A Deep Learning Inversion Method for Aquifer Permeability Coefficient Field Based on Uncertainty Quantification

Abstract

This invention relates to the field of groundwater hydrological parameter inversion technology, and discloses a deep learning inversion method for aquifer permeability coefficient fields based on uncertainty quantification, comprising the following steps: Step 1, Data Acquisition and Preprocessing: Acquire spatial head field data from inter-well pumping tests, integrate the head fields formed by multiple pumping tests into the model input tensor, and perform normalization preprocessing on the head field data; acquire the corresponding real permeability coefficient field data, perform logarithmic transformation and normalization on the permeability coefficient values, and use them as model training labels; Step 2, Model Construction: Construct a deep residual U-Net neural network model H-UQNet with a built-in Dropout layer. This invention achieves high-precision and high-efficiency permeability coefficient field inversion. Through the built-in uncertainty quantification mechanism and geostatistical constraints, it significantly improves the reliability and geological rationality of the results, and has important theoretical research value and broad engineering application prospects.

Description

Technical Field

[0001] This invention relates to the field of groundwater hydrological parameter inversion technology, and in particular to a deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification. Background Technology

[0002] Effective management, sustainable utilization, and pollution prevention of groundwater resources heavily rely on the accurate characterization of hydrogeological parameters of aquifers. Among these, the permeability coefficient (or hydraulic conductivity, K) is a key parameter describing groundwater flow capacity. Obtaining high-resolution, heterogeneous permeability coefficient fields is fundamental for groundwater numerical simulation and risk assessment. Currently, one of the main technical means to obtain permeability coefficient field information is in-situ well-to-well disturbance pumping tests. These tests infer aquifer characteristics by pumping water at a specific rate in one or more wells and monitoring the spatiotemporal changes in hydraulic head (H) in a series of observation wells within the aquifer. These hydraulic head response data indirectly reflect the distribution of permeability properties of the subsurface medium.

[0003] To retrieve continuous permeability fields from these sparse and indirect head observation data, researchers have developed various methods. Traditional methods mainly rely on inversion techniques that combine numerical simulation with optimization algorithms (such as gradient methods and genetic algorithms). In recent years, with the development of artificial intelligence technology, machine learning, especially deep learning methods represented by convolutional neural networks, has shown great potential in processing spatial data such as head fields, providing new solutions for aquifer parameter inversion.

[0004] At present, aquifer parameter inversion still has the following defects and shortcomings:

[0005] 1. Lack of Systematic Uncertainty Quantification: Traditional inversion methods and most deep learning-based models ultimately aim to provide a single, deterministic permeability coefficient field as the "best" prediction. However, due to the sparsity of observational data, measurement errors, and the imperfections of the models themselves, inversion results inevitably contain uncertainty. Existing technologies generally lack a built-in, efficient uncertainty quantification (UQ) mechanism, failing to provide a confidence assessment of the prediction results. This makes it impossible to guarantee the reliability of the prediction results when used for high-risk decisions (such as nuclear waste disposal and pollution remediation scheme design), limiting their engineering application value.

[0006] 2. Insufficient geological plausibility of prediction results: Many deep learning models trained based on pixel-level errors (such as mean square error) can achieve low numerical errors, but the permeability fields they generate may not be visually or geostatistically plausible. For example, unnatural checkerboard artifacts may appear, or there may be a lack of spatial continuity and correlation structure that conforms to specific geological laws, resulting in a significant discrepancy with the morphology of real geological bodies.

[0007] 3. Limited robustness to sparse data: In practical engineering, the number of observation wells is often limited, resulting in a highly sparse input head data field. Traditional deterministic deep learning models face challenges in predicting robustness and generalization ability when dealing with such sparse inputs, easily leading to overfitting or unreasonable interpolation results.

[0008] 4. The contradiction between computational cost and inference efficiency: Traditional simulation-optimization-based inversion methods require tens of thousands of forward model calculations, resulting in extremely high computational costs and taking days or even weeks. Although deep learning methods are fast in the inference stage, some methods that can quantify uncertainty (such as deep ensembles) require training multiple models independently, which increases the training cost exponentially and limits their ease of application. Summary of the Invention

[0009] To address the technical problems mentioned in the background section, the present invention aims to provide a deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification. This method can provide high-precision permeability coefficient field prediction using only spatial hydraulic head data, while systematically quantifying the uncertainty of the prediction results.

[0010] This invention is achieved using the following technical solution: a deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification, comprising the following steps:

[0011] Step 1: Obtain the spatial head field data of the pumping test between wells, integrate the head fields formed by multiple pumping tests into the model input tensor, and perform normalization preprocessing on the head field data; obtain the corresponding real data of the permeability coefficient field, and perform logarithmic transformation and normalization on the permeability coefficient values ​​as model training labels.

[0012] Step 2: Construct a deep residual U-Net neural network model H-UQNet with a built-in Dropout layer as the prediction model for the inversion of the permeability coefficient field;

[0013] Step 3: Train the H-UQNet model using a hybrid loss function, which includes multiple loss terms based on observation error and geostatistical constraints;

[0014] Step 4: Keep the Dropout layer enabled during the inference phase of the trained H-UQNet model, perform multiple forward propagations on the same set of head field input data to obtain multiple permeability coefficient field prediction samples; calculate the mean prediction result and spatial variance uncertainty field of the permeability coefficient field based on the multiple permeability coefficient field prediction samples.

[0015] Step 5: Output the mean prediction result of the permeability coefficient field and the corresponding spatial variance uncertainty field, and visualize it to characterize the spatial distribution characteristics and prediction uncertainty of the aquifer permeability coefficient.

[0016] Furthermore, in step one, Z-score normalization is performed on each head field data channel, and the true value of the permeability coefficient field is first taken as the natural logarithm and then normalized to the range of 0 to 1 using Min-Max.

[0017] Furthermore, in step two, the neural network model H-UQNet adopts an encoder-decoder architecture, which consists of multiple stacked residual convolutional blocks. Each residual convolutional block includes two convolutional layers, two batch normalization layers, at least one nonlinear activation function, a Dropout layer, and a shortcut residual connection that adds the input signal to the output of the convolutional layer.

[0018] The fast residual connection adds a convolutional layer to adjust the number of input signal channels to match the dimension of the output channels when the number of input and output channels of the residual convolutional block is inconsistent.

[0019] Furthermore, the encoder path of the neural network model H-UQNet includes an initial convolutional layer for expanding the number of channels of the input head data to a predetermined initial number of channels, and sequentially includes at least two levels of residual convolutional blocks and corresponding downsampling operations to extract features at different scales.

[0020] Furthermore, the decoder path of the neural network model H-UQNet includes upsampling processes corresponding to each level of the encoder. Each upsampling level enlarges the spatial size of the feature map through transposed convolution, and then concatenates it with the feature map of the corresponding encoder stage through skip connections before inputting it into the residual convolution block of the decoder to fuse and reconstruct features.

[0021] Furthermore, the output layer of the neural network model H-UQNet includes a final convolutional layer, which is used to reduce the number of feature channels of the decoder output to 1 and generate a preliminary permeation coefficient field prediction result. Then, the prediction result is reshaped to the target spatial resolution through an upsampling operation, and the output value is mapped to the 0-1 range through the Sigmoid activation function.

[0022] The final convolutional layer is a 1×1 convolutional layer; the upsampling operation uses bilinear interpolation.

[0023] Furthermore, the encoder path includes two levels of residual convolutional blocks and two downsampling operations, and a bridging residual convolutional block is set at the end of the encoder as a bottleneck layer; the decoder path includes two upsampling operations corresponding to the two levels of encoder, each upsampling operation doubles the spatial resolution of the feature map through transposed convolution, and then concatenates it with the output feature map of the corresponding encoder layer via skip connections before inputting it into the decoder residual convolutional block.

[0024] Furthermore, in step three, the hybrid loss function used is composed of a weighted average of the mean absolute error loss, structural similarity loss, and variogram loss.

[0025] Furthermore, the variogram loss calculates and penalizes the difference between the predicted permeability field and the true permeability field on the empirical semivariogram, thereby forcing the model to generate a permeability field that conforms to a predetermined geostatistical spatial structure.

[0026] Furthermore, in step five, the mean permeability coefficient field and the variance uncertainty field are displayed in the form of a color-coded map to intuitively represent the spatial distribution and uncertainty of the permeability coefficient.

[0027] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0028] This invention systematically integrates an uncertainty quantification mechanism in aquifer parameter inversion. By maintaining the Dropout layer active during the inference phase and employing Monte Carlo Dropout technology to perform multiple forward propagations on the same input, multiple permeability coefficient field prediction samples are generated, and then the mean field and spatial variance field are calculated. This method not only provides the "best estimate" result but also gives the prediction confidence at each spatial location, significantly improving the reliability and practicality of the inversion results, and is particularly suitable for high-risk decision-making scenarios such as pollution remediation or nuclear waste disposal.

[0029] This invention comprehensively optimizes prediction accuracy and geological plausibility by introducing a hybrid loss function. This loss function integrates multiple loss terms, including mean absolute error, structural similarity, and variogram, effectively avoiding the checkerboard artifacts or structural inconsistencies caused by traditional methods that rely solely on pixel-level errors. This ensures that the generated permeability field conforms to actual geological laws both visually and statistically.

[0030] The H-UQNet model proposed in this invention possesses powerful feature extraction and reconstruction capabilities. The model adopts an encoder-decoder architecture, incorporating residual connections and skip connections, which can effectively capture multi-scale spatial features while preserving detailed information, significantly improving robustness and generalization ability to sparse and noisy data.

[0031] This invention achieves a good balance between computational efficiency and inference performance. Compared with traditional simulation-optimization-based inversion methods that require tens of thousands of forward modeling calculations, this method can quickly generate prediction and uncertainty maps once training is complete, greatly improving inversion efficiency while avoiding the problem of multiplied training costs caused by multi-model integration.

[0032] In summary, this invention not only achieves high-precision and high-efficiency permeability coefficient field inversion, but also significantly improves the reliability and geological rationality of the results through the built-in uncertainty quantification mechanism and geostatistical constraints, thus possessing important theoretical research value and broad engineering application prospects. Attached Figure Description

[0033] Figure 1 This is a flowchart of the inversion method proposed in this invention;

[0034] Figure 2 The image shows the prediction results obtained by testing the inversion method proposed in this invention.

[0035] Figure 3 The above are prediction diagrams of multiple sets of uncertain K-field results obtained by testing using the inversion method proposed in this invention. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0037] Example 1:

[0038] Please combine Figure 1 - Figure 3 The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification proposed in this embodiment mainly includes the following steps: data acquisition and preprocessing, H-UQNet model construction, model training, and probabilistic prediction and uncertainty quantification based on Monte Carlo Dropout, as detailed below:

[0039] Step 1: Data acquisition and preprocessing, the detailed steps of which are as follows.

[0040] This step aims to prepare standardized input and output data for the neural network model.

[0041] Data source: Head data (H) obtained through numerical simulation or field tests. A second independent inter-well interference pumping test was conducted. Each observation well records steady-state or transient head values, forming... A spatial water head field.

[0042] Data formatting: The spatial water head fields are integrated into a four-dimensional tensor, with dimensions (batch size, ...). (Grid height, grid width). As the number of input channels.

[0043] Data preprocessing.

[0044] (1) Head data (H): Z-score normalization (subtracting the mean and dividing the standard deviation) is performed independently on the channel corresponding to each pumping test to make its distribution adapt to the network input.

[0045] (2) Permeability coefficient field (K, as label): Take the logarithm (log(k)) of the true permeability coefficient value, and then perform Min-Max normalization to... The interval is used to match the activation function of the network's output layer.

[0046] Step 2: Construct the H-UQNet model. The detailed steps are as follows.

[0047] The core of this step is to construct a deep residual U-Net network with a built-in Dropout mechanism to achieve uncertainty quantification. In this solution, the model is named H-UQNet, and its specific structure is a residual U-Net customized for this task. Its technical features are detailed below:

[0048] The model is an encoder-decoder architecture, in which all core building blocks are residual convolutional blocks.

[0049] In detail, as the basic unit of the model, each residual convolutional block contains: two The system consists of a convolutional layer; two batch normalization layers; a ReLU activation function; a Dropout layer to introduce randomness for uncertainty quantization; and a shortcut connection. The shortcut connection directly appends the input to the two convolutional layers, forming a residual structure. If the number of channels in the input and output are different, a shortcut connection is used to pass the input to the output. The number of channels in the convolutional layer is adjusted to match the dimension.

[0050] Encoder path structure.

[0051] (1) Initial convolutional layer: The input head data tensor first passes through a... Convolutional layers, reducing their channel count from The number of pumping tests was expanded to 64 to obtain the initial feature map.

[0052] (2) First encoding stage: The initial feature map is input into a residual convolutional block (64 input channels, 64 output channels) for feature extraction, and then processed through a... The max pooling layer (MaxPool2d) is used for downsampling, which halves the spatial dimension of the feature map.

[0053] (3) Second encoding stage: The downsampled feature map is input into the second residual convolutional block (64 input channels, 128 output channels), and then passed through another... The max-pooling layer is then downsampled a second time.

[0054] Bridge or bottleneck layer: The feature map after two downsampling (smallest size, most channels) is fed into a bridged residual convolutional block (128 input channels, 256 output channels) to process and integrate the highest-order abstract features.

[0055] Decoder path structure.

[0056] (1) First decoding stage: The output of the bridging layer first passes through a... The transposed convolutional layer (ConvTranspose2d) is upsampled, doubling its spatial dimension and reducing the number of channels from 256 to 128. Then, through skip connections, the upsampled feature map is concatenated with the output feature map of the second residual convolutional block in the encoder path (i.e., the feature map before the second downsampling) along the channel dimension. The concatenated feature map (128+128=256 channels) is fed into a residual convolutional block (256 input channels, 128 output channels) for feature fusion and refinement.

[0057] (2) Second decoding stage: The output of the first decoding stage is again passed through a... The transposed convolutional layer performs upsampling, doubling the spatial dimension and reducing the number of channels from 128 to 64. This upsampled result is then concatenated with the output feature map of the first residual convolutional block in the encoder path via a second skip connection, channel-wise. The concatenated feature map (64+64=128 channels) is then fed into the second decoder residual convolutional block (128 input channels, 64 output channels).

[0058] Output and size adjustment.

[0059] The model output stage of this scheme employs a strategy of convolution followed by interpolation to generate the final high-resolution permeability coefficient field. The technical features of this step are as follows:

[0060] (1) Final convolutional layer: The output feature map of the second decoding stage is passed through a... The final convolutional layer reduces its channel count to 1, yielding a preliminary prediction result that is close to the spatial resolution of the input head field.

[0061] (2) Size reshaping: Due to dimensional changes during encoding and decoding, the spatial size of the initial prediction result may not be the target size (e.g., Therefore, this invention employs a bilinear interpolation upsampling layer to accurately reshape the preliminary prediction results to the final target size.

[0062] Final activation: The resized prediction field is finally passed through a sigmoid activation function, which constrains all pixel values ​​to . The range is a perfect match for the preprocessed label data format.

[0063] Step 3: Model training, the detailed steps of which are as follows.

[0064] A hybrid loss function is used to guide model training in order to simultaneously optimize prediction accuracy and geological plausibility.

[0065] The hybrid loss function It is a weighted sum of multiple loss terms:

[0066]

[0067] In the above formula , , These are the weighting coefficients for each item, and The value ranges from [0.3, 0.9]. The value ranges from [0.1, 0.5]. The value is [0.05, 0.3]. In this scheme, Take 0.6, Take 0.25, Take 0.15.

[0068] (Mean Absolute Error Loss): Ensures the basic accuracy of the prediction results in pixel values. It is calculated as the average of the absolute values ​​of the differences between the predicted and actual values.

[0069] (Structural similarity loss): This measure assesses the similarity between the predicted and real fields in terms of brightness, contrast, and structure, ensuring visual fidelity of local structures and effectively avoiding the limitations of relying solely on structural similarity loss. Potential ambiguity issues.

[0070] (Variation function loss): Another key technical feature of this invention. By calculating and penalizing the difference between the predicted field and the actual field in the empirical variogram, the model is forced to generate a permeability coefficient field that conforms to specific geostatistical laws (such as spatial continuity and correlation). This loss ensures the geological rationality of the prediction results, making their shape closer to that of real geological bodies.

[0071] The calculation process is as follows:

[0072] In summary, pixel coordinates are randomly sampled on the image grid; the Euclidean distance between the sampled point pairs is calculated and assigned to corresponding distance buckets; the mean squared difference of paired pixel values ​​is calculated within each bucket to obtain the semivariogram curves of the predicted and true fields; the difference between the predicted and true curves is compared in each bucket, and the final loss value is obtained by weighted MAE / MSE aggregation.

[0073] In detail,

[0074] 1. Definition of semivariogram: For a two-dimensional spatial field , The variance of the difference between the values ​​of a random field Z at two points x and x+h that are ℎ apart; semivariogram. It is defined as half of the variance.

[0075]

[0076] 2. Empirical semivariogram calculation: In model training, for a batch of prediction fields... and real scene Calculate their empirical semivariograms respectively. and The calculation process is as follows:

[0077] Randomly sample a large number of pixel coordinates on the image grid.

[0078] Calculate the Euclidean distance between all pairs of points. They are then categorized according to preset distance intervals (buckets).

[0079] For each distance bin Calculate the average of the squared differences of pixel values ​​for all pairs of points within the bucket, and use this as the semivariogram value at that distance.

[0080]

[0081] in, It is the distance All point pairs within the bucket The set, It is the number of point pairs in the set.

[0082] 3. Loss Calculation: The loss value is calculated by comparing the empirical semivariogram curves of the predicted field and the actual field. This loss function can be expressed as:

[0083]

[0084] in:

[0085] This refers to the batch size.

[0086] This represents the number of buckets at a distance from the distribution point.

[0087] It is a function that measures the difference, and can be taken as mean absolute error (MAE) or mean squared error (MSE).

[0088] This is a weighting coefficient used to balance the impact of uneven distribution of quantities on the midpoints of buckets at different distances; it can typically be set to... .

[0089] Step 4: Model inference and posterior sample generation, the detailed steps of which are as follows.

[0090] Prediction is performed using a pre-trained H-UQNet model, and uncertainty is quantified using Monte Carlo Dropout (MC Dropout) technology.

[0091] The probabilistic prediction method includes the following steps:

[0092] 1. Activate Dropout layers: Set the trained model to evaluation mode, but force all Dropout layers in the model to be enabled (by setting them to training mode). This operation makes the forward propagation of the network random even during the prediction phase.

[0093] 2. Perform T random forward propagations: For the same set of input head data H, repeat the process T times (e.g., Forward propagation. Due to the random dropout behavior of neurons in the Dropout layer, each propagation passes through a slightly different subnetwork, thus generating a slightly different permeability field prediction. .

[0094] 3. Collect posterior samples: Calculate the T predicted results... As a set, this set is considered as an approximate sample of the posterior probability distribution of the true permeability coefficient field.

[0095] 4. Calculate statistics and quantify uncertainty: Based on this posterior sample set, perform statistical calculations for each spatial pixel:

[0096] (1) Best estimation field (mean field): This mean field serves as the final, most probable prediction of the permeability coefficient field.

[0097] (2) Uncertainty field (variance field): .

[0098] The aforementioned uncertainty field (variance field) is one of the core outputs of this scheme, which intuitively demonstrates the confidence level of the model's predictions at different spatial locations. Furthermore, the larger the variance, the higher the uncertainty of the model's prediction at that location (e.g., areas far from the observation well), and vice versa.

[0099] Step 5: Predict application and performance verification, the detailed steps of which are as follows.

[0100] This step applies the posterior sample set generated in step 4 to aggregate the final prediction results, quantify uncertainty, visualize the results, and comprehensively verify the model's overall performance. The prediction application and performance verification method includes the following specific steps.

[0101] 1. Prediction result aggregation:

[0102] Calculation of the best estimation field (mean field): For the posterior sample set For each sample, perform an arithmetic mean calculation pixel-by-pixel to obtain a mean field. This mean field incorporates information from all possible predictions and is used as the optimal single-point estimate of the true permeability coefficient field.

[0103] Uncertainty field (variance field): For T samples in the posterior sample set, the variance is calculated pixel-by-pixel to obtain a variance field. .

[0104] 2. Results are visualized.

[0105] The calculated mean and variance fields are visualized using color-coded maps to show the spatial distribution trend of the permeability coefficient and the spatial distribution of uncertainty in the prediction results, respectively.

[0106] A number of samples are randomly selected from the posterior sample set for visualization to visually demonstrate the diversity and range of possibilities in the model's predictions.

[0107] 3. Result Verification:

[0108] Accuracy verification: through comparison Figure 2 The real K field in part a) and Figure 2 As can be clearly seen from the predicted mean K-field in section b), the optimal estimated field (mean field) output by this method closely matches the real field in terms of spatial structure. Whether it is the location, shape, size, or connectivity of the high-permeability region (red area in the figure) or the low-permeability region (blue area in the figure), the prediction results accurately reproduce the real situation. Figure 2 The absolute error plot in section d) visually quantifies the accuracy of the prediction. Figure 2 The majority of the region appears dark, indicating extremely low prediction error. The error only slightly increases in the boundary regions where the permeability gradient changes most dramatically. Figure 2 The lighter-colored portion of part d) is in accordance with physical laws and within acceptable limits. Therefore, Figure 2 This strongly demonstrates that the method of the present invention has extremely high prediction accuracy.

[0109] Validation of the effectiveness of uncertainty quantification: Figure 2 The prediction variance (uncertainty) in section c) demonstrates the core advantage of the probabilistic prediction of this invention. The uncertainty of the prediction is not uniform random noise, but rather exhibits a spatialized pattern related to the intrinsic structure of the problem. Figure 2 The grid-like bright spots (high variance regions) in section c) clearly identify areas with low confidence in the model predictions. This may correspond to the interpolation regions between the observed data points, which is in line with the objective laws of geological inversion.

[0110] Figure 2 The non-uniformity of part c) demonstrates that the inversion method proposed in this scheme can effectively perform self-assessment, that is, it can give confident predictions with low variance in areas with strong data constraints and clear patterns, while giving cautious predictions with high variance in areas with insufficient information and high inference difficulty. This capability provides key quantitative evidence for risk assessment in practical engineering.

[0111] Geological rationality verification: Figure 3 The results presented visually validate the geological rationality of the model's generated results. Each random sample maintains spatial statistical characteristics consistent with the real field, such as smoothness, characteristic scale, and natural morphology, without any computational artifacts or unnatural structures. These 12 samples collectively constitute a stable description of the real field (e.g., the high-permeability core in the lower left corner appears consistently in all samples), but exhibit reasonable and subtle differences in local details (such as the precise boundaries and internal morphology of low-permeability channels).

[0112] Through the above technical solutions, the H-UQNet model proposed in this embodiment not only simplifies the optimization process in traditional methods but also significantly improves computational efficiency and prediction accuracy. More importantly, this invention innovatively provides a quantification of the systematic uncertainty of the prediction results, elevating the prediction from a single "point estimate" to a complete "probability distribution estimate." This provides crucial quantitative evidence for groundwater risk assessment, decision support, and model reliability analysis, possessing significant theoretical value and broad engineering application prospects.

[0113] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification, characterized in that, Includes the following steps: Step 1: Obtain the spatial head field data of the pumping test between wells, integrate the head fields formed by multiple pumping tests into the model input tensor, and perform normalization preprocessing on the head field data; obtain the corresponding real data of the permeability coefficient field, and perform logarithmic transformation and normalization on the permeability coefficient values ​​as model training labels. Step 2: Construct a deep residual U-Net neural network model H-UQNet with a built-in Dropout layer as the prediction model for the inversion of the permeability coefficient field; Step 3: Train the H-UQNet model using a hybrid loss function, which includes multiple loss terms based on observation error and geostatistical constraints; Step 4: Keep the Dropout layer enabled during the inference phase of the trained H-UQNet model and perform multiple forward propagations on the same set of head field input data to obtain multiple permeability coefficient field prediction samples. The mean prediction result and spatial variance uncertainty field of the permeability coefficient field are calculated based on the multiple permeability coefficient field prediction samples. Step 5: Output the mean prediction result of the permeability coefficient field and the corresponding spatial variance uncertainty field, and visualize it to characterize the spatial distribution characteristics and prediction uncertainty of the aquifer permeability coefficient. In step two, the neural network model H-UQNet adopts an encoder-decoder architecture, which consists of multiple stacked residual convolutional blocks. Each residual convolutional block includes two convolutional layers, two batch normalization layers, at least one nonlinear activation function, a Dropout layer, and a shortcut residual connection that adds the input signal to the output of the convolutional layer. The fast residual connection adds a convolutional layer to adjust the number of input signal channels to match the dimension of the output channels when the number of input and output channels of the residual convolutional block is inconsistent. The encoder path of the neural network model H-UQNet includes an initial convolutional layer to expand the number of channels of the input head data to a predetermined initial number of channels, and sequentially includes at least two levels of residual convolutional blocks and corresponding downsampling operations to extract features at different scales. The decoder path of the neural network model H-UQNet includes upsampling processes corresponding to each stage of the encoder. Each upsampling stage enlarges the spatial size of the feature map through transposed convolution, and then concatenates it with the feature map of the corresponding encoder stage through skip connections before inputting it into the residual convolution block of the decoder to fuse and reconstruct features.

2. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 1, characterized in that, In step one, Z-score normalization is performed on each head field data channel, and the true value of the permeability coefficient field is first taken as the natural logarithm and then normalized to the range of 0 to 1 using Min-Max.

3. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 1, characterized in that, The output layer of the neural network model H-UQNet includes a final convolutional layer, which reduces the number of feature channels of the decoder output to 1 and generates a preliminary permeation coefficient field prediction result. Then, the prediction result is reshaped to the target spatial resolution through an upsampling operation, and the output value is mapped to the 0-1 range through the Sigmoid activation function. The final convolutional layer is a 1×1 convolutional layer; the upsampling operation uses bilinear interpolation.

4. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 3, characterized in that, The encoder path includes two levels of residual convolutional blocks and two downsampling operations, with a bridging residual convolutional block at the end of the encoder as a bottleneck layer. The decoder path includes two upsampling operations corresponding to the encoder. Each upsampling operation doubles the spatial resolution of the feature map through transposed convolution, and then concatenates it with the output feature map of the corresponding encoder layer via skip connections before inputting it into the decoder residual convolutional block.

5. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 1, characterized in that, In step three, the hybrid loss function used consists of three weighted components: mean absolute error loss, structural similarity loss, and variogram loss.

6. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 5, characterized in that, The variogram loss calculates and penalizes the difference between the predicted permeability field and the true permeability field on the empirical semivariogram, thereby forcing the model to generate a permeability field that conforms to a predetermined geostatistical spatial structure.

7. The deep learning inversion method for aquifer permeability coefficient field based on uncertainty quantification according to claim 1, characterized in that, In step five, the mean permeability coefficient field and the variance uncertainty field are displayed in the form of a color-coded map to intuitively represent the spatial distribution and uncertainty of the permeability coefficient.

Images