An aero-magnetic data deep learning forward and inverse method and system under physical constraint

Abstract

The application discloses a kind of physical constraint under aero-magnetic data deep learning forward and inversion method and system. Including: constructing the forward network based on deep operator network DeepONet, the network adopts logarithm-linear dual space mechanism, to log resistivity profile and flight height as input, predicts log quadratic field;Inversion network is constructed, with the input of observed transient electromagnetic response and flight height, and the output of resistivity profile;When training, the forward network is pre-trained to convergence under data-driven loss and Maxwell equation physical constraint loss;Inversion network is trained under the joint supervision of its model space loss and physical data space loss, wherein the physical data space loss is obtained by comparing the re-synthesized response with the original observation after inputting the resistivity profile predicted by the inversion network into the frozen forward network.The application deeply integrates physical laws into data-driven models, and realizes the integrated efficient solution of aero-magnetic forward and inversion.

Description

Technical Field

[0001] This invention relates to the field of aerospace electromagnetics, and in particular to a method and system for deep learning forward and inverse modeling of aerospace electromagnetic data under physical constraints. Background Technology

[0002] Airborne Electromagnetic (AEM) is a highly efficient non-contact exploration technology widely used in underground resource exploration, groundwater assessment, and geological hazard investigation. It involves transmitting transient electromagnetic fields into the ground via an airborne transmitting coil and receiving the secondary field response induced by the underground medium, thereby inverting the distribution of underground resistivity or conductivity. Traditional AEM data processing relies on deterministic optimization methods such as Gauss-Newton and Occam inversion, which have significant drawbacks: firstly, the inversion process requires iteratively solving forward operators and calculating the Jacobian matrix, with the computational load increasing exponentially with the number of measurement points, making it difficult to meet the real-time processing requirements of large-scale, high-density survey data; secondly, traditional methods are highly dependent on the initial model, easily getting trapped in local optima, and exhibit poor stability in noisy environments.

[0003] In recent years, deep learning technology, with its powerful nonlinear mapping capabilities, has been introduced into the field of airborne electromagnetic exploration. Data-driven methods based on models such as Convolutional Neural Networks (CNN) and Long Short-Term Memory Networks (LSTM) can achieve rapid mapping from measurement point responses to resistivity models through offline training, with inference speeds reaching millisecond levels, significantly improving inversion efficiency. However, existing deep learning methods are generally purely data-driven, with obvious limitations: First, network training relies on large-scale, high-quality labeled datasets, and generalization ability drops sharply when dataset coverage is insufficient; second, the network does not learn electromagnetic physics laws, and the output resistivity profiles and response curves often violate Maxwell's equations; third, the model has poor interpretability, making it difficult to meet the reliability and traceability requirements of geological exploration.

[0004] To address the physical inconsistency issues inherent in purely data-driven methods, the industry has begun incorporating physical constraints into deep learning frameworks, leading to methods such as Physically Guided Neural Networks (PGNNs) and Physically Information Neural Networks (PINNs). These methods incorporate the residuals of partial differential equations into the loss function, ensuring the network output satisfies fundamental physical laws. However, in the field of airborne electromagnetics, existing technologies still face significant bottlenecks: First, the dynamic range of airborne electromagnetic signals spans multiple orders of magnitude, making direct automatic differentiation in linear space prone to gradient explosion or numerical collapse; second, the presence of field singularities near the emission source hinders stable fitting by neural networks; third, existing PINN models are mostly optimized for single geological models or single survey lines, failing to achieve universal operator learning across altitudes and strata; and fourth, forward and inverse modeling are independent, making integrated and universal solutions difficult.

[0005] In summary, existing airborne electromagnetic data processing methods generally suffer from low computational efficiency, insufficient physical consistency, poor numerical stability, and weak generalization ability, failing to simultaneously meet the engineering requirements of rapid exploration, high-precision imaging, and physical compliance. Therefore, developing an airborne electromagnetic deep learning forward and inverse modeling technique that combines high-speed inference, strong generalization, and high physical consistency has become a key technical problem urgently needing to be solved in this field. Summary of the Invention

[0006] To address the problems identified in existing technologies, this invention provides a deep learning-based forward and inverse modeling method and system for airborne electromagnetic data under physical constraints. Based on DeepONet, a physical information neural operator is constructed. An automatic differentiation stability is ensured through a logarithmic-linear dual-space mechanism, and physical constraints from Maxwell's equations are incorporated to achieve efficient integrated solutions for airborne electromagnetic forward and inverse modeling. This invention features fast computation speed, strong generalization ability, and high physical consistency, and can be widely applied to airborne electromagnetic exploration scenarios such as mineral exploration, groundwater investigation, and geological disaster monitoring.

[0007] A deep learning method for forward and inverse retrieval of airborne electromagnetic data under physical constraints includes the following steps: Step S1: Construct a forward modeling network. The forward modeling network adopts the DeepONet architecture, which includes a branch network and a trunk network. The branch network takes the logarithmic resistivity profile of the strata and the altitude of the aircraft as inputs, and outputs an environmental feature vector. The backbone network takes logarithmic depth and logarithmic time as input and outputs a spatiotemporal feature vector. The environmental feature vector and the spatiotemporal feature vector are fused to obtain the logarithmic quadratic field prediction value; Step S2: Construct an inversion network, which takes the observed transient electromagnetic response time series and flight altitude as input and directly outputs a logarithmic resistivity profile; Step S3: Training the forward modeling network: Using a dataset containing stochastic stratigraphic models and their numerical forward modeling labels, and introducing the residuals of Maxwell's diffusion equation as physical constraints, the forward modeling network is pre-trained until convergence; Step S4: Train the inversion network: Freeze all weights of the pre-trained forward network in step S3, feed the predicted resistivity profile of the inversion network into the frozen forward network to resynthesize the response, and construct the physical data space loss and the model space loss together to supervise the training of the inversion network.

[0008] A physically constrained deep learning forward and inverse modeling system for airborne electromagnetic data, used to execute the forward and inverse modeling method of this invention, comprising: Forward modeling network module: Based on the DeepONet architecture, it is used to map the resistivity model and flight altitude to transient electromagnetic secondary field response; Inversion network module: used to map the observed transient electromagnetic response and flight altitude into a resistivity profile; Physical constraint embedding module: used to introduce Maxwell's equation residuals as a loss term during forward modeling network training; Physically Guided Closed-Loop Module: Used to call and freeze the pre-trained forward modeling network module during inversion network training, and resynthesize the response from the inversion results to construct the physical consistency loss.

[0009] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement a deep learning forward and inverse method for airborne electromagnetic data under physical constraints.

[0010] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a deep learning forward and inverse method for airborne electromagnetic data under weighted physical constraints.

[0011] Compared with the prior art, the present invention has the following beneficial effects: 1. High-speed inference and strong generalization ability: This invention learns the mapping between function spaces through DeepONet. Once training is complete, both forward and inverse inference can achieve millisecond-level predictions, completely breaking free from the constraints of iterative solutions in traditional methods. Combined with training using a stochastic stratum model, the network possesses the ability to learn general operators across heights and strata.

[0012] 2. Excellent Physical Consistency: This invention deeply integrates the logarithmic-linear dual-space mechanism with the residual constraints of Maxwell's equations into forward modeling training; and pioneers an integrated closed loop of "forward modeling prior learning - inversion physics guidance," forcing the inversion results to satisfy the electromagnetic diffusion equation. Experimental data show that the physical loss can be as low as 0.000712, ensuring that the prediction results strictly conform to physical laws.

[0013] 3. High precision and numerical stability: By employing dual-space decoupling and the use of infinitely differentiable activation functions such as tanh, the gradient explosion problem caused by the large dynamic range of airborne electromagnetic signals and singularities in the source field is effectively solved. The mean absolute percentage error (MAPE) of the inversion model on the test set can be controlled within 6%, enabling accurate reconstruction of underground anomalies such as low-resistivity and high-resistivity layers.

[0014] 4. Integrated Framework: This invention enables forward and inverse modeling to share the same set of physical constraints, forming an integrated closed loop of "forward modeling prior learning - inverse modeling physical guidance", which simplifies the processing flow and improves the overall performance and interpretability of the system. Attached Figure Description

[0015] Figure 1 This is a schematic diagram of the core architecture of the forward model of this invention; Figure 2 This is a structural diagram of the branch net of the present invention; Figure 3 This is a structural diagram of the trunk net of the present invention; Figure 4 This is a schematic diagram of the overall inversion function of the present invention; Figure 5 This is a structural diagram of the inverted backbone network of the present invention; Figure 6 This is a schematic diagram of the inversion results of a theoretical data example according to the present invention, which shows the good inversion effect of low-resistivity layer and high-resistivity layer. Detailed Implementation

[0016] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings: 1 Overall Framework The overall framework of the Aeronautical Electromagnetic Data Physical Information Neural Operator (AEM-PINO) proposed in this invention consists of two major modules: a forward modeling network and an inverse modeling network. The two share the same set of physical constraint mechanisms, forming an integrated closed loop of "forward modeling prior learning - inverse modeling physical guidance".

[0017] The forward modeling network employs a deep operator network architecture, comprising a branch network and a trunk network. The branch network takes logarithmic resistivity profile and flight altitude as inputs and outputs an environmental feature vector; the trunk network takes logarithmic depth and logarithmic time as inputs and outputs a spatiotemporal feature vector. The two are fused through latent space dot product and biased to obtain a logarithmic quadratic field prediction. The loss function of the forward modeling network consists of two weighted terms: data-driven loss L... data (Mean squared error between predicted surface response and numerical forward modeling labels) and physical constraint loss L pde (Calculate the residuals of Maxwell's diffusion equation on randomly collocated points, and use a log-linear double spatial gradient penetration mechanism to ensure numerical stability).

[0018] The inversion network is a neural network based on the Transformer architecture. Its inputs are the observed transient electromagnetic response time series and flight altitude, and it directly outputs a logarithmic resistivity profile. Its loss function also includes two terms: model space loss L... m (Mean square error between predicted and true resistivity, used only if tags exist) and physical data space loss L d (The mean square error of feeding the predicted resistivity into a frozen pre-trained forward modeling network to resynthesize the response and comparing it with the original observations).

[0019] In practical applications, a large number of random stratigraphic models and their numerical forward modeling labels, along with physical collocation constraints, are first used to independently train a forward modeling network until convergence. Then, all weights of the forward modeling network are frozen and used as a fixed physical arbiter for training the inversion network—the gradient of the inversion network only updates its own parameters, and the forward modeling network does not participate in gradient backpropagation. During the inference phase, the trained inversion network can directly image resistivity profiles from measured curves within milliseconds, while the forward modeling network can serve as a fast surrogate model, providing electric field predictions for any stratigraphic model at any spatiotemporal point. This framework achieves a deep integration of physical laws and data-driven approaches, significantly improving the resolution and generalization ability of airborne electromagnetic inversion.

[0020] 1.1 Forward Model Network Structure To establish a continuous mapping from the subsurface resistivity model to the transient electromagnetic secondary field, this invention employs DeepONet, a deep operator network, as the core architecture of the forward modeling surrogate. Unlike traditional numerical methods, DeepONet can learn the mapping between infinite-dimensional function spaces. Once trained, it can provide instantaneous predictions at arbitrary spatiotemporal coordinates, completely eliminating the constraints of grid discretization and iterative solutions. Considering the extremely large dynamic range and sensitivity to derivatives of physical equations in airborne electromagnetic data, this invention designs a logarithmic-linear dual-space decoupled forward modeling network, the overall structure of which is as follows: Figure 1 As shown.

[0021] 1.2 Branch Net like Figure 2 The diagram shows the structure of the forward branching network, Branch Net. The function of the branching network is to encode the macroscopic geological environment function into a latent vector. Its input vector is composed of two concatenated parts: Logarithmic resistivity profile of the formation: , where N layer The dimension (i.e., the number of discrete layers, which is 100 in this example). The altitude of the aircraft above the ground H∈R + .

[0022] Therefore, the input dimension of the branch network is N. layer + 1. The network consists of three stacked fully connected layers, with 256 hidden neurons. The activation function is Gaussian error linear unit (GELU), and the output layer dimension is p=128, generating an environmental feature vector: V branh =B(log10ρ,H)∈R p ; Where log10ρ is the resistivity profile and H is the one-dimensional flight altitude.

[0023] The GELU activation function has good nonlinear expression capabilities and is smooth throughout, making it suitable for encoding complex variations in resistivity with depth.

[0024] Detailed descriptions of each layer are as follows: Input: N layer + 1 (number of resistivity logs for each layer + flight altitude); First fully connected layer: 256 neurons, GELU activated; Second fully connected layer: 256 neurons, GELU activated; Output layer: 128 neurons, no activation function; The number of parameters is approximately (N) layer + 1)*256+256*256+256*128.

[0025] 1.3 Trunk Net: Spatiotemporal Coordinate Positioning like Figure 3 The diagram shows the Trunk Net network structure for forward modeling. The trunk network is responsible for mapping the spatiotemporal coordinates to be queried into corresponding basis function vectors. Its inputs are depth coordinates z and time coordinates t, both of which are linear physical quantities. To enhance the network's ability to capture high-frequency components, the inputs are first converted to logarithmic coordinates: z log =log 10 (z+ z ) and t log =log 10 (t+ t ); in z , t To prevent singularities, small positive numbers are used. The backbone network consists of five fully connected layers, with the hidden layer also having 256 neurons, but the hyperbolic tangent (tanh) activation function is chosen. This is because subsequent physical constraints require calculating the second and first partial derivatives of the electric field with respect to depth and time, and tanh is an infinitely differentiable function, ensuring the numerical stability of the automatic differentiation process. The output layer dimension is consistent with the branch network, resulting in the spatiotemporal feature vector: .

[0026] 1.4 Loss Function The core of training a physical information neural network lies in the construction of its loss function. Traditional deep learning networks rely solely on the residual between observed data and predicted values ​​for driving operation. However, the AEM-PINO framework proposed in this invention constructs a multi-objective collaborative optimization loss function system by transforming Maxwell's equations and boundary conditions into regularization constraints. The total loss function L... total Data constraint term L data and the residual term L of the physical partial differential equation pde Composed of linear weighted combinations: L total =λ data L data +λ pde L pde ; where λ data and λ pde These are the hyperparameter weights that balance the accuracy of data fitting and physical consistency.

[0027] (1) Data-Driven Loss The data-driven loss term aims to force the transient electromagnetic response curve output by the operator network at a specific altitude of aircraft flight (i.e., z = -H above the ground) to maintain a high degree of fit with known forward simulations or actual field observations in the dataset. Since both the input features and target labels have been transformed to logarithmic space, the data loss term is defined using the mean squared error (MSE) on a logarithmic scale. ; In the formula, B is the total number of samples in the current training batch (Batch Size); N t This represents the number of sampling points in the time channel; This indicates that the network has a certain relationship with the i-th sample at height H. i Time step t j The predicted log-quadratic field value; The labels represent the corresponding true logarithmic electromagnetic field response. By calculating the MSE in logarithmic space, the network can automatically apply equal fitting penalties to early and late channels at different orders of magnitude, avoiding the defect of traditional linear space loss functions that easily ignore weak late channels.

[0028] (2) Physically-Informed PDE Loss Physical constraint term L pdeThe calculation does not require any real label data. Its core idea is to randomly scatter tens of thousands of virtual spatiotemporal coordinate points (physical collocation points) in a continuous space including the air layer and the underground medium, and force the network predictions to strictly satisfy Maxwell's diffusion equation at these collocation points.

[0029] For each environment sample in the current training batch, the system operates in the physical linear spatiotemporal domain [0, Z]. max ]×[0,T max N are randomly selected uniformly from within. c virtual coordinate point (z) lin,m ,t lin,m The physical residual loss term is defined as the squared mean of the equation residuals at all collocation points:

[0030] Where Nc is the number of physical collocations, μ0 is the permeability, and σ is the conductivity.

[0031] 2. Inversion Model The inversion model adopts an overall architecture that combines data-driven and physical constraint-based optimization, such as... Figure 4 The model takes the airborne electromagnetic observation response curve and flight altitude as input. After being fused through an input stitching layer, it is fed into the inversion backbone network, directly outputting the predicted subsurface resistivity profile. During training, the model simultaneously introduces dual supervision signals: on the one hand, it constructs the model spatial loss by comparing with the actual resistivity profile; on the other hand, it inputs the predicted profile into a frozen PINO forward modeling network to generate a predicted response and compare it with the observation data, constructing the physical data loss. By weightedly fusing the two losses, the model strictly satisfies the physical laws of the electromagnetic diffusion equation while fitting the data, effectively improving the accuracy, stability, and physical consistency of the inversion results.

[0032] 2.1 Inversion Model Network Structure To rapidly invert the subsurface resistivity structure from observed transient electromagnetic responses, this invention designs an inversion network F based on a transform structure. inv Its structure is as follows Figure 5 As shown.

[0033] (1) Input layer The input to the inversion network consists of two parts: first, a batch of electromagnetic time-domain response sequences, with a dimension of (batch, response_dim), which is the core observation data of airborne electromagnetic exploration; second, the flight altitude corresponding to each sample, with a dimension of (batch, 1), representing the single-dimensional numerical feature of flight altitude. Here, batch represents the number of samples during training or inference.

[0034] (2) Pretreatment layer Since the Transformer requires the input to be in sequence form (batch, seq_len, feature_dim), the time-domain response needs to be reshaped: the one-dimensional electromagnetic response sequence is split into seq_len time steps, and the feature dimension of each time step is input_size = ceil(response_dim / seq_len); if response_dim is not divisible by seq_len, the last dimension is truncated to ensure shape matching.

[0035] For flight altitude features, the original 1-dimensional altitude feature is first expanded to altitude_expand_dim dimension (default 5 dimensions) through a linear transformation layer. Then, the altitude feature is copied to each time step through a dimension expansion operation to obtain an expanded feature with dimensions (batch, seq_len, altitude_expand_dim), thus ensuring that it is aligned with the dimension of the time domain response sequence.

[0036] (3) Feature fusion and projection First, the reshaped temporal response sequence (dimension (batch, seq_len, input_size)) and the expanded altitude feature sequence (dimension (batch, seq_len, altitude_expand_dim)) are concatenated in the last dimension to obtain a fused feature with dimension (batch, seq_len, input_size + altitude_expand_dim). Then, the concatenated fused feature is projected onto the d_model dimension (default 128) required by the Transformer through the linear layer input_proj, outputting a feature sequence with dimension (batch, seq_len, d_model).

[0037] (4) Location encoding and Dropout To compensate for the lack of built-in positional information in the Transformer architecture, a learnable positional encoding parameter with shape (1, seq_len, d_model) is introduced and added to the input feature sequence. Then, a Dropout operation (with a default probability of 0.1) is performed on the feature sequence to reduce the risk of model overfitting and enhance generalization ability.

[0038] (5) Transformer encoder layer This network stacks four Transformer encoder layers. Each encoder layer contains a multi-head self-attention mechanism with eight attention heads and a feedforward network with a hidden layer dimension of 512. The multi-head self-attention mechanism captures long-distance dependencies between features at different locations within the sequence, while the feedforward network performs a non-linear transformation on the attention output. Each encoder layer also includes layer normalization and Dropout operations to ensure training stability. The input and output dimensions of the entire encoder module are consistent, both being three-dimensional tensors of (batch, seq_len, d_model).

[0039] (6) Global average pooling layer Global average pooling (averaging along the seq_len dimension) is performed on the sequence features output by the Transformer encoder to compress the sequence features into global features, changing the dimension from (batch, seq_len, d_model) to (batch, d_model), reducing the dimension and aggregating the sequence information.

[0040] (7) Output layer The global features are mapped to the resistivity profile dimension resistivity_dim through the linear layer fc_out, and the final output is (batch, resistivity_dim), which is the resistivity profile inversion result for each sample.

[0041] 2.2 Physically Guided Closed Loop The inversion network is not trained independently, but works in conjunction with a pre-trained forward network. During training, the weights of the forward network are completely frozen and used only as fixed physical operators. Specifically: The resistivity profile ρ predicted by the inversion network pred The forward modeling network, frozen with flight altitude H as input, is resynthesized to predict the response E. pred .

[0042] Calculate the physical data space loss L d The mean square error between the resynthesized response after feeding the predicted resistivity into the frozen forward modeling network and the original observed response is L. d = .

[0043] If a true resistivity label exists (i.e., supervised learning), and the model spatial loss L is calculated... m To predict the mean square error between the resistivity profile and the true resistivity label, i.e. .

[0044] The total loss is L=λ m ×L m + λd ×L d , where λ m , λ d This is the weighting coefficient. In this example, we take λ. m =0.7, λ d =0.3.

[0045] This physical guidance mechanism forces the inversion results to not only approximate the statistical prior, but also to be consistent with the forward physical process, thereby significantly suppressing the non-uniqueness of ill-conditioned inverse problems and improving the resolution of deep resistivity.

[0046] After training, the inversion model of this invention exhibits excellent performance in terms of accuracy, generalization, and physical consistency. The mean absolute percentage error (MAPE) of the model on the training and validation sets is 5.37% and 5.85%, respectively, with the deviation controlled within 6%, indicating that the predicted results match the actual resistivity profile well. The small difference between training and validation errors indicates that the model has no significant overfitting. Furthermore, the physical loss is only 0.000712, proving that the predicted results strictly satisfy the physical constraints of the electromagnetic diffusion equation. A comparison of the predicted curves and the actual curves on the validation set shows a high degree of overlap in shape and trend, directly confirming the model's dual accuracy at both the numerical and physical levels. This provides high-precision and high-reliability resistivity imaging results for airborne electromagnetic exploration. Figure 6 An example of the inversion result of theoretical data is given. In this underground resistivity model, there is a low-resistivity layer and a high-resistivity layer. This invention can accurately invert these two resistivity anomaly layers.

Claims

1. A deep learning forward and inverse method for airborne electromagnetic data under physical constraints, characterized in that, Includes the following steps: Step S1: Construct a forward modeling network, which adopts the DeepONet architecture of deep operator networks, including branch networks and backbone networks; The branch network takes the logarithmic resistivity profile of the strata and the altitude of the aircraft as inputs, and outputs an environmental feature vector. The backbone network takes logarithmic depth and logarithmic time as input and outputs a spatiotemporal feature vector. The environmental feature vector and the spatiotemporal feature vector are fused to obtain the logarithmic quadratic field prediction value; Step S2: Construct an inversion network, which takes the observed transient electromagnetic response time series and flight altitude as input and directly outputs a logarithmic resistivity profile; Step S3: Training the forward modeling network: Using a dataset containing stochastic stratigraphic models and their numerical forward modeling labels, and introducing the residuals of Maxwell's diffusion equation as physical constraints, the forward modeling network is pre-trained until convergence; Step S4: Train the inversion network: Freeze all weights of the forward modeling network trained in step S3, feed the predicted resistivity profile of the inversion network into the frozen forward modeling network to resynthesize the response, and construct the physical data space loss and the model space loss together to supervise the training of the inversion network.

2. The method according to claim 1, characterized in that, In step S1, the input of the branch network consists of N dimensions. layer The logarithmic resistivity profile log10ρ and the one-dimensional flight altitude H are spliced ​​together; the input of the backbone network is the depth coordinate z and the time coordinate t after logarithmic transformation. log =log10(z+ z) and t log =log10(t+ t), where z, t is a positive number to prevent the singularity from being too small.

3. The method according to claim 1, characterized in that, In step S3, the total loss function L for training the forward network is... total For data-driven loss L data With physical partial differential equation loss L pde Weighted sum: L total =λ data L data +λ pde L pde ; Where, λ data and λ pde These are the hyperparameter weights that balance the accuracy of data fitting and physical consistency.

4. The method according to claim 3, characterized in that, The data-driven loss L data Calculation of mean square error based on logarithmic space: ； Where B is the total number of samples in the current training batch; N t This represents the number of sampling points in the time channel; This indicates that the network has a certain relationship with the i-th sample at height H. i Time step t j The predicted log-quadratic field value; The corresponding true logarithmic electromagnetic field response label.

5. The method according to claim 3, characterized in that, The physical partial differential equation loss L pde Represented as: Where Nc is the number of physical collocations, μ0 is the permeability, and σ is the conductivity.

6. The method according to claim 1, characterized in that, In step S4, the model space loss L m To predict the mean square error between the resistivity profile and the true resistivity label; the physical data spatial loss L d The mean square error between the resynthesized response after feeding the predicted resistivity into the frozen forward modeling network and the original observed response; the total loss of the inversion network is L=λ m ×L m + λ d ×L d , where λ m , λ d These are the weighting coefficients.

7. The method according to claim 1, characterized in that, The inversion network is built on the Transformer architecture and includes: an input layer, a preprocessing layer, a feature fusion and projection layer, a position encoding and dropout layer, a Transformer encoder layer, a global average pooling layer, and an output layer; wherein, the output layer is used to map global features to the resistivity profile dimension.

8. A deep learning forward and inverse modeling system for airborne electromagnetic data under physical constraints, used to perform the method according to any one of claims 1 to 7, characterized in that, include: Forward modeling network module: Based on the DeepONet architecture, it is used to map the resistivity model and flight altitude to transient electromagnetic secondary field response; Inversion network module: used to map the observed transient electromagnetic response and flight altitude into a resistivity profile; Physical constraint embedding module: used to introduce Maxwell's equation residuals as a loss term during forward modeling network training; Physically Guided Closed-Loop Module: Used to call and freeze the pre-trained forward modeling network module during inversion network training, and resynthesize the response from the inversion results to construct the physical consistency loss.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the deep learning forward and inverse method for airborne electromagnetic data under physical constraints as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the program implements the deep learning forward and inverse method for airborne electromagnetic data under physical constraints as described in any one of claims 1 to 7.

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