Underground water seepage parameter inversion method combining FNO and ensemble Kalman filtering

Abstract

The invention relates to the technical field of underground water seepage parameter inversion, in particular to an FNO and ensemble Kalman filtering combined underground water seepage parameter inversion method, which comprises the following steps of: 1, establishing a two-dimensional Darcy flow equation seepage model, and generating a training set and a test set of permeability and a hydraulic water head; step 2, constructing a Fourier neural operator network, and performing offline training on the Fourier neural operator network to obtain a Fourier neural operator substitution model; step 3, under an ensemble Kalman filtering framework, using an adaptive correction algorithm to optimize model parameters, and obtaining an adaptive correction Fourier neural operator network based on ensemble Kalman filtering; step 4, performing ensemble Kalman filtering posterior sampling to generate a posterior sample; and step 5, extracting permeability parameters from the posterior sample. According to the method, the Fourier neural operator and the ensemble Kalman filtering are combined, and the precision and efficiency of underground water seepage parameter inversion are improved by optimizing network parameters through a self-adaptive correction algorithm.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of groundwater seepage parameter inversion, in particular to a groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter. BACKGROUND

[0002] Groundwater seepage parameter inversion is one of the core problems in hydrology and environmental science, and is crucial for understanding groundwater flow and pollutant migration paths. Solving this problem has important practical application value for water resource management and environmental protection. However, groundwater seepage parameter inversion usually involves highly nonlinear and ill-posed partial differential equations, which are extremely challenging to solve.

[0003] Existing numerical simulation-based methods, such as the Finite Element Method (FEM) and the Finite Difference Method (FDM), although can simulate groundwater flow by solving partial differential equations, have obvious limitations. These methods are computationally expensive, especially when dealing with complex aquifer structures and large-scale observation data, requiring a large amount of computational resources and time, resulting in low efficiency. In addition, these methods have difficulty in dealing with nonlinear relationships and uncertainties, making it difficult to accurately capture the complex relationship between parameters and observation data, thereby limiting the inversion accuracy. At the same time, these methods are highly dependent on initial conditions and prior information, and deviations in initial conditions or prior information can lead to inaccurate inversion results. Finally, these traditional methods usually only provide point estimates of parameters, making it difficult to quantify the uncertainty of the inversion results, limiting their reliability in practical applications.

[0004] Although Bayesian methods can provide point estimates and uncertainty quantification of parameters, traditional Markov Chain Monte Carlo (MCMC) methods are computationally intensive and not suitable for large-scale problems. In recent years, with the rapid development of machine learning technology, neural networks have shown great potential in solving complex nonlinear problems. In particular, the Fourier Neural Operator (FNO), as a new type of neural network architecture, can learn the mapping relationship between operators and quickly generate solutions to partial differential equations, with high computational efficiency and strong generalization ability. At the same time, the Ensemble Kalman Filter (EnKF), as a posterior sampling method based on the Bayesian framework, can effectively handle uncertainty and provide uncertainty quantification of parameters through the propagation and update of ensemble members.

[0005] The Chinese patent application with publication number CN118862715A discloses a groundwater seepage prediction method, system, device and medium based on transfer learning and conditional embedding operator theory, wherein the method comprises the following steps: training a Fourier neural operator network in an offline case, obtaining a trained Fourier neural operator network by minimizing a regression loss; fine-tuning the trained Fourier neural operator network using transfer learning to obtain an adaptive Fourier neural operator network; updating the adaptive Fourier neural operator network using conditional embedding operator theory to obtain an adaptive Fourier neural operator network based on CEOD (Conditional embedding operator discrepancy); and solving to obtain a prediction result. The method replaces the traditional numerical method and simple neural network for solving partial differential equations based on transfer learning and conditional embedding operator theory, combines Fourier transform with neural operators, and uses fast Fourier transform in the Fourier domain to speed up the solution of differential operators in partial differential equations. However, although the Fourier neural operator network improves the calculation efficiency, the complexity of the model itself and the calculation cost in the training process are still high, the uncertainty of the inversion result cannot be quantified, and thus the groundwater seepage parameter inversion cannot be efficient, accurate and reliable. SUMMARY

[0006] In view of the low precision and efficiency of groundwater seepage parameter inversion in the prior art, the present application provides a groundwater seepage parameter inversion method combining FNO and ensemble Kalman filtering.

[0007] The present application is achieved by the following technical solutions: A groundwater seepage parameter inversion method combining FNO and ensemble Kalman filtering, comprising the following steps: Step 1: establishing a two-dimensional Darcy flow equation seepage model to generate a training set and a test set of permeability and hydraulic head; Step 2: constructing a Fourier neural operator network, using the training set and the test set of permeability and hydraulic head to perform offline training on the Fourier neural operator network to obtain a Fourier neural operator surrogate model ; Step 3: for the Fourier neural operator surrogate model , using an adaptive correction algorithm to optimize the model parameters under the framework of ensemble Kalman filtering to obtain an adaptive correction Fourier neural operator network based on ensemble Kalman filtering; Step 4: using the adaptive correction Fourier neural operator network based on ensemble Kalman filtering to perform ensemble Kalman filtering posterior sampling to generate posterior samples; Step 5: extracting the permeability parameters from the posterior samples.

[0008] Preferably, in step 1, the training and test sets for permeability are generated by two different Gaussian probability measures, and the training and test sets for hydraulic head are generated by the finite element method.

[0009] Preferably, in step 2, the Fourier neural operator network includes two locally fully connected neural operators and several Fourier neural operators; The first locally fully connected neural operator performs high-dimensional mapping on the input data to obtain representative features; Fourier neural operators perform Fourier transform and linear transform on the characterizable features to obtain Fourier domain features. The second locally fully connected neural operator remaps the Fourier domain features back to the same function space as the input data, generating an output consistent with the input data.

[0010] Preferably, in step 3, the specific steps for obtaining the adaptive modified Fourier neural operator network based on Kalman filtering are as follows: Step 31, Initialization: Assume there exists a high-fidelity model. ; Step 32, Online Calculation: Using Fourier Neural Operators to Replace the Model As a solution to the positive problem, an ensemble Kalman filter is run, and the ensemble Kalman filter is applied in each iteration. After that, check whether the Fourier neural operator replacement model needs to be corrected. Record the sample state at that moment as ; Step 33, Model Update Judgment: Calculate the Fourier neural operator replacement model respectively. and high-fidelity models exist The relative error at the point is used as the judgment criterion to obtain the updated alternative model; Step 34: Use the updated alternative model as the forward problem solver to perform ensemble Kalman filtering sampling to obtain posterior samples; Step 35: Repeat steps 32-34 until the stopping criterion is met to obtain the adaptive modified Fourier neural operator network based on ensemble Kalman filtering.

[0011] Preferably, in step 33, if the relative error is greater than a given threshold... Then in Random selection within the unit sphere sample points ,in, Let k represent the sample points, be a constant, and solve using the finite element method. 1 sample point, to obtain ; Then construct an updated training set ; using the updated training set re-training the Fourier neural operator surrogate model , updating the Fourier neural operator surrogate model of the last two layers of the network parameters, the output model is denoted as , and let ; If the relative error does not exceed a given threshold , continue to use the Fourier neural operator surrogate model as the updated surrogate model.

[0012] Preferably, in step 4, during posterior sampling, a corresponding set Kalman filter sampling method is selected for different scenes: For inverse problems, iterative regularization set Kalman filter sampling is used; For nonlinear problems, iterative regularization set Kalman filter sampling with statistical linearization is used; For complex problems with uncertain or changing prior distribution, adaptive iterative regularization set Kalman filter sampling with statistical linearization is used.

[0013] Preferably, in step 5, the process of extracting permeability parameters from posterior samples is: First, collect all posterior samples obtained by set Kalman filter posterior sampling; Then, average the permeability values in all posterior samples to calculate the posterior mean permeability ; Finally, the posterior mean permeability is output as the inversion result for subsequent analysis and application.

[0014] A groundwater seepage parameter inversion system combining FNO and set Kalman filter, comprising a data generation module, an initial model processing module, a model optimization module, a posterior sampling module and a permeability extraction module, the data generation module is used to establish a two-dimensional Darcy flow equation seepage model, and generate a training set and a test set of permeability and hydraulic head; the initial model processing module is used to construct a Fourier neural operator network, and the Fourier neural operator network is trained offline using the training set and the test set of permeability and hydraulic head to obtain a Fourier neural operator surrogate model ; the model optimization module is used to optimize the Fourier neural operator surrogate model In the ensemble Kalman filtering framework, the model parameters are optimized using an adaptive correction algorithm to obtain an adaptive correction Fourier neural operator network based on ensemble Kalman filtering; the sampling module is used to perform ensemble Kalman filtering posterior sampling using the adaptive correction Fourier neural operator network based on ensemble Kalman filtering to generate posterior samples; and the permeability extraction module is used to extract the permeability parameters from the posterior samples.

[0015] An electronic device comprises a memory and a processor, the memory stores a computer program, and the processor implements the steps of the groundwater seepage parameter inversion method combined with FNO and ensemble Kalman filtering when executing the computer program.

[0016] A storage medium has a computer program stored thereon, and the computer program implements the steps of the groundwater seepage parameter inversion method combined with FNO and ensemble Kalman filtering when executed by a processor.

[0017] Compared with the prior art, the following beneficial effects are achieved: The groundwater seepage parameter inversion method combined with FNO and ensemble Kalman filtering of the present application uses a Fourier neural operator as a surrogate model for the forward problem, which significantly reduces the forward problem solving time. At the same time, ensemble Kalman filtering (EnKF) and its variants reduce the sample size required for posterior sampling, thereby greatly improving the overall efficiency of the inversion.

[0018] The Fourier neural operator can learn and capture the complex nonlinear relationship between the parameters and the observation data. Ensemble Kalman filtering further improves the inversion accuracy by effectively handling parameter uncertainty and adaptively adjusting the prior distribution and regularization parameters. In particular, the adaptive iterative regularization ensemble Kalman filter (Adaptive IREKF-SL) can accurately and quickly perform posterior sampling when dealing with prior distributions containing hyperparameters. At the same time, by introducing a regularization mechanism, overfitting and semi-convergence problems are effectively avoided, and the stability of the inversion process is enhanced. Ensemble Kalman filtering shows strong adaptability, and in the presence of noise or inaccurate prior information, it can still obtain stable and reliable inversion results.

[0019] The combination of FNO and ensemble Kalman filtering in the groundwater seepage parameter inversion method combined with FNO and ensemble Kalman filtering reduces the consumption of computing resources, reduces the hardware cost, and improves the practicality and economy of the inversion method. Compared with traditional numerical methods, FNO and AFNO can improve the computational efficiency while maintaining high accuracy, verifying the effectiveness and superiority of the present application in the problem of groundwater seepage parameter inversion. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1is a flow chart of a groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter of the application; Figure 2 (a) is the true value of the random parameter (b) (c) (d) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when IREKF is applied as the posterior sampling algorithm; Figure 3 (a) is the true value of the random parameter (b) (c) (d) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when IREKF is applied as the posterior sampling algorithm; Figure 4 (a) (b) (c) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when IREKF is applied as the posterior sampling algorithm; Figure 5 (a) (b) (c) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when IREKF is applied as the posterior sampling algorithm; Figure 6 (a) (b) (c) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when IREKF-SL is applied as the posterior sampling algorithm; Figure 7 (a) (b) (c) are respectively and the posterior mean values obtained by FEM, FNO and AFNO inversion when Adaptive IREKF-SL is applied as the posterior sampling algorithm. DETAILED DESCRIPTION

[0021] The application will be further described in detail below with specific examples, which are an explanation of the application rather than a limitation.

[0022] The application discloses a groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter, referring to Figure 1 , comprising the following steps: Step 1, a two-dimensional Darcy flow equation seepage model is established, and a training set and a test set of permeability and hydraulic head are generated.

[0023] The two-dimensional Darcy flow equation is a second-order elliptic partial differential equation for describing the seepage law of low-speed fluid flow in porous media, and the equation form is as follows:

[0024]

[0025] where, is the underground permeability, reflecting the water conductivity of different positions of the porous medium; is the hydraulic head, representing the pressure potential energy of the fluid at the point; is the source term. The porous medium region takes a square domain , the source term , .

[0026] The training set and test set of the permeability are generated by two different Gaussian probability measures, and the training set and test set of the hydraulic head are generated by the finite element method.

[0027] Specifically, in the training set, the permeability is generated by a Gaussian probability measure, and the covariance matrix of the Gaussian measure is , is the Laplace matrix, where, represents the unit matrix.

[0028] The hydraulic head is the finite element solution in the grid state discretized as . When the training data is obtained by using the finite element method, the stiffness matrix has uncertainty, and the uncertainty parameter is taken.

[0029] In the test set, the true value of the permeability is generated by another Gaussian probability measure (covariance matrix ).

[0030] The measured value of the hydraulic head is the finite element solution when the true value of the permeability is discretized as , and the uncertainty parameter of the stiffness matrix is .

[0031] Specifically, the finite element method is used to solve the two-dimensional Darcy flow equation numerically to generate the training set and test set of hydraulic head. The finite element method is a method for solving partial differential equations by dividing the solution domain into a finite number of smaller elements and converting the partial differential equation into a system of algebraic equations. In the solving process, a stiffness matrix is constructed, which is obtained by discretizing the partial differential equation and contains information about the physical properties (such as material properties and geometric shapes) in the equation. In the offline process, a system of linear equations is obtained. In actual calculations, the construction of the stiffness matrix involves numerical integration of spatial derivatives in the equation, which can introduce numerical errors. To simulate this uncertainty, an uncertainty parameter is introduced when constructing the stiffness matrix, which has different values in the training set and test set, respectively, and This uncertainty parameter can evaluate the performance of the model when facing different degrees of numerical uncertainty.

[0032] Additive Gaussian noise with a mean of 0 to the numerical value of the hydraulic head calculated by the finite element method. Take the uniform grid point position for data measurement, for example, 100 measurement points, the measurement point coordinates are The measurement point data belongs to noisy data, where, and represent the row and column indices of the measurement point in the grid. Specifically, and The value range of both is from 1 to 10, which means that in the grid of , every 10 grid points are taken as a measurement point, thus obtaining measurement points.

[0033] The training set and test set are generated in the same way, but different covariance matrices are used to introduce uncertainty, which can enhance the generalization and robustness of the model. The training set contains 300 random parameters and their corresponding finite element solutions, and the test set contains 100 random parameters and their corresponding finite element solutions.

[0034] Step 2, construct the Fourier neural operator network, use the training set and test set of permeability and hydraulic head to offline train the Fourier neural operator network, obtain the Fourier neural operator surrogate model .

[0035] The Fourier neural operator network (FNO) includes two local fully connected neural operators and several Fourier neural operators. The first local fully connected neural operator performs high-dimensional mapping on the input data to obtain representable features. The Fourier neural operator performs Fourier transform and linear transform on the representable features to obtain Fourier domain features. The second local fully connected neural operator re-maps the Fourier domain features back to the same function space as the input data to generate an output result consistent with the input data.

[0036] Specifically, the input data is first mapped to the frequency domain through fast Fourier transform, then the frequency domain features between specified frequencies are obtained through a filter, and then the frequency domain features are converted from the frequency domain to the physical domain through inverse Fourier transform to obtain the conversion result. At the same time, the linear transformation of the frequency domain features is obtained, and the results of the Fourier transform and the linear transformation are added and processed through a nonlinear activation function to obtain the final hidden features of the Fourier layer. The final hidden features are passed layer by layer, and each layer will perform the same feature extraction and nonlinear activation function processing until the last layer, generating Fourier domain features. The second local fully connected neural operator projects the Fourier domain features back to the function space of the input data to complete the mapping from features to output.

[0037] At the data processing level, the parameters of the partial differential equation are input into the AFNO network , which is composed of a local fully connected neural operator , a Fourier neural operator , and another local fully connected neural operator . The specific process is as follows: After the parameters pass through the first local fully connected neural operator , a large number of representable features are extracted; The representable features are extracted through Fourier transform and linear transform, respectively. The features extracted by Fourier transform are denoted as first extracted features , and the features extracted by linear transform are denoted as second extracted features ; The first extracted features and the second extracted features are added, and then processed through a nonlinear activation function to obtain the final hidden features of the Fourier layer ; These hidden features are passed layer by layer to the next layer of the network. Each layer will perform similar feature extraction and processing until the last layer.

[0038] the Fourier domain feature generated by the last Fourier layer input to the second local fully connected neural operator to obtain the numerical solution corresponding to the parameter data .

[0039] Mathematically, the process can be described as:

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] wherein, represents a characterizable feature, is a feature extracted by Fourier transform, is a feature extracted by linear transform, is a nonlinear activation function, is a Fourier transform, is an inverse Fourier transform, is a periodic transform, is a parameter related to linear transform, generally represents the output feature of the last Fourier layer, and the output of each layer in the layer-by-layer transmission process is the input of the next layer. Therefore, is the output of the previous layer of , and finally is obtained through such layer-by-layer transmission and transformation.

[0046] The present application uses the GELU function as the nonlinear activation function, and the expression is wherein , is the error function.

[0047] Step 3, for the Fourier neural operator substitute model Within the framework of ensemble Kalman filtering, an adaptive correction algorithm is used to optimize model parameters, resulting in an adaptively corrected Fourier neural operator network (AFNO) based on ensemble Kalman filtering. The AFNO further optimizes network parameters based on FNO through an adaptive correction algorithm to improve solution accuracy in specific posterior distribution regions. During posterior exploration, new training points are automatically selected to fine-tune the FNO model.

[0048] Within the ensemble Kalman filter framework, adaptive correction algorithms (such as Adam) are used to optimize network parameters, particularly the last two layers. The specific process is as follows: Step 31, Initialization: Assume there exists a high-fidelity model. This high-fidelity model It closely resembles the true positive model, and sets the initial learning rate. Maximum number of adaptive corrections and error threshold The initial iteration step size of the ensemble Kalman filter Set it to 0.1; Step 32, Online Calculation: Using Fourier Neural Operators to Replace the Model As a solution to the positive problem, an ensemble Kalman filter is run, and the ensemble Kalman filter is applied in each iteration. After that, check whether the Fourier neural operator replacement model needs to be corrected. Record the sample state at that moment as ; Step 33, Model Update Judgment: Calculate the Fourier neural operator replacement model respectively. and high-fidelity models exist The relative error at the point is used as the judgment criterion to obtain the updated alternative model; Where the relative error is greater than a given threshold Then in Random selection within the unit sphere sample points ,in, Let k represent the sample points, be a constant, and solve using the finite element method. 1 sample point, to obtain ; Then construct an updated training set Use the updated training set Retraining the Fourier neural operator replacement model Update the Fourier neural operator replacement model The parameters of the last two network layers are denoted as the output model. and order ; If the relative error does not exceed the given threshold Then, the Fourier neural operator will continue to be used as an alternative model. As an updated alternative model; Step 34: Use the updated alternative model as the forward problem solver to perform ensemble Kalman filtering sampling to obtain posterior samples; Step 35: Repeat steps 32-34 until the stopping criterion is met to obtain the adaptive modified Fourier neural operator network based on ensemble Kalman filtering.

[0049] Step 4: Use an adaptive modified Fourier neural operator network based on ensemble Kalman filtering to perform ensemble Kalman filtering posterior sampling and generate posterior samples.

[0050] During posterior sampling, the appropriate ensemble Kalman filter sampling method is selected for different scenarios: (i) For the inverse problem, iterative regularized set Kalman filtering is used for sampling.

[0051] Iterative Regularized Ensemble Kalman Filtering (IREKF) builds upon traditional Ensemble Kalman Filtering (EnKF) by introducing a regularization term and a stopping criterion (such as the Morozov difference principle) to avoid overfitting and semi-convergence. It iteratively updates the particle set to gradually approximate the posterior distribution. The detailed calculation steps are as follows: (1) Input: Let a priori The initial set of particles generated, , , For the observation noise covariance matrix. Start the loop. (The loop continues...) It is the number of particles; Represents the covariance matrix of a normal distribution; It is a parameter between 0 and 1, which is usually used to control the convergence speed or step size of the algorithm; It is greater than The parameters are typically used to determine the stopping criterion or the number of iterations for an algorithm; It is an iteration counter that starts from 0 and increments by 1 with each iteration until the stopping criterion is met.

[0052] (2) Prediction Steps. Define the forward solution for the particle: . Indicates the first Parameter values ​​of individual particles Through mapping The predicted observations are obtained by transforming the data into the observation space. Using this set, the sample mean and the mean of the forward solution can be defined as follows: , .

[0053] (3) Analysis step. Define the sample covariance matrix as follows: , .

[0054] Compute the Kalman gain , for the step size. denote the covariance matrix between the parameters and the observation . denote the covariance matrix of the observation itself.

[0055] (4) Update each particle in the set according to the following formula (i.e., the Kalman filtering equation): . denote the actual observation of the th particle at the th iteration. The observation here is the actual measurement data compared with the model-predicted observation .

[0056] (5) Let denote the noise level. If , stop the loop and output the posterior sample .

[0057] (B) For nonlinear problems, the iterative regularized ensemble Kalman filter with statistical linearization is used for sampling. Based on IREKF, statistical linearization processing is further introduced to avoid the collapse of the ensemble by modifying the calculation method of the Kalman gain. This method approximates the gradient of the forward model through statistical linearization, enhancing the stability and accuracy of the algorithm. The detailed calculation steps are as follows: (1) Input: Let be the initial particle set generated by the prior , , , be the observation noise covariance matrix. Start the loop for .

[0058] (2) Prediction step. Define the forward solution of the particle: . Through this set, the sample mean and the mean of the forward solution can be defined as follows: , .

[0059] (3) Analysis step. Define the sample covariance matrix as follows: , .

[0060] Compute , Kalman gain , is the step size.

[0061] (4) Update each particle in the set according to the following formula (i.e., the Kalman filter equation): .

[0062] (5) Let denote the noise level. If , stop the loop and output the posterior sample .

[0063] (Three) For complex problems with uncertain or changing prior distributions, an adaptive iterative regularized ensemble Kalman filter with statistical linearization is used for sampling. Based on IREKF-SL, it is assumed that the prior distribution is random and a hyperparameter is introduced. By adaptively updating the hyperparameter, the adaptability and accuracy of the algorithm are further improved. This method can dynamically adjust the regularization parameter during the iteration process, better handling the uncertainty of the prior distribution. The detailed calculation steps are as follows: (1) Input: Initialize the hyperparameter , let be the initial particle set generated by the prior , , , is the observation noise covariance matrix. is a hyperparameter used to adjust the tightness of the prior distribution, which affects the initialization process of the particle filter algorithm. For start the loop.

[0064] (2) Prediction step. Define the forward solution of the particle as: . Through this set, the sample mean and the forward solution mean can be defined as follows: , .

[0065] (3) Analysis step. Define the sample covariance matrix as follows: , .

[0066] Compute , Kalman gain , is the step size.

[0067] (4) Update each particle in the set according to the following formula (i.e., Kalman filtering equation): .

[0068] (5) Update the hyperparameters according to the hyperparameter distribution using the following formula : If the hyperparameters follow an exponential distribution , the update formula is ; If the hyperparameters follow a normal distribution , the update formula is ; Let denote the noise level. If , stop the loop and output the posterior samples .

[0069] Through the above steps, the three methods provide efficient solutions for groundwater seepage parameter inversion in different scenarios. IREKF is suitable for inverse problems, IREKF-SL is suitable for nonlinear problems, and Adaptive IREKF-SL performs well in complex problems with uncertain or changing prior distributions.

[0070] Step 5: Extract the permeability parameter from the posterior samples. The detailed process is as follows: First, collect all the posterior samples obtained by ensemble Kalman filtering. These samples contain multiple possible permeability values.

[0071] Then, average the permeability values in all posterior samples to calculate the posterior mean permeability , that is:

[0072] where is the number of posterior samples, is the permeability value in the th posterior sample. The posterior mean permeability represents the most likely value of the permeability under the given observation data and model assumptions.

[0073] Finally, output the calculated posterior mean permeability as the inversion result for subsequent analysis and application.

[0074] Step 6: Numerical verification The finite element method (FEM), FNO and AFNO are used as forward solvers, combined with IREKF, IREKF-SL and Adaptive IREKF-SL for posterior sampling. The performance of the results in terms of accuracy and efficiency is explored when solving the inverse problem of the two-dimensional Darcy flow equation under different particle numbers and noise levels. The training set and test set data are generated using the Gaussian probability measure. The training set contains 300 groups of random parameters and their finite element solutions, and the test set contains 100 groups.

[0075] The two sets of data are generated in the same way, but different covariance matrices are used to introduce uncertainty to enhance the generalization and robustness of the model. The filtering method is selected from IREKF, IREKF-SL and Adaptive IREKF-SL. The particle number , initial iteration step , maximum adaptive correction number are set.

[0076] Figure 2 and Figure 3 show the performance comparison of different forward solvers for the two-dimensional Darcy flow equation inversion problem under different particle numbers. When the particle number is small , the posterior mean obtained by IREKF using FEM as the forward solver has a significant error compared with the true value. If FNO is used as the forward solver, not only is there a large deviation between the posterior mean and the true value, but the error is further increased compared with the posterior mean obtained by FEM. Even if AFNO is introduced for model correction, there is still a large difference between the inversion result and the true value and the FEM inversion solution. It is speculated that the main reason for this phenomenon is that the lack of particle number leads to inaccurate estimation of sample variance, affecting the calculation of Kalman gain, and thus reducing the accuracy of parameter estimation.

[0077] When the particle number is large After the increase of the number of particles, the error between the inversion results obtained by using FEM as the forward solver and the true value is significantly reduced. This shows that FEM solver can provide more accurate estimation results as the number of particles increases. However, there is still a large error between the posterior mean obtained by using FNO as the forward solver and the true value as well as the posterior mean obtained by FEM. This error is mainly due to the difference in data distribution between the training set and the test set, which leads to the lack of generalization ability of the FNO model. However, after the modification of the AFNO model, the error between the posterior mean obtained by using AFNO as the forward solver and the posterior mean obtained by FEM is significantly reduced and is closer to the true value. This result shows that the AFNO model can effectively improve the accuracy of inversion, and by correcting the inaccuracy of the FNO model in the posterior distribution space, it can enhance the generalization ability and inversion accuracy of the model.

[0078] Figure 4 and Figure 5 The results show that under different noise levels, for the two-dimensional Darcy flow equation inversion problem, using FEM, FNO and AFNO as the forward solver and combining IREKF for posterior sampling. Under low noise level conditions, Figure 4 The posterior mean obtained by the three forward solvers is shown. The results show that compared with FEM, the inversion solution obtained by FNO as the forward solver has a large error. However, AFNO significantly improves the inversion accuracy after introducing the adaptive correction algorithm, and the error between the inversion solution obtained by FEM is significantly reduced. This verifies that AFNO can effectively improve the inversion accuracy under low noise level and obtain inversion results close to FEM. Under high noise level conditions, Figure 5 The posterior mean obtained by the three forward solvers through IREKF sampling under the same experimental setup is shown. It can be seen that the error between the FNO inversion solution and the FEM inversion solution is large. Although the error between the inversion solution of AFNO and FEM is reduced after introducing the adaptive correction algorithm, compared with the low noise level condition, the correction effect of AFNO is not significant enough. This may be because the increase of noise level leads to the reduction of AFNO correction effect. In general, AFNO model can effectively improve the inversion accuracy under low noise level, but under high noise level, the correction effect may become insignificant due to noise interference. This shows that in practical application, noise level is one of the important factors affecting the inversion accuracy, and AFNO as an improved forward solver also has different performances under different noise levels.

[0079] In the study of the inversion problem of the two-dimensional Darcy flow equation, a statistical comparison of sampling time and accuracy was conducted for three forward problem solvers. Experimental results show that the total time for using the FNO substitute model and performing IREKF sampling with this model is approximately 597 seconds; while after introducing an adaptive correction mechanism, the training and posterior sampling of the AFNO substitute model requires 804 seconds; in contrast, solving the forward problem using the traditional FEM and performing IREKF sampling requires approximately 1835 seconds. These data indicate that although FNO has some errors, it is about 3 times more time efficient than FEM, and these errors can be effectively corrected by the adaptive algorithm, thus increasing the overall computational cost only slightly. Furthermore, AFNO saves more than 2 times the time compared to FEM while providing higher accuracy sampling results.

[0080] Figure 6 and Figure 7 The effects of two different ensemble Kalman filtering algorithms—IREKF-SL and AdaptiveIREKF-SL—on the two-dimensional Darcy flow equation inversion problem are demonstrated. In the experiment, the number of particles ( ) and noise level ( The values ​​were fixed to ensure comparability of results. The experiments only compared the closeness of the posterior means obtained from FNO and AFNO to FEM, rather than comparing them to the true values. In the IREKF-SL method, after a certain number of iterations (e.g., 10), the relative error between the posterior means obtained from FNO and FEM is calculated to determine whether adaptive correction is needed. Figure 6 The posterior means obtained by the three methods, FEM, FNO, and AFNO, under this algorithm are shown. The results indicate that AFNO, compared to the uncorrected FNO, is more effective in reducing the error between the posterior means and those obtained by FEM, demonstrating the improvement effect of AFNO in enhancing inversion accuracy. In the Adaptive IREKF-SL method, the initial hyperparameters are set to 1 and assumed to follow an exponential distribution. During each iteration, the relative error is calculated to determine whether correction is needed. Figure 7 The diagram shows the posterior means obtained by the three methods, FEM, FNO, and AFNO, under this algorithm. Similar to IREKF-SL, AFNO also shows an improvement over FNO under the Adaptive IREKF-SL algorithm, making the posterior mean closer to the posterior mean obtained by FEM.

[0081] In addition, the posterior sampling time of IREKF-SL and Adaptive IREKF-SL algorithm is about 1069 seconds and 357 seconds respectively when FEM is used as the forward solver. When FNO surrogate model is used, the sampling time of IREKF-SL is about 421 seconds, while the sampling time of Adaptive IREKF-SL is further reduced to about 126 seconds. After introducing the adaptive correction mechanism, the sampling time of AFNO surrogate model is about 489 seconds for IREKF-SL and about 112 seconds for Adaptive IREKF-SL. It is concluded that the improved posterior sampling algorithm (i.e. IREKF-SL and Adaptive IREKF-SL) significantly accelerates the iteration speed of the ensemble particles and reduces the overall sampling time. Compared with the FEM method, FNO and AFNO as surrogate models significantly save the calculation time without introducing high solving error, thereby verifying the effectiveness of AFNO in improving the efficiency and accuracy of inversion. These results show that AFNO combined with the improved posterior sampling algorithm can not only provide accuracy comparable to the traditional FEM method, but also greatly reduce the computational cost, which has important practical value for solving the inversion problem with large computational cost.

[0082] The groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter of the application is developed around the solution of partial differential equation inverse problem, an alternative model is constructed based on Bayesian inversion theory, and the problem is solved by posterior sampling of target parameters. In view of the large sample size required by MCMC, three variants of ensemble Kalman filter are proposed for posterior sampling.

[0083] In the forward problem solving part, in order to reduce the calculation amount, a neural operator surrogate model is introduced from the idea of machine learning surrogate model, which can learn operators in infinite-dimensional space. In order to further improve the calculation efficiency, Fourier neural operator is used as the forward problem surrogate model under the Bayesian framework, and is combined with ensemble Kalman filter for sampling, and an online adaptive correction mechanism is also introduced, which allows Fourier neural operator to continuously fit new data to obtain better posterior estimation.

[0084] In summary, the Fourier neural operator and ensemble Kalman filter are combined in the application, the network parameters are optimized by the adaptive correction algorithm, the accuracy and efficiency of groundwater seepage parameter inversion are effectively improved, and the application has important application value.

[0085] The application further discloses a groundwater seepage parameter inversion system combining a Fourier neural operator and an ensemble Kalman filter, which comprises a data generation module, an initial model processing module, a model optimization module, a posterior sampling module and a permeability extraction module. The initial model processing module is used for building a Fourier neural operator network and performing offline training to obtain a Fourier neural operator substitute model The model optimization module is used for optimizing model parameters of the Fourier neural operator substitute model

[0086] The application further discloses an electronic device comprising a memory and a processor, wherein the memory stores a computer program, and the processor realizes the steps of the groundwater seepage parameter inversion method combining the Fourier neural operator and the ensemble Kalman filter when executing the computer program.

[0087] The application further discloses a storage medium, wherein the storage medium stores a computer program, and the computer program realizes the steps of the groundwater seepage parameter inversion method combining the Fourier neural operator and the ensemble Kalman filter when being executed by a processor.

[0088] The above merely describes the preferred embodiments of the application, and is not intended to limit the technical scheme of the application in any way, and those skilled in the art should understand that, without departing from the spirit and principle of the application, the technical scheme can be simply modified and replaced, and these modifications and replacements also belong to the protection scope of the claims.

Claims

1. A method for groundwater seepage parameter inversion combining FNO and ensemble Kalman filter, characterized in that, The method comprises the following steps: Step 1, a two-dimensional Darcy flow equation seepage model is established, and a training set and a test set of permeability and hydraulic head are generated; Step 2, constructing the Fourier neural operator network, using the training set and the test set of permeability and hydraulic head to train the Fourier neural operator network offline, and obtaining the Fourier neural operator surrogate model ; Step 3, replace the Fourier neural operator model with the Fourier neural operator surrogate model In the ensemble Kalman filter framework, the model parameters are optimized using an adaptive correction algorithm to obtain an adaptive correction Fourier neural operator network based on the ensemble Kalman filter. Step 4, ensemble Kalman filter posterior sampling is performed on the adaptive modified Fourier neural operator network based on the ensemble Kalman filter, and posterior samples are generated; Step 5, the permeability parameter is extracted from the posterior samples.

2. The method of groundwater seepage parameter inversion combining FNO and ensemble Kalman filter according to claim 1, characterized in that, In step 1, the training set and the test set of the permeability are generated by two different Gaussian probability measures, and the training set and the test set of the hydraulic head are generated by the finite element method.

3. The method of claim 1, wherein, In step 2, the Fourier neural operator network comprises two local full connection neural operators and a plurality of Fourier neural operators; The first local full connection neural operator performs high-dimensional mapping on the input data to obtain a representable feature; The Fourier neural operator performs Fourier transform and linear transform on the representable feature to obtain a Fourier domain feature; The second local full connection neural operator maps the Fourier domain feature back to the same function space as the input data to generate an output result consistent with the input data.

4. The method of groundwater seepage parameter inversion combining FNO and ensemble Kalman filter according to claim 1, characterized in that, In step 3, the specific steps of obtaining the adaptive modified Fourier neural operator network based on the Kalman filter are as follows: Step 31, Initialization: Assume that there is one high-fidelity model ; Step 32, online computation: use the Fourier neural operator surrogate model As a forward problem solver, the ensemble Kalman filter is run, the ensemble Kalman filter is run times, after which it is checked whether the Fourier neural operator surrogate model needs to be corrected , the sample state at this moment is recorded as ; Step 33, model update judgment: respectively calculate the relative error of the Fourier neural operator surrogate model and the high-fidelity model at the points in the training set, and obtain the updated surrogate model as the judgment criterion of the relative error. In ​ Step 34, using the updated substitute model as a forward problem solver, performing ensemble Kalman filter sampling to obtain posterior samples; Step 35, repeating steps 32-34 until the stopping criterion is met, and obtaining the adaptive modified Fourier neural operator network based on the ensemble Kalman filter.

5. The method of groundwater flow parameter inversion combining FNO and ensemble Kalman filter according to claim 4, characterized in that, In step 33, if the relative error is greater than a given threshold , then randomly select sample points in the unit sphere , where represents a sample point, k is a constant, and the finite element method is used to solve sample points to obtain ; Afterwards, the updated training set is constructed ; using the updated training set re-training the Fourier neural operator surrogate model , updating the Fourier neural operator surrogate model of the last two layers of the network, the output model is denoted by , and let ; If the relative error does not exceed a given threshold then the Fourier Neural Operator surrogate model is continued to be used as the updated surrogate model.

6. The method of groundwater seepage parameter inversion combining FNO and ensemble Kalman filter according to claim 1, characterized in that, In step 4, during posterior sampling, corresponding ensemble Kalman filter sampling methods are selected for different scenarios: For inverse problems, iterative regularization ensemble Kalman filter sampling is used; For nonlinear problems, iterative regularization ensemble Kalman filter sampling with statistical linearization is used; For complex problems with uncertain or changing prior distribution, adaptive iterative regularization ensemble Kalman filter sampling with statistical linearization is used.

7. The method of groundwater flow parameter inversion combining FNO and ensemble Kalman filter according to claim 6, characterized in that, In step 5, the process of extracting the permeability parameter from the posterior samples is as follows: First, collect all posterior samples obtained by ensemble Kalman filter posterior sampling; Then, the permeability values in all the posterior samples are averaged to calculate the posterior mean permeability ; Finally, the posterior mean permeability As an output of the inversion result, for subsequent analysis and application.

8. A system for groundwater seepage parameter inversion combining FNO and ensemble Kalman filter, characterized in that, The method comprises a data generation module, an initial model processing module, a model optimization module, a posterior sampling module and a permeability extraction module, the data generation module is used to establish a two-dimensional Darcy flow equation seepage model, and generate a training set and a test set of permeability and hydraulic head; the initial model processing module is used to construct a Fourier neural operator network, and perform offline training on the Fourier neural operator network by using the training set and the test set of permeability and hydraulic head, to obtain a Fourier neural operator surrogate model ; the model optimization module is used to optimize model parameters of the Fourier neural operator surrogate model under a set Kalman filtering framework, and use an adaptive correction algorithm to obtain an adaptive correction Fourier neural operator network based on set Kalman filtering; the sampling module is used to perform set Kalman filtering posterior sampling by using the adaptive correction Fourier neural operator network based on set Kalman filtering, to generate posterior samples; and the permeability extraction module is used to extract permeability parameters from the posterior samples. 9.An electronic device comprising a memory and a processor, the memory storing a computer program, wherein, The processor executes the computer program to realize the steps of the groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter in any one of claims 1-7.

10. A storage medium having stored thereon a computer program, characterized in that The computer program is executed by the processor to realize the steps of the groundwater seepage parameter inversion method combining FNO and ensemble Kalman filter in any one of claims 1-7.

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