in the local convolution branch, the convolution operation used is a 1×1 convolution operation.

[0011] Furthermore, in the adaptive fusion module, the specific methods for fusing the global Fourier features and local convolutional features corresponding to the gravity field data include: Obtain the global Fourier feature weights and local convolutional feature weights corresponding to the depth and spatial location of the gravitational field; The global Fourier feature weights and local convolution feature weights are normalized using softmax so that the sum of the two weights is 1, resulting in the normalized global Fourier feature weights and normalized local convolution feature weights. The global Fourier feature is multiplied by the weights of the normalized global Fourier feature, and the local convolutional feature is multiplied by the weights of the normalized local convolutional feature. The two multiplication results are added together to obtain the fused feature.

[0012] Furthermore, the spatial attention module comprises, in sequence, a convolutional layer with a kernel size of 1, a GeLU activation function layer, another convolutional layer with a kernel size of 1, a Sigmoid activation function layer, and a multiplicative output layer; wherein: The first convolutional layer with a kernel size of 1 is used to compress the number of channels of the fused features to 1 / 4 of the original. The gelu activation function layer is used to introduce non-linearity into the output of the first convolutional layer with a kernel size of 1; The second convolutional layer with a kernel size of 1 is used to restore the output of the Gelu activation function layer to the original channel dimension; The Sigmoid activation function layer is used to generate attention weights in the [0,1] interval based on the output of the second convolutional layer with a kernel size of 1; The multiplication output layer multiplies the fused features with the attention weights output from the Sigmoid activation function layer, producing features for density anomaly identification.

[0013] The beneficial effects of adopting the above-mentioned further solutions are as follows: Unlike common attention mechanisms, this spatial attention module does not use pooling or downsampling, preserving the spatial resolution of the original feature maps and avoiding information loss. This spatial attention module achieves spatially adaptive weighting of the feature maps through simple 1×1 convolutions and sigmoid activation, enabling the network to "focus" on the regions most critical to the inversion task. Working in conjunction with the adaptive fusion module, it forms an advanced architecture with both depth and spatial adaptation.

[0014] Furthermore, the activation module employs the GELU nonlinear activation function.

[0015] Furthermore, there are 11 spatial adaptive operator layers and 5 cross-layer residual jump connections; the corresponding values ​​of p are 3, 5, 7, 9, and 11; pk=2.

[0016] A method for training an FNO deep learning model for gravity inversion is provided, comprising the following steps: Set the batch size, learning rate, optimizer, and termination training conditions; The scalar gravity field data of the known three-dimensional density model and the coordinate grid representing the spatial location are used as inputs to the FNO deep learning model for gravity inversion, and the corresponding output three-dimensional density model is obtained. Construct a loss function and calculate the loss value based on the real 3D density model and the corresponding output 3D density model; The FNO deep learning model for gravity inversion is optimized using an optimizer based on the loss value until the training termination condition is met, thus completing the training of the FNO deep learning model for gravity inversion.

[0017] Furthermore, the expression for the loss function is:

[0018]

[0019] in Represents the loss function; For structural similarity loss, , , For the first A predicted three-dimensional density model for each sample; For the first A true three-dimensional density model of a sample; is the normalized density constant; N is the total number of coordinate grids; For the first In the nth sample The true 3D density values ​​of each coordinate grid; For the first In the nth sample Predicted 3D density values ​​for each coordinate grid; These are the weighting coefficients; This represents the consistency loss in physical forward modeling. Gravity forward modeling anomaly data for a true 3D density model; To predict gravity forward modeling anomaly data for a three-dimensional density model.

[0020] The technical effects of adopting the above further solution are as follows: Structural similarity loss represents the degree of similarity between the predicted and actual results, ensuring that the predicted model and the actual model are consistent in shape and location. This loss term can segment and delineate the clear edges of small target objects even when the number of target and background voxels is unbalanced. Physical forward modeling consistency loss forces the model's predicted 3D density model to reconstruct the input gravity data through known forward modeling physics. This is the key to embedding physical constraints into the data-driven model. By introducing this constraint term, the inversion process can be stabilized, the model's robustness enhanced, and the quality of the inverted density model improved.

[0021] A gravity inversion method based on an FNO deep learning model for gravity inversion is provided, comprising the following steps: Acquire scalar gravity field data and coordinate grids representing spatial locations, and input them into a trained FNO deep learning model for gravity inversion. Obtain the three-dimensional density model output by the FNO deep learning model for gravity inversion to complete the gravity inversion.

[0022] The beneficial effects of this invention are as follows: This gravity inversion method adaptively fuses local convolutional branch results and global Fourier branch sensing results, and can dynamically adjust its "receptive field" according to the physical characteristics of the gravity field (sensitivity decreases with depth). This enables differentiated and accurate extraction of shallow fine anomalies and deep regional anomalies, and easily inverts deep anomalies under the interference of shallow anomalies. The entire inversion process only requires a single forward propagation and takes very little time (seconds). It overcomes the bottleneck of high computational cost of traditional iterative inversion, and significantly improves inversion accuracy, especially in the recovery of large-scale structures and complex geological boundaries. Attached Figure Description

[0023] Figure 1 This is a structural block diagram of the FNO deep learning model used for gravity inversion in an embodiment of the present invention; Figure 2 The random walk density model and corresponding forward gravity data constructed in this embodiment of the invention; Figure 3 The loss curves of the training set and validation set in this embodiment of the invention are compared with the standard FNO. Figure 4 This is a comparison between a certain inversion result of FNO on the test set and the true result in an embodiment of the present invention; and planar views of the true model and the inversion result at 400m are captured respectively; Figure 5 This is a comparison between a certain inversion result of FNO on the test set and the actual result in an embodiment of the present invention. Planar views of the actual model and the inversion result at 400m are also shown. Figure 6The present invention provides actual gravity data and inversion results for the Vinton Salt Dome in the United States. Detailed Implementation

[0024] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0025] like Figure 1 As shown, the FNO deep learning model for gravity inversion includes sequentially connected feature lifting layers, at least two spatial adaptive operator layers, a feature projection layer placed after the last spatial adaptive operator layer, and at least one cross-layer residual skip connection; each spatial adaptive operator layer includes a global Fourier branch, a local convolutional branch, an adaptive fusion module, a spatial attention module, and an activation module; wherein: The feature enhancement layer is used to concatenate the input scalar gravity field data with the coordinate grid representing the spatial location and project it into a high-dimensional feature space to obtain the high-dimensional features corresponding to the gravity field data. The global Fourier branch uses the fast Fourier transform to perform a linear transformation on the high-dimensional features corresponding to the gravity field data in the frequency domain to capture low-frequency trends and outputs the global Fourier features corresponding to the gravity field data. The local convolution branch is used to perform convolution operations on the high-dimensional features corresponding to the gravity field data to capture high-frequency trends and output the local convolution features corresponding to the gravity field data. The adaptive fusion module is used to fuse the global Fourier features and local convolutional features corresponding to the gravity field data, and output the fused features corresponding to the gravity field data; the fusion weights are adaptively adjusted according to the gravity field depth and spatial location. The spatial attention module is used to generate attention weights corresponding to the fused features through the spatial attention mechanism, multiply the fused features by the corresponding attention weights, and output the features for density anomaly identification. The activation module uses the Gelu nonlinear activation function to activate the output of the spatial attention module. The feature projection layer predicts the three-dimensional density model corresponding to the gravity field data by mapping the output of the activation module back to the target output dimension, thus completing gravity inversion. A cross-layer residual jump connection is used to fuse the input of the k-th spatial adaptive operator layer and the output of the p-th spatial adaptive operator layer, and the fusion result is used as the input of the next layer structure of the p-th spatial adaptive operator layer; pk≥1.

[0026] In some embodiments, the feature enhancement layer is a fully connected layer, and its data processing procedure is exemplarily as follows: First, a normalized two-dimensional coordinate grid (x, y) is generated for each spatial location of the input tensor; Subsequently, the gravity data channel and the two coordinate grid channels are concatenated along the channel dimension to obtain a 3-channel feature tensor; Finally, a fully connected layer upscales the 3-channel features to a higher feature dimension (width = 128 in this embodiment). This step maps the low-dimensional input to a high-dimensional representation space suitable for subsequent complex transformations.

[0027] In the global Fourier branch, specific methods for capturing low-frequency trends by using the Fast Fourier Transform to linearly transform the high-dimensional features corresponding to gravity field data in the frequency domain include: The high-dimensional features corresponding to the gravity field data are transformed to the frequency domain by fast Fourier transform. The transformed frequency domain feature data is then multiplied point-by-point with a set of learnable complex weight tensors. The result of the point-by-point multiplication is then transformed back to the spatial domain by inverse Fourier transform to obtain the global Fourier features corresponding to the gravity field data. Among them, some feature data are 32 low-frequency Fourier modes in the X direction and 17 low-frequency Fourier modes in the Y direction.

[0028] In the local convolution branch, the convolution operation used is a 1×1 convolution operation.

[0029] In the adaptive fusion module, the specific method for fusing the global Fourier features and local convolutional features corresponding to the gravity field data includes the following steps: A1. Obtain the global Fourier feature weights and local convolution feature weights corresponding to the depth and spatial location of the gravitational field; in particular, the data at the same depth will be processed in two branches, so each depth will also correspond to two weights, thus obtaining the weight matrix corresponding to each depth. A2. Normalize the global Fourier feature weights and local convolution feature weights using softmax so that the sum of the two weights is 1, thus obtaining the normalized global Fourier feature weights and normalized local convolution feature weights. A3. Multiply the global Fourier feature with the normalized global Fourier feature weights, multiply the local convolution feature with the normalized local convolution feature weights, and add the two multiplication results to obtain the fused feature.

[0030] In some embodiments, the spatial attention module includes a convolutional layer with a kernel size of 1, a Gelu activation function layer, another convolutional layer with a kernel size of 1, a Sigmoid activation function layer, and a multiplication output layer connected in sequence; wherein: The first convolutional layer with a kernel size of 1 is used to compress the number of channels of the fused features to 1 / 4 of the original. The gelu activation function layer is used to introduce non-linearity into the output of the first convolutional layer with a kernel size of 1; The second convolutional layer with a kernel size of 1 is used to restore the output of the Gelu activation function layer to the original channel dimension; The Sigmoid activation function layer is used to generate attention weights in the [0,1] interval based on the output of the second convolutional layer with a kernel size of 1; The multiplication output layer multiplies the fused features with the attention weights output from the Sigmoid activation function layer, producing features for density anomaly identification.

[0031] For example, there are 11 spatial adaptive operator layers and 5 cross-layer residual jump connections; the values ​​of p are 3, 5, 7, 9, and 11; pk=2. In this case, since the 5th cross-layer residual jump fuses the input of the 9th spatial adaptive operator layer with the output of the 11th spatial adaptive operator layer (i.e., the last spatial adaptive operator layer), the 5th cross-layer residual jump uses the fusion result as the input of the feature projection layer.

[0032] In some embodiments, the feature projection layer consists of two fully connected layers, and the dimensions of the predicted 3D density model are [batch size, 16, 32, 32].

[0033] In some embodiments, the method for training an FNO deep learning model for gravity inversion includes the following steps: B1. Set the batch size, learning rate, optimizer, and termination training conditions; B2. Take the scalar gravity field data of the known three-dimensional density model and the coordinate grid representing the spatial position as input to the FNO deep learning model for gravity inversion, and obtain the corresponding output three-dimensional density model. B3. Construct a loss function and calculate the loss value based on the real 3D density model and the corresponding output 3D density model; The FNO deep learning model for gravity inversion is optimized using an optimizer based on the loss value until the training termination condition is met, thus completing the training of the FNO deep learning model for gravity inversion.

[0034] Optionally, the expression for the loss function is:

[0035]

[0036] in Represents the loss function; For structural similarity loss, , , For the first A predicted three-dimensional density model for each sample; For the first A true three-dimensional density model of a sample; The normalized density constant can be taken as 1 g / cm³. 3 N is the total number of coordinate grids; For the first In the nth sample The true 3D density values ​​of each coordinate grid; For the first In the nth sample Predicted 3D density values ​​for each coordinate grid; These are the weighting coefficients; This represents the consistency loss in physical forward modeling. Gravity forward modeling anomaly data for a true 3D density model; To predict gravity forward modeling anomaly data for a three-dimensional density model.

[0037] Structural similarity loss is insensitive to class imbalance between the foreground (anomaly) and background, effectively driving the network to accurately characterize the geometry and spatial location of the anomaly. It is particularly suitable for scenarios where the target object typically occupies only a small portion of the volume in gravity inversion. Physical forward modeling consistency loss forces the model's predicted 3D density model to reconstruct the input gravity data using known forward modeling physics, which is crucial for embedding physical constraints into the data-driven model. This loss term fundamentally limits the space of the model's output solution, confining it to a set of feasible solutions that satisfy basic physical laws. This greatly alleviates the ambiguity of the inversion problem and improves the physical interpretability and generalization ability of the inversion results.

[0038] In some embodiments, the gravity inversion method based on the FNO deep learning model for gravity inversion includes the following steps: Acquire scalar gravity field data and coordinate grids representing spatial locations, and input them into a trained FNO deep learning model for gravity inversion. Obtain the three-dimensional density model output by the FNO deep learning model for gravity inversion to complete the gravity inversion.

[0039] In one embodiment of the invention, the underground space is divided into 16×32×32 uniform grids (representing the Z, X, and Y directions respectively), each grid having a size of 50m×50m×50m. The background medium density is set to 0 g / cm³, and the anomaly density is set to 1 g / cm³. Then, a three-dimensional density model is generated using a random walk method. Within the aforementioned 16×32×32 grid, one or two starting points consisting of 2×2×2=8 adjacent small squares are randomly selected and moved along random directions (up, down, left, right, forward, and backward) within the geological constraints to generate irregularly shaped density anomalies. Figure 2 An example of a randomly generated 3D density model is shown. In this embodiment, a total of 50,000 3D density models were constructed as the subsequent training and validation sets. The test set consists of random combinations of regular prisms and inclined stepped models at different locations and depths to simulate actual geological anomaly morphology.

[0040] Subsequently, gravity anomaly data for the three-dimensional density model were calculated using the gravity forward modeling formula. According to the principle of potential field superposition, the gravity anomaly value at any observation point can be obtained by superimposing the gravity anomalies generated by each rectangular prism element at that point. In a spatial rectangular coordinate system, the center coordinates of a certain prism are... Gravity anomaly generated at observation point Q(x,y,z) It can be represented as:

[0041] in It is the gravitational constant; Density; , , , .

[0042] The loss curves of the training set and validation set during the training process in this embodiment are as follows: Figure 3 As shown, the loss of this model converges rapidly and tends to stabilize. A comparison with the standard FNO algorithm reveals that the results of this model (spatial adaptive FNO) are significantly better than the standard FNO algorithm.

[0043] After the model training is completed, the model parameters are saved, and the random model in the test set is inverted for verification.

[0044] Qualitative analysis: Figure 4 and Figure 5 The inversion results for some test sets are shown. It can be seen that the model / gravity inversion method successfully recovered the shape and spatial location of the anomaly, and is in high agreement with the real model.

[0045] Quantitative analysis: The model's relative L2 error E and the coefficient of determination R for data fitting were used. 2Two metrics were used for evaluation. On the test set, this inversion method outperformed the traditional gravity inversion method in terms of model reconstruction error, while maintaining a very high data fit, thus verifying its effectiveness.

[0046] This gravity inversion method was applied to actual gravity data from the Vinton Salt Dome in the United States, such as... Figure 6 As shown, the inversion results clearly depict the location and spatial distribution of the salt domes, which is consistent with known geological understanding, further proving the practicality of this model / gravity inversion method.

[0047] In summary, this invention overcomes the shortcomings of traditional inversion methods, such as strong dependence on the initial model, low computational efficiency, weak generalization ability of traditional deep learning models, and difficulty in capturing global features, when performing gravity inversion. It achieves intelligent gravity data inversion with high precision, high efficiency, and strong generalization ability.

Claims

1. A deep learning model for FNO (Focus on Noise Reversal) for gravity inversion, characterized in that, It includes sequentially connected feature lifting layers, at least two spatial adaptive operator layers, a feature projection layer placed after the last spatial adaptive operator layer, and at least one cross-layer residual skip connection; each spatial adaptive operator layer includes a global Fourier branch, a local convolution branch, an adaptive fusion module, a spatial attention module, and an activation module; wherein: The feature enhancement layer is used to concatenate the input scalar gravity field data with the coordinate grid representing the spatial location and project it into a high-dimensional feature space to obtain the high-dimensional features corresponding to the gravity field data. The global Fourier branch uses the fast Fourier transform to perform a linear transformation on the high-dimensional features corresponding to the gravity field data in the frequency domain to capture low-frequency trends and outputs the global Fourier features corresponding to the gravity field data. The local convolution branch is used to perform convolution operations on the high-dimensional features corresponding to the gravity field data to capture high-frequency trends and output the local convolution features corresponding to the gravity field data. The adaptive fusion module is used to fuse the global Fourier features and local convolutional features corresponding to the gravity field data, and output the fused features corresponding to the gravity field data; the fusion weights are adaptively adjusted according to the gravity field depth and spatial location. The spatial attention module is used to generate attention weights corresponding to the fused features through the spatial attention mechanism, multiply the fused features by the corresponding attention weights, and output the features for density anomaly identification. The activation module is used to activate the output of the spatial attention module; The feature projection layer predicts the three-dimensional density model corresponding to the gravity field data by mapping the output of the activation module back to the target output dimension, thus completing gravity inversion. A cross-layer residual jump connection is used to fuse the input of the k-th spatial adaptive operator layer and the output of the p-th spatial adaptive operator layer, and the fusion result is used as the input of the next layer structure of the p-th spatial adaptive operator layer; pk≥1.

2. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, In the global Fourier branch, specific methods for capturing low-frequency trends by using the Fast Fourier Transform to linearly transform the high-dimensional features corresponding to gravity field data in the frequency domain include: The high-dimensional features corresponding to the gravity field data are transformed to the frequency domain by fast Fourier transform. The transformed frequency domain feature data is then multiplied point-by-point with a set of learnable complex weight tensors. The result of the point-by-point multiplication is then transformed back to the spatial domain by inverse Fourier transform to obtain the global Fourier features corresponding to the gravity field data. Among them, some feature data are 32 low-frequency Fourier modes in the X direction and 17 low-frequency Fourier modes in the Y direction.

3. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, In the local convolution branch, the convolution operation used is a 1×1 convolution operation.

4. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, In the adaptive fusion module, the specific methods for fusing the global Fourier features and local convolutional features corresponding to gravity field data include: Obtain the global Fourier feature weights and local convolutional feature weights corresponding to the depth and spatial location of the gravitational field; The global Fourier feature weights and local convolution feature weights are normalized using softmax so that the sum of the two weights is 1, resulting in the normalized global Fourier feature weights and normalized local convolution feature weights. The global Fourier feature is multiplied by the weights of the normalized global Fourier feature, and the local convolutional feature is multiplied by the weights of the normalized local convolutional feature. The two multiplication results are added together to obtain the fused feature.

5. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, The spatial attention module consists of a convolutional layer with a kernel size of 1, a GeLU activation function layer, another convolutional layer with a kernel size of 1, a Sigmoid activation function layer, and a multiplicative output layer, connected in sequence; wherein: The first convolutional layer with a kernel size of 1 is used to compress the number of channels of the fused features to 1 / 4 of the original. The gelu activation function layer is used to introduce non-linearity into the output of the first convolutional layer with a kernel size of 1; The second convolutional layer with a kernel size of 1 is used to restore the output of the Gelu activation function layer to the original channel dimension; The Sigmoid activation function layer is used to generate attention weights in the [0,1] interval based on the output of the second convolutional layer with a kernel size of 1; The multiplication output layer is used to multiply the fused features with the attention weights output by the Sigmoid activation function layer, and output the features used for density anomaly identification.

6. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, The activation module uses the GELU non-linear activation function.

7. The FNO deep learning model for gravity inversion according to claim 1, characterized in that, There are 11 spatial adaptive operator layers and 5 cross-layer residual jump connections; the values ​​of p are 3, 5, 7, 9, and 11; pk=2.

8. A training method for the FNO deep learning model for gravity inversion as described in any one of claims 1 to 7, characterized in that, Includes the following steps: Set the batch size, learning rate, optimizer, and termination training conditions; The scalar gravity field data of the known three-dimensional density model and the coordinate grid representing the spatial location are used as inputs to the FNO deep learning model for gravity inversion, and the corresponding output three-dimensional density model is obtained. Construct a loss function and calculate the loss value based on the real 3D density model and the corresponding output 3D density model; The FNO deep learning model for gravity inversion is optimized using an optimizer based on the loss value until the training termination condition is met, thus completing the training of the FNO deep learning model for gravity inversion.

9. The training method according to claim 8, characterized in that, The expression for the loss function is: in Represents the loss function; For structural similarity loss, , , For the first A predicted three-dimensional density model for each sample; For the first A true three-dimensional density model of a sample; is the normalized density constant; N is the total number of coordinate grids; For the first In the nth sample The true 3D density values ​​of each coordinate grid; For the first In the nth sample Predicted 3D density values ​​for each coordinate grid; These are the weighting coefficients; This represents the physical forward modeling consistency loss. Gravity forward modeling anomaly data for a true 3D density model; To predict gravity forward modeling anomaly data for a three-dimensional density model.

10. A gravity inversion method based on the FNO deep learning model for gravity inversion according to any one of claims 1 to 7, characterized in that, Includes the following steps: Acquire scalar gravity field data and coordinate grids representing spatial locations, and input them into a trained FNO deep learning model for gravity inversion. Obtain the three-dimensional density model output by the FNO deep learning model for gravity inversion to complete the gravity inversion.

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