High-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy

Abstract

The invention is suitable for the technical field of hydrogeology, and provides a high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy, and the method comprises the steps: learning a mapping relation between a model parameter and a model response from a large amount of high-dimensional data through a deep neural network; through coupling Monte Carlo stochastic simulation, monitoring stations are arranged based on a multi-objective optimization algorithm of information entropy, more reliable observation data are provided for a parameter inversion process, and finally, the inversion precision of a permeability field is improved. According to the method, the high-dimensional parameter inversion problem in the heterogeneous aquifer is solved, and good calculation efficiency and recognition precision are shown. Specifically, a layout scheme of monitoring stations is optimized, errors of inversion results are reduced by efficiently utilizing monitoring data, and accuracy and reliability of inversion are improved; inversion of the high-dimensional permeability field in the heterogeneous aquifer is achieved, technical support is provided for the problem of complex solute transport simulation, and wide practical application prospects are shown.

Description

Technical Field

[0001] The invention belongs to the technical field of hydrogeology, and in particular relates to a high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy. Background Art

[0002] Permeability field is an important parameter used in hydrogeology to describe the characteristics of aquifer media. It directly determines the spatial distribution of groundwater flow and solute transport. However, in actual geological environments, permeability fields usually have significant spatial heterogeneity and uncertainty, which poses great challenges to its accurate quantitative description and inversion identification. Traditional permeability field inversion methods, such as gradient method, genetic algorithm and Markov chain Monte Carlo (MCMC) method, all rely on repeated iterations of numerical models and optimization algorithms. Although these methods can solve low-dimensional inversion problems to a certain extent, when faced with high-dimensional complex systems, they often have problems such as high computational cost, slow convergence speed and easy to fall into local optimal solutions, making it difficult to achieve accurate inversion of the permeability field of heterogeneous aquifers. In addition, the traditional permeability field inversion process is often limited by limited monitoring data. Therefore, how to reasonably optimize the monitoring network layout to maximize information acquisition has always been a major problem in permeability field inversion. Most of the existing methods are based on experience and trial and error, and it is difficult to effectively improve the inversion accuracy under complex geological conditions.

[0003] In view of the above problems, the present invention proposes a high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy. Summary of the invention

[0004] The purpose of the present invention is to provide a high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy, aiming to solve the problems raised in the above background technology.

[0005] The purpose of the present invention is achieved through the following technical solutions: A high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy includes the following steps: Step 1: Establish a numerical model of groundwater flow and solute transport in a heterogeneous aquifer basin; Step 2: Based on the prior information of the permeability field and the geological statistics, the high-dimensional permeability field is generated using the Karhunen-Loève expansion method, and at the same time, it is characterized into low-dimensional parameters to be estimated; Step 3: Based on the numerical model in step 1, simulate the dynamic response of the hydraulic head and solute concentration corresponding to the permeability field in step 2 to form a training data set and a test data set of permeability field-dynamic response sample pairs; Step 4: normalize the dynamic responses in the training data set and the test data set in step 3; Step 5: Use the training data set obtained in step 4 to learn the mapping relationship from the high-dimensional permeability field to the hydraulic head or solute concentration response, construct a surrogate model for forward prediction based on the convolutional residual neural network, and compare the error between the surrogate model output and the numerical model output on the test data set obtained in step 4 to evaluate its prediction accuracy; Step 6: Use the surrogate model trained in step 5 to randomly generate a water head response or solute concentration response data set for entropy analysis, and use the multi-objective optimization criterion of maximum information entropy and minimum redundancy to optimize the pre-selected monitoring sites to obtain a single optimized monitoring network; Step 7: Repeat step 6 using Monte Carlo simulation, perform statistical analysis on the results, and determine the optimal monitoring network. Step 8: Using the single optimized monitoring network obtained in step 6 and the ensemble optimized monitoring network obtained in step 7, a data assimilation algorithm is used to fit the model prediction and actual observation data at the monitoring station, and the posterior low-dimensional parameters to be estimated are updated; Step 9: Use the root mean square error to quantify the accuracy of the posterior estimation of the low-dimensional parameter to be estimated; Step 10: Based on the posterior low-dimensional estimated parameters obtained in step 8, the posterior estimate of the high-dimensional permeability field is reconstructed using the inverse process of the Kahunen-Loève expansion.

[0006] Furthermore, in step 4, the normalization process is performed using a 0-1 normalization method; the 0-1 normalization method is as follows: set up , indicating that N sample The set of model responses obtained by random sampling simulations; Indicates j A sample is a dimension N m ×1 vector; then j Samples Monitoring sites in i The normalized calculation formula of the model response is: ; In the formula, Indicates monitoring site i The model response at Indicates j Samples Monitoring sites in i The normalized model response is Respectively represent the minimum and maximum values ​​of the model response set; Finally, we get the normalized model response set .

[0007] Furthermore, in step 5, the neural network structure of the alternative model is constructed by a ResNet-18 network based on two-dimensional convolution; the neural network structure is composed of an initial convolution module, four residual convolution modules, a fully connected module and an output layer connected in sequence; wherein the initial convolution module consists of a convolution layer, a batch normalization layer and a pooling layer; each residual convolution module consists of two residual blocks; and the fully connected module consists of a pooling layer and a flattening layer; The activation function connecting each layer is the RELU function: ; The activation function of the output layer is the Sigmoid function: .

[0008] Furthermore, in step 5, a substitution model for forward prediction based on a convolutional residual neural network is trained, and its loss function is an objective function based on the L1 norm: ; In the formula, represents the connection weight parameter between each layer in the forward neural network, represents the alternative model output, represents the numerical model output, is a hyperparameter that controls the strength of regularization, N Indicates the number of monitoring sites.

[0009] Furthermore, in step 5, the construction and training of the alternative model are both implemented under the PyTorch framework.

[0010] Furthermore, in step 6, the multi-objective optimization criteria for maximum information entropy and minimum redundancy are as follows: ; In the formula, It means that the optimized monitoring system should maximize the effective information and minimize the redundant information. represents the weight parameter that controls the amount of effective information, represents the weight parameter that controls the amount of redundant information, represents the joint entropy of the selected monitoring positions, represents the sum of the mutual information between the selected monitoring position and the unselected monitoring position, represents the total correlation of the selected monitoring positions; The multi-objective optimization criterion of maximum information entropy and minimum redundancy is as follows: first, the monitoring location with the highest information entropy is selected. As the number of selected monitoring locations increases, the joint entropy of the selected monitoring locations gradually increases until the termination condition is met: ,in, represents the joint entropy of the selected monitoring positions, represents the joint entropy of the pre-selected monitoring locations, It represents the joint entropy ratio of the selected monitoring position and the pre-selected monitoring position.

[0011] Furthermore, in step 7, the Monte Carlo simulation method is specifically implemented as follows: step 6 is repeated 100 times; for the head response data set, the set of all monitoring locations that appear in the 100 optimization results is used as the final set optimized monitoring network; for the solute concentration response data set, the number of occurrences in the 100 optimization results is used as the final set optimized monitoring network. The set of monitoring locations is used as the final set to optimize the monitoring network.

[0012] Furthermore, in step 9, the root mean square error quantification method is as follows: ; In the formula, represents the root mean square error, represents the number of model parameters to be estimated, represents the reference value of the model parameter to be estimated, is the average value of the posterior set of model parameters to be estimated.

[0013] Compared with the prior art, the present invention has the following beneficial effects: The present invention proposes an efficient high-dimensional permeability field inversion method by integrating Monte Carlo sampling, information entropy theory and deep learning technology. This method shows good computational efficiency and recognition accuracy in solving the problem of high-dimensional parameter inversion in heterogeneous aquifers, and has important theoretical guidance significance and practical application potential. Specifically, the present invention can efficiently utilize monitoring data and reduce the error of inversion results, thereby improving the accuracy and reliability of inversion. In addition, this method realizes the inversion of high-dimensional permeability fields in heterogeneous aquifers, provides strong technical support for complex solute transport simulation problems, and shows broad practical application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 Flow chart of the method of the present invention.

[0015] Figure 2 Schematic diagram of the module structure of the forward neural network.

[0016] Figure 3 The results of 100 Monte Carlo simulations of entropy optimization of the monitoring network based on the head response and solute concentration response data sets; (a) is the result of 100 Monte Carlo simulations of entropy optimization of the monitoring network based on the head response data set; (b) is the result of 100 Monte Carlo simulations of entropy optimization of the monitoring network based on the solute concentration response data set.

[0017] Figure 4 Schematic diagram of the error evaluation results between the low-dimensional estimated parameters and the true values ​​based on head observation inversion.

[0018] Figure 5 Schematic diagram of the error evaluation results between the low-dimensional estimated parameters and the true values ​​based on the inversion of solute concentration observations.

[0019] Figure 6 Schematic diagram of the posterior estimation results of the high-dimensional permeability field based on head observation inversion; (a) is the head-preselected monitoring network; (b) is the head-single MIMR optimized monitoring network, Rsp=0.95; (c) is the head-single MIMR optimized monitoring network, Rsp=0.97; (d) is the head-single MIMR optimized monitoring network, Rsp=0.99; (e) is the head-collective MIMR optimized monitoring network.

[0020] Figure 7 Schematic diagram of the high-dimensional permeability field posterior estimation results based on solute concentration observation inversion; (a) is the solute concentration-preselected monitoring network; (b) is the solute concentration-single MIMR optimized monitoring network, Rsp=0.95; (c) is the solute concentration-single MIMR optimized monitoring network, Rsp=0.97; (d) is the solute concentration-single MIMR optimized monitoring network, Rsp=0.99; (e) is the solute concentration-collective MIMR optimized monitoring network. DETAILED DESCRIPTION

[0021] In order to have a clearer understanding of the technical features, purposes and beneficial effects of the present invention, the technical solution of the present invention is now described in detail below, but it should not be construed as limiting the applicable scope of the present invention.

[0022] The specific implementation of the present invention is described in detail below in conjunction with specific embodiments.

[0023] One embodiment of the present invention provides a high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy, and its flow chart is as follows: Figure 1 As shown, the method comprises the following steps: Step 1: Establish a numerical model of groundwater flow and solute transport in a heterogeneous aquifer basin; In step 1, the groundwater flow and solute concentration transport simulation program is TOUGHREACT.

[0024] Step 2: Based on the prior information of the permeability field and the geological statistics, the high-dimensional permeability field is generated using the Karhunen-Loève expansion method, and at the same time, it is characterized into low-dimensional parameters to be estimated; In step 2, random sampling is achieved through the Latin hypercube sampling method.

[0025] Step 3: Based on the numerical model in step 1, simulate the dynamic response of the hydraulic head and solute concentration corresponding to the permeability field in step 2 to form a training data set and a test data set of permeability field-dynamic response sample pairs; In step 3, the model parameters in the input file of the TOUGHREACT program are rewritten by writing a program. The TOUGHREACT program can call the executable program and read the results in the output file, thereby obtaining a sample set consisting of model parameters and model responses.

[0026] Step 4: normalize the dynamic responses in the training data set and the test data set in step 3; In step 4, the normalization process is performed using a 0-1 normalization method; the 0-1 normalization method is as follows: set up , indicating that N sample The set of model responses obtained by random sampling simulations; Indicates j A sample is a dimension N m ×1 vector; then j Samples Monitoring sites in i The normalized calculation formula of the model response is: ; In the formula, Indicates monitoring site i The model response at Indicates j Samples Monitoring sites in i The normalized model response is Respectively represent the minimum and maximum values ​​of the model response set; Finally, we get the normalized model response set .

[0027] Step 5: Use the training data set obtained in step 4 to learn the mapping relationship from the high-dimensional permeability field to the hydraulic head or solute concentration response, construct a surrogate model for forward prediction based on the convolutional residual neural network, and compare the error between the surrogate model output and the numerical model output on the test data set obtained in step 4 to evaluate its prediction accuracy; In step 5, the construction and training of the alternative model are both implemented under the PyTorch framework.

[0028] The neural network structure of the alternative model is constructed by a ResNet-18 network based on two-dimensional convolution; the neural network structure is composed of an initial convolution module, four residual convolution modules, a fully connected module and an output layer connected in sequence; among them, the initial convolution module consists of a convolution layer, a batch normalization layer and a pooling layer; each residual convolution module consists of two residual blocks; the fully connected module consists of a pooling layer and a flattening layer; The activation function connecting each layer is the RELU function: ; The activation function of the output layer is the Sigmoid function: .

[0029] The input of the neural network is the heterogeneous high-dimensional permeability field in the basin, and the grid is divided into 40×80. Therefore, the input is a 1×40×80 matrix data (the number of channels is 1 and the matrix size is 40×80). The output of the neural network is N d The data first passes through the initial convolution module to extract parameter features, and then increases the network depth through multiple residual blocks. At the same time, the gradient vanishing problem is reduced by adding the output of the current residual block to the input of the next residual block. Multiple downsampling is performed using a convolutional layer with a stride of 2 to reduce the size of the feature matrix. After that, the feature map is compressed to a fixed size and flattened through an adaptive average pooling layer, and finally output through a fully connected module.

[0030] The module structures of the neural network are as follows: Figure 2 As shown in the figure, taking the initial convolution layer as an example, Conv2d represents a two-dimensional convolution layer (2D Convolution); k represents the size of the convolution kernel (Kernel), k=7 represents the size of the convolution kernel is 7x7; s represents the stride, that is, the step length of each sliding of the convolution kernel, s=2 represents the convolution kernel slides 2 units each time; p represents padding, that is, adding additional grids at the boundary of the input feature matrix to maintain the size of the convolution output or reduce the size change, p=3 represents adding 3 layers of additional grids at the boundary of the input feature matrix; the number sequence 64×20×40 represents the number of channels is 64, and the size of the feature matrix is ​​20×40. Maxpool2d represents the maximum pooling layer. Adaptive AvgPool2d represents the adaptive average pooling layer.

[0031] Train a surrogate model for forward prediction based on a convolutional residual neural network, and its loss function is an objective function based on the L1 norm: ; In the formula, represents the connection weight parameter between each layer in the forward neural network, represents the alternative model output, represents the numerical model output, is a hyperparameter that controls the strength of regularization, N Indicates the number of monitoring sites.

[0032] Step 6: Use the surrogate model trained in step 5 to randomly generate a water head response or solute concentration response data set for entropy analysis, and use the multi-objective optimization criterion of maximum information entropy and minimum redundancy to optimize the pre-selected monitoring sites to obtain a single optimized monitoring network; In step 6, the multi-objective optimization criteria for maximum information entropy and minimum redundancy are as follows: ; In the formula, It means that the optimized monitoring system should maximize the effective information and minimize the redundant information. represents the weight parameter that controls the amount of effective information, represents the weight parameter that controls the amount of redundant information, represents the joint entropy of the selected monitoring positions, represents the sum of the mutual information between the selected monitoring position and the unselected monitoring position, Represents the total correlation of the selected monitoring locations.

[0033] The multi-objective optimization criterion of maximum information entropy and minimum redundancy is as follows: first, the monitoring location with the highest information entropy is selected. As the number of selected monitoring locations increases, the joint entropy of the selected monitoring locations gradually increases. This process will continue until the termination condition is met: ,in, represents the joint entropy of the selected monitoring positions, represents the joint entropy of the pre-selected monitoring locations, It represents the joint entropy ratio of the selected monitoring location and the pre-selected monitoring location, which is determined based on actual research and is usually above 95%.

[0034] Step 7: Repeat step 6 using Monte Carlo simulation, perform statistical analysis on the results, and determine the optimal monitoring network. In step 7, the Monte Carlo simulation method is specifically implemented as follows: step 6 is repeated 100 times; for the head response data set, the set of all monitoring locations that appear in the 100 optimization results is used as the final set optimized monitoring network; for the solute concentration response data set, the number of occurrences in the 100 optimization results The set of monitoring locations is used as the final set to optimize the monitoring network.

[0035] Step 8: Using the single optimized monitoring network obtained in step 6 and the ensemble optimized monitoring network obtained in step 7, a data assimilation algorithm is used to fit the model predictions and actual observation data at the monitoring stations, and the posterior low-dimensional parameters to be estimated are updated.

[0036] Step 9: Use the root mean square error to quantify the accuracy of the posterior estimation of the low-dimensional parameter to be estimated; In step 9, the root mean square error quantification method is as follows: ; In the formula, represents the root mean square error, represents the number of model parameters to be estimated, represents the reference value of the model parameter to be estimated, is the average value of the posterior set of model parameters to be estimated.

[0037] Step 10: Based on the posterior low-dimensional estimated parameters obtained in step 8, the posterior estimate of the high-dimensional permeability field is reconstructed using the inverse process of the Kahunen-Loève expansion.

[0038] Example 1. First, a numerical model of groundwater and conservative solute transport in a two-dimensional heterogeneous saturated freshwater aquifer medium is established. The area size is 10×20m, the left and right boundaries are set to constant water heads h=1.1m and h=1.0m, respectively, and the upper and lower boundaries are non-flow boundaries. A linear solute source with a length of 5m is generalized along the y-axis direction on the left boundary, with an initial solute concentration of 0mol / L and a solute release concentration of 5mol / L. The model area is evenly discretized into 40×80 units, and the permeability of each grid in the TOUGHREACT simulator is given by the following formula: ; In the formula, ; is the permeability correction value, which follows a logarithmic Gaussian distribution in this embodiment; The TOUGHREACT simulator was used to dynamically simulate the solute transport process within 10 days. Data were collected every 2 days for a total of 5 time steps. The head distribution at time step t=5T and the solute concentration distribution at time t=[1, 2, 3, 4, 5]T were recorded.

[0039] The above step 2 is the low-dimensional parameter to be estimated after dimensionality reduction in this embodiment, i.e., the KLE term , and its prior information is , the number of parameters is 200; the mean value of the permeability field is 0, the variance is 2, and the correlation length in the x-axis direction is , the relevant length in the y-axis direction .

[0040] In the above step 3 in this embodiment, the number of samples of the training sample set is 10,000, and the number of samples of the test sample set is 2,000.

[0041] The neural network input in step 5 in this embodiment is a 40×80 permeability parameter field, and the output is the head response at the monitoring site at t=5T, or the sum of the solute concentrations observed at the monitoring site during t=1~5T; the algorithm is implemented in Python, and the regression model of the deep neural network is built using the PyTorch framework. In order to update the weight parameters efficiently and robustly, the Adam optimizer is used, and the learning rate is set to 0.0001. The model was trained for 300 epochs on an NVIDIA GeForce RTX 3070 GPU.

[0042] In the above step 6 in this embodiment, the size of the head response and solute concentration response data set used for entropy optimization is 2000.

[0043] The Monte Carlo simulation results of step 7 in this embodiment are as follows: Figure 3 As shown in the figure, it indicates the frequency of each pre-selected monitoring station being selected in 100 tests. It can be seen that for the head data, the selected monitoring stations are distributed more concentratedly, and all the monitoring stations that appear are selected to form the collective monitoring network (a total of 37 monitoring stations); for the solute concentration data, the selected monitoring stations are distributed more dispersedly, and the monitoring stations that appear 4 times or more are selected as the collective monitoring network (a total of 57 monitoring stations).

[0044] The number of particles in the iterative local update smoother in step 8 in this embodiment is N e =2000.

[0045] The error evaluation result of the low-dimensional estimated parameter (KLE term) based on the head observation and solute concentration observation inversion in step 9 in this embodiment is as follows: Figure 4 , Figure 5 As shown in Figure 2, it can be seen that the a posteriori estimates of assimilated water head observations at the ensemble MIMR optimized monitoring network are better than those at the pre-selected monitoring network and the single MIMR optimized monitoring network; the a posteriori estimates of assimilated solute concentration observations at the ensemble MIMR optimized monitoring network are comparable to those at the pre-selected monitoring network, but the number of monitoring stations is reduced.

[0046] The posterior estimation result of the high-dimensional permeability field based on the head observation and the solute concentration observation inversion in step 10 in this embodiment is as follows: Figure 6 , Figure 7As shown. The results of the reference field, two randomly sampled posterior fields, the mean field of 2000 posterior fields, and the deviation field between the mean field and the reference field are compared. The black dots in the sub-figure reference field represent the locations of monitoring stations in each scenario. It can be seen that with the increase in the amount of information, the inversion reliability of the permeability field also increases. The posterior estimation results obtained by the single MIMR monitoring network optimization method have large uncertainty. The proposed ensemble MIMR monitoring network optimization method realizes the efficient use of head measurement and solute concentration measurement, and improves the reliability and robustness of the inversion identification of the permeability field.

[0047] The above are only preferred embodiments of the present invention. It should be pointed out that, for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention. These should also be regarded as the protection scope of the present invention. These will not affect the effect of the implementation of the present invention and the practicality of the patent.

Claims

1. A high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy, characterized in that: The following steps are involved: Step 1: Establish a numerical model of groundwater flow and solute transport in a heterogeneous aquifer basin; Step 2: Based on the prior information of the permeability field and the geological statistics, the high-dimensional permeability field is generated using the Karhunen-Loève expansion method, and at the same time, it is characterized into low-dimensional parameters to be estimated; Step 3: Based on the numerical model in step 1, simulate the dynamic response of the hydraulic head and solute concentration corresponding to the permeability field in step 2 to form a training data set and a test data set of permeability field-dynamic response sample pairs; Step 4: normalize the dynamic responses in the training data set and the test data set in step 3; Step 5: Use the training data set obtained in step 4 to learn the mapping relationship from the high-dimensional permeability field to the hydraulic head or solute concentration response, construct a surrogate model for forward prediction based on the convolutional residual neural network, and compare the error between the surrogate model output and the numerical model output on the test data set obtained in step 4 to evaluate its prediction accuracy; Step 6: Use the surrogate model trained in step 5 to randomly generate a water head response or solute concentration response data set for entropy analysis, and use the multi-objective optimization criterion of maximum information entropy and minimum redundancy to optimize the pre-selected monitoring sites to obtain a single optimized monitoring network; Step 7: Repeat step 6 using Monte Carlo simulation, perform statistical analysis on the results, and determine the optimal monitoring network. Step 8: Using the single optimized monitoring network obtained in step 6 and the ensemble optimized monitoring network obtained in step 7, a data assimilation algorithm is used to fit the model prediction and actual observation data at the monitoring station, and the posterior low-dimensional parameters to be estimated are updated; Step 9: Use the root mean square error to quantify the accuracy of the posterior estimation of the low-dimensional parameter to be estimated; Step 10: Based on the posterior low-dimensional estimated parameters obtained in step 8, the posterior estimate of the high-dimensional permeability field is reconstructed using the inverse process of the Kahunen-Loève expansion.

2. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 4, the normalization process is performed using a 0-1 normalization method; the 0-1 normalization method is as follows: set up , indicating that N sample The set of model responses obtained by random sampling simulations; Indicates j A sample is a dimension N m ×1 vector; then j Samples Monitoring sites in i The normalized calculation formula of the model response is: ； In the formula, Indicates monitoring site i The model response at Indicates j Samples Monitoring sites in i The normalized model response is Respectively represent the minimum and maximum values ​​of the model response set; Finally, we get the normalized model response set .

3. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In the step 5, the neural network structure of the alternative model is constructed by a ResNet-18 network based on two-dimensional convolution; the neural network structure is composed of an initial convolution module, four residual convolution modules, a fully connected module and an output layer connected in sequence; wherein the initial convolution module consists of a convolution layer, a batch normalization layer and a pooling layer; each residual convolution module consists of two residual blocks; and the fully connected module consists of a pooling layer and a flattening layer; The activation function connecting each layer is the RELU function: ； The activation function of the output layer is the Sigmoid function: 。 4. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 5, a substitution model for forward prediction based on a convolutional residual neural network is trained, and its loss function is an objective function based on the L1 norm: ； In the formula, represents the connection weight parameter between each layer in the forward neural network, represents the alternative model output, represents the numerical model output, is a hyperparameter that controls the strength of regularization, N Indicates the number of monitoring sites.

5. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 5, the construction and training of the alternative model are both implemented under the PyTorch framework.

6. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 6, the multi-objective optimization criteria for maximum information entropy and minimum redundancy are as follows: ； In the formula, It means that the optimized monitoring system should maximize the effective information and minimize the redundant information. represents the weight parameter that controls the amount of effective information, represents the weight parameter that controls the amount of redundant information, represents the joint entropy of the selected monitoring positions, represents the sum of the mutual information between the selected monitoring position and the unselected monitoring position, represents the total correlation of the selected monitoring positions; The multi-objective optimization criterion of maximum information entropy and minimum redundancy is as follows: first, the monitoring location with the highest information entropy is selected. As the number of selected monitoring locations increases, the joint entropy of the selected monitoring locations gradually increases until the termination condition is met: ,in, represents the joint entropy of the selected monitoring positions, represents the joint entropy of the pre-selected monitoring locations, It represents the joint entropy ratio of the selected monitoring position and the pre-selected monitoring position.

7. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 7, the Monte Carlo simulation method is specifically implemented as follows: step 6 is repeated 100 times; for the head response data set, the set of all monitoring locations that appear in the 100 optimization results is used as the final set optimized monitoring network; for the solute concentration response data set, the number of occurrences in the 100 optimization results The set of monitoring locations is used as the final set to optimize the monitoring network.

8. The high-dimensional permeability field inversion method based on Monte Carlo sampling and information entropy according to claim 1 is characterized in that: In step 9, the root mean square error quantification method is as follows: ； In the formula, represents the root mean square error, represents the number of model parameters to be estimated, represents the reference value of the model parameter to be estimated, is the average value of the posterior set of model parameters to be estimated.

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