Groundwater model parameter estimation method combining deep learning and local set update

Abstract

This invention discloses a groundwater model parameter estimation method combining deep learning and local set updating, comprising: constructing a groundwater numerical model; constructing a prior parameter sample set; obtaining the corresponding model output set; acquiring observation data; determining the number and proportion of local sets; randomly selecting local center samples and calculating similarity; forming a local prior sample set; updating all local prior sample sets one by one to a local posterior sample set; randomly selecting posterior samples to obtain the overall posterior sample set; replacing the prior sample set with the posterior sample set, and iteratively outputting the parameter estimation results. By combining a deep learning update mechanism with a local set update strategy, this invention effectively addresses the challenges of high-dimensionality, strong nonlinearity, non-Gaussian parameter distribution, and the "different parameters, same effect" problem in groundwater model parameter estimation, significantly improving the computational efficiency and accuracy of data assimilation algorithms in complex groundwater system parameter estimation tasks.

Description

Technical Field

[0001] This invention belongs to the field of hydrology and earth science, and relates to groundwater model parameter estimation technology, specifically to a groundwater model parameter estimation method that combines deep learning and local set updates. Background Technology

[0002] Numerical simulation of groundwater is fundamental to groundwater resource management and pollution prevention, and obtaining accurate hydrogeological parameters (such as permeability fields and pollution source parameters) is crucial for constructing high-fidelity models. Data assimilation techniques, by fusing multi-source observation data with physical models, have been widely applied to groundwater parameter inversion. However, traditional ensemble data assimilation frameworks based on Gaussian assumptions exhibit significant limitations when dealing with complex groundwater systems that exhibit typical non-Gaussian characteristics and the "different parameters, same effect" problem.

[0003] Specifically, influenced by complex geological processes, aquifers often develop highly heterogeneous structures such as paleochannels and fractures, resulting in highly non-Gaussian distributions of parameters such as permeability coefficients. To overcome this challenge, introducing nonlinear mapping strategies based on deep learning has become an effective approach. These strategies utilize the powerful nonlinear feature extraction and high-dimensional pattern learning capabilities of deep neural networks to replace update operators in traditional data assimilation that are constrained by linear assumptions, effectively approximating the true posterior distribution of the non-Gaussian parameter field. However, existing deep learning update mechanisms typically rely on global mapping patterns. When faced with the problem of parameter ambiguity caused by extremely limited observational information, they often fall into a single pattern, making it difficult to effectively explore and cover all potential posterior modes.

[0004] On the other hand, sparse observational data easily leads to the "different parameters, same effect" problem, where multiple different parameter combinations often produce equally good observational fitting results, resulting in a significant multimodal characteristic in the posterior probability distribution of the parameters. To alleviate this problem, a local set update strategy has been proposed and shown application potential. This strategy dynamically constructs local sets of a specific size for samples and updates them independently by quantitatively assessing the similarity between samples and between the model output and the observed values. This mechanism can guide samples from different regions to converge to their nearest posterior mode, greatly promoting the exploration of multimodal posterior distributions and demonstrating high flexibility in solving the problem of multiple solutions. However, existing local update frameworks still use linear operators based on the Kalman formula in the underlying computation. Once the local parameter space still exhibits strong non-Gaussian characteristics, its update performance will significantly degrade.

[0005] In practical, complex groundwater parameter inversion tasks (such as the joint inversion of high-dimensional non-Gaussian permeability fields and pollution source parameters), the strong non-Gaussianity of the parameters and the "different parameters, same effect" problem are usually highly intertwined and mutually influential. Although existing nonlinear mapping strategies and multimodal exploration strategies each have their strengths in addressing single challenges, there is still a lack of a unified framework in the field that can simultaneously overcome these two types of underlying technical bottlenecks. In fact, simple global nonlinear mapping is inadequate in exploring multiple solutions when information is scarce; while simple local set updates are hampered by the inherent limitations of linear operators and cannot accurately reconstruct complex non-Gaussian structures.

[0006] Therefore, how to break through the underlying barriers of existing technologies within a single framework, achieve deep coupling between nonlinear representation capabilities and multimodal space exploration capabilities, and enable algorithms to simultaneously cope with the combined challenges of high dimensionality, strong nonlinearity, non-Gaussianity, and "different parameters with the same effect" is a key technical problem that urgently needs to be solved in the current field of data assimilation. Summary of the Invention

[0007] Purpose of the invention: To address the technical bottleneck of existing single data assimilation algorithms in simultaneously dealing with the strong non-Gaussian characteristics of groundwater system parameters and the challenge of multiple solutions for "different parameters with the same effect," this invention provides a groundwater model parameter estimation method that combines deep learning and local ensemble updating. Its core innovation lies in breaking through the limitations of traditional assimilation methods in terms of nonlinear characterization accuracy and multi-peak posterior space exploration capabilities. By constructing a deep learning-driven local hybrid updating framework, it achieves efficient and robust characterization of the high-dimensional parameter distribution of complex groundwater systems.

[0008] Technical Solution: To achieve the above objectives, this invention provides a groundwater model parameter estimation method combining deep learning and local set updates, comprising the following steps:

[0009] S1: Based on the hydrogeological conditions of the study area, construct a groundwater numerical model and set the corresponding boundary conditions and physical parameters;

[0010] S2: Based on the geostatistical characteristics and historical observation data of the study area, construct a sample set of prior parameters at the initial time of the iteration;

[0011] S3: Substitute each parameter sample in the prior parameter sample set into the groundwater numerical model and run the forward simulation program to obtain the corresponding model output set;

[0012] S4: Obtain observational data for the study area;

[0013] S5: Determine the number of local sets and the local proportion in the local set update strategy;

[0014] S6: Based on the number of local sets, randomly select local center samples from the prior parameter sample set and calculate the similarity with the prior samples;

[0015] S7: For each local center sample, select the prior sample with the highest similarity according to the local proportion to form a local prior sample set;

[0016] S8: Through a deep learning-based update process, transfer learning is used to update all local prior sample sets one by one to local posterior sample sets.

[0017] S9: Randomly draw posterior samples from all local posterior sample sets to obtain the overall posterior sample set;

[0018] S10: Replace the prior sample set with the posterior sample set, repeat the iterative steps until the maximum number of iterations is reached, and output the parameter estimation results.

[0019] Further, the prior parameter sample set in step S2 is represented as follows: , This represents the physical parameters, and the superscript "(0)" indicates the initial time of the iteration process. This represents the total number of parameter samples contained in the set, which is used to characterize the prior distribution of groundwater model parameters.

[0020] Furthermore, the implementation process of forward simulation in step S3 is as follows:

[0021] A1: Obtain the steady-state hydraulic head by numerically solving the groundwater steady-flow equation. and pore water flow velocity :

[0022]

[0023] as well as

[0024]

[0025] in, In the Cartesian coordinate system, the first... Each spatial coordinate direction For along the first Permeability coefficient in direction Represents the porosity of the aquifer medium;

[0026] A2: Calculate the pollutant concentration in the aquifer by numerically solving the convection-dispersion equation. :

[0027]

[0028] in, For time, The volumetric flow rate per unit volume of the aquifer. The solute concentration of the pollution source. In the Cartesian coordinate system, the first... Each coordinate direction; their product defines the pollutant release intensity. ;Diffusion tensor The calculation formula is as follows:

[0029]

[0030]

[0031]

[0032] in, and These represent longitudinal and lateral dispersion, respectively. and These represent the pore water velocity components along the longitudinal and transverse coordinate directions, respectively. The molecular diffusion coefficient;

[0033] After the above forward modeling calculation, the model output set consisting of the state variables corresponding to each parameter sample is obtained.

[0034] Furthermore, the observation data in step S4 includes groundwater head data or solute concentration data obtained at preset observation well locations and at specific times;

[0035] Based on the accuracy characteristics of the observation instruments, the corresponding observation error covariance matrix is ​​constructed. ;in, The diagonal elements are used to characterize the error variance of the measurement data from each observation well, while the off-diagonal elements are used to characterize the correlation between different observation data.

[0036] Furthermore, in step S5, the number of local sets in the local set update strategy is determined. With local proportions ;

[0037] Local set number Defined as the total number of local sets generated from the prior sample set;

[0038] Local proportions Defined as the ratio of the number of samples contained in a single local set to the total number of prior samples; the number of samples contained in a local set is given by the formula... The calculation shows that, among which This represents the number of samples within the local set. Indicates local proportions, This represents the total number of prior samples. This represents the floor operator.

[0039] Further, step S6 includes:

[0040] Randomly draw from the prior sample set Each sample is used as a local center sample;

[0041] For each local center sample Filter out those with the highest similarity 1 sample to construct the corresponding local prior sample set;

[0042] Similarity is determined by a joint measure of the model output space bias and the parameter space distance: for the In the next iteration, the joint metric is obtained through the following composite cost function. Perform quantification calculations:

[0043]

[0044] in, Used to measure model output Compared with actual observation data The deviation between them, where the matrix This represents the observation error covariance matrix; while The quantization parameter vector is used to quantize local center samples. Compared with prior samples The distance between them, of which Represents the parametric covariance matrix; and They are respectively and The maximum value is used as a normalization factor to ensure that both variance indicators are scaled to the maximum value. Within the range; These are weighting parameters used to adjust the relative weight contributions of the model output spatial distance and the parameter spatial distance in the composite cost function.

[0045] Further, step S7 includes: based on the aforementioned composite cost function The introduction, The smaller the value, the higher the similarity. Therefore, for each local center sample... Its local set is obtained through in Selected from Minimum value The local set is constructed from samples and denoted as . .

[0046] Furthermore, in step S8, the parameter estimation process based on deep learning learns the nonlinear mapping from the innovation vector to the update vector by training a deep neural network model, and adopts a transfer learning strategy to update the local prior set to the local posterior set one by one.

[0047] The definitions of the innovation vector and the update vector originate from the set smoother algorithm based on Kalman update, namely:

[0048]

[0049] in, For the iteration ordinal number, and They represent according to the set and The calculated sample cross-covariance matrix and autocovariance matrix; hyperparameters It is the first The observation error inflation coefficient of the next iteration satisfies the constraint condition. ,in This represents the total number of iterations. This is a random disturbance term that conforms to the distribution of observation error;

[0050] From a functional perspective, an update step based on the Kalman formula is constructed from... arrive The explicit mapping, i.e.:

[0051]

[0052] in, Known as the innovation vector, This is called the update vector. For the first The next iteration uses the mapping operator from the innovation vector to the update vector constructed by the Kalman formula;

[0053] For the first The iteration of the ... For each local set, a deep learning update process based on transfer learning strategy is used to update the local prior set. Update them one by one to the local posterior set .

[0054] Furthermore, the deep learning update process based on transfer learning strategy in step S8 includes:

[0055] B1: Constructing the training dataset: by using local prior sets and Exhaustive pairing and bias calculation of all samples in the dataset generate a model without the need for additional forward modeling. The first set of training data, i.e., the second set of training data The iteration of the ... The training dataset generated by each local set is defined as follows:

[0056]

[0057] in A random sample representing the observation error;

[0058] B2: Model Training: Employing a transfer learning strategy, build or load the initial deep neural network model, and then... The training setup shown learns the nonlinear mapping relationship from the observation data space to the parameter space in the training data, and obtains the nonlinear update operator. ;

[0059] B3: Parameter Update: Calculate the observation innovation vector Inputting this into the trained neural network model yields the parameter update vector: and with the prior local set The update is completed by addition, resulting in the posterior local set. Then, by substituting the model parameters into the numerical model, the local posterior output set can be obtained. .

[0060] Furthermore, the transfer learning strategy in step B2 refers to updating the local sample set only for the first time (i.e., (At that time) A blank deep neural network model was used for training. To be continued in the next update When a local sample set The model obtained from training the previous local dataset is directly loaded. As the initial model, and based on the current training set Hyperparameter fine-tuning and weight updates are performed to achieve knowledge transfer across local sets.

[0061] Furthermore, the implementation of step S9 specifically includes:

[0062] For the results obtained after step S8 update A local posterior sample set is used to reconstruct the global set using a random sampling strategy;

[0063] Specifically, samples are randomly drawn from all local posterior sample sets and merged to reconstruct a set containing... The first sample The global posterior sample set of the next iteration ;in, This is the total number of samples in the initial prior parameter set constructed in step S2, thus ensuring that the size of the parameter set remains consistent before and after iteration.

[0064] Furthermore, the implementation of step S10 specifically includes:

[0065] Determine the current iteration number of the assimilation update. Has the preset maximum number of iterations been reached? :

[0066] like Then the currently obtained global posterior sample set will be... As the initial prior sample set for the next iteration (i.e., assignment replacement), update the iteration step number. Then return to step S5 to re-extract local center samples and construct local sets, starting a new round of iterative updates;

[0067] like If the iteration loop terminates, the current global posterior sample set is used. The output is the final posterior parameter sample set, ending the parameter estimation process.

[0068] Unlike previous approaches that relied solely on global deep learning mapping or traditional local set research methods limited by linear Kalman operators, this invention innovatively proposes a local set coupled with deep learning update strategy. This strategy deeply integrates a local set update mechanism, which possesses strong posterior multimodal exploration advantages and can effectively alleviate the "different parameters with the same effect" problem, with a deep learning operator that has strong nonlinear fitting and non-Gaussian feature restoration capabilities. This ensures that the method of this invention can address the challenge of the simultaneous existence of parameter non-Gaussianity and the "different parameters with the same effect" problem in the process of groundwater parameter estimation.

[0069] Beneficial effects: Compared with the prior art, the unique advantage of this invention lies in: the proposed deep learning-based ensemble smoother (ES) (DL) Algorithms and Iterative Local Update Mechanisms (ILUES) (K) Based on the algorithm, this invention achieves a deep leap and integration of underlying methodologies. This invention fully leverages the advantages of previous ILUES... (K) The mechanism enhances its local update capabilities in exploring multimodal posterior distributions and addressing the "different parameters, same effect" problem, while also inheriting and expanding upon previous ES (Extreme Emissions) mechanisms. (DL) The framework leverages the powerful nonlinear mapping capabilities of deep learning operators to accurately characterize strong non-Gaussian features.

[0070] Through this deep coupling, the local set and deep learning coupled update framework of this invention not only effectively overcomes the limitations of traditional ES... (K) Algorithms and ILUES that simply combine linear operators (K) The algorithm overcomes the performance degradation bottleneck in non-Gaussian scenarios, while also effectively avoiding pure Elasticsearch. (DL)The algorithm significantly improves its ability to stably represent complex multi-solution spaces by addressing the mode collapse that can easily occur during global updates when observation information is extremely scarce.

[0071] Furthermore, this invention delves into the specific impacts of the number and proportion of local sets on algorithm performance and computational cost, striving to find a globally optimal solution that balances high-precision assimilation with controlled computational burden. This is achieved through comparison with Elasticsearch's single-update mechanism. (DL) ILUES (K) and traditional ES (K) The algorithm underwent systematic comparative verification, demonstrating that this invention not only overcomes the most complex and comprehensive challenges arising from the interplay of high dimensionality, strong nonlinearity, non-Gaussian characteristics, and the "different parameters with the same effect" problem, but also exhibits extremely high computational accuracy and broad applicability for single, conventional problems (such as pure non-Gaussian parameter estimation and pure "different parameters with the same effect" problems). This innovative fusion paradigm greatly enhances the robustness of the data assimilation algorithm, marking a further improvement in the applicant's technical system in the field of complex hydrogeological parameter inversion, and providing a highly advanced and reliable comprehensive technical solution for estimating parameters in complex groundwater models. Attached Figure Description

[0072] Figure 1 This is a flowchart of the method of the present invention;

[0073] Figure 2 This is a diagram showing the setup and results analysis of Example 1. Wherein, (a) shows the setup and results analysis from... (a) and (b) are prior samples randomly drawn from ES; (e) are respectively samples drawn from ES. (K) ES (DL) ILUES (K) and ILUES (DL) The obtained posterior samples; (f) is the root mean square error between the model output of each algorithm and the observed values;

[0074] Figure 3 This is a schematic diagram of the setup for Example 2. Wherein, (a) is the setup used for generating data via the direct sampling method. (a) Training images generated by random field implementation; (b) Reference images used in Example 2. (c) Location of observation wells (red circles) and injection / pumping wells (blue lower / upper triangles); (d) Prior information. The mean field of the field;

[0075] Figure 4 The diagram shows the U-Net architecture used in Examples 2 and 3. Specifically, (a) shows the ES architecture in Example 2. (DL) and ILUES (DL) (a) The network used; (b) The network used in Example 3.

[0076] Figure 5 For example, ES in Example 2 (K) ES (DL) ILUES (K) and ILUES (DL) Algorithm estimation results and analysis diagrams. Where (a, d, g, j) represents the estimated average permeability field. (b, e, h, k) represents the posterior time frame. A histogram of field statistics, where the black dashed line indicates the reference permeability coefficient value. ( )and ( (c, f, i, l) represents the standard deviation field of the estimation results. The color scales on the left and right sides correspond to the average values, respectively. Field and standard deviation field.

[0077] Figure 6 A schematic diagram of Example 3 is provided. (a) shows the location of the model observation point; (b) shows the reference channelization of Example 3. (c) shows the prior and reference source intensity parameters, where the blue area represents the distribution of source intensity parameters estimated from 500 prior samples, and the black dots represent reference values; (d) shows the prior... The mean field of the field.

[0078] Figure 7 Example 3 is based on ES (K) ES (DL) ILUES (K) and ILUES (DL) The obtained estimate Field analysis diagram. Where (a–d) represents... The posterior estimate of the mean field is given by (e–h), where (e–h) is the corresponding standard deviation field. ILUES (DL) The results are based on and Configuration.

[0079] Figure 8 For ES-based (K) ES (DL) ILUES (K) and ILUES (DL) Posterior marginal density plots of the estimated eight pollution source parameters. The black vertical lines represent the ground truth, the dashed lines represent the baseline algorithm, and the green solid lines represent ILUES. (DL) All results are based on 10 replicate experiments. (DL) The results are based on and Configuration.

[0080] Figure 9 For example, ILUES in Example 3 (DL) and ILUES (K) Different local proportions The root-mean-square error (RMSE) is set as follows. Where (a) is the estimated RMSE. Field and Reference (a) RMSE between fields; (b) RMSE between model output and observations. Green dots represent ILUES. (DL) The blue dots represent ILUES. (K) The triangle represents the average of 10 repeated trials for each algorithm.

[0081] Figure 10 For ES (K) ES (DL) ILUES (K) and ILUES (DL) Different The root mean square error box plot is set below. Where (a) is... Indicates estimation Field and Reference The RMSE between fields, (b) is This represents the RMSE between the simulated output and the observed values. Each box plot shows the median, interquartile range (25%–75%), minimum and maximum values ​​(excluding outliers), and outliers. The horizontal axis represents... Purple, yellow, blue, and green correspond to ES respectively. (K) ES (DL) ILUES (K) and ILUES (DL) The blue superscript at the top indicates the average value of the box plot. Detailed Implementation

[0082] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0083] Example 1:

[0084] This embodiment aims to verify the groundwater model parameter estimation method (hereinafter referred to as ILUES) proposed in this invention, which combines deep learning and local set updating. (DL)This embodiment examines the effectiveness and robustness of the algorithm in addressing the strong "different parameters with the same effect" problem. To this end, a typical function model with infinitely many peak posterior modes is constructed. This model exhibits significant multiple-solution characteristics, enabling direct and effective quantitative evaluation of the algorithm's performance boundary in overcoming the "different parameters with the same effect" problem.

[0085] Furthermore, to fully verify the advancement of the present invention, the ILUES proposed in this invention will be used in this embodiment and subsequent embodiments. (DL) The algorithm is systematically compared and analyzed with three mainstream benchmark algorithms. The benchmark algorithms specifically include: ES... (K) Algorithms (i.e., traditional global ensemble smoothers based on the Kalman formula, such as the ESMDA algorithm), ES (DL) Algorithms (ensemble smoothers that perform global updates based solely on deep learning), and ILUES (K) Algorithm (a set smoother that combines local set updates with the Kalman formula).

[0086] Based on the above, this embodiment provides a groundwater model parameter estimation method combining deep learning and local set updates, the flowchart of which is as follows. Figure 1 As shown, the specific steps include the following:

[0087] S1: Establish a numerical model for groundwater;

[0088] In this embodiment, a low-dimensional nonlinear model with an infinitely multimodal posterior distribution is used to evaluate the algorithm's ability to solve the "different parameters, same effect" problem. The model is defined as follows:

[0089]

[0090] For this function, given the observation values An infinite number of parameter combinations It can produce the same reasonable model output as... Consistent, among which This represents observation noise. Therefore, the posterior distribution forms a distribution centered at the origin with a radius of... The circle. Accurately recovering the posterior distribution of this circle is a rigorous test of the algorithm's ability to handle "different parameters with the same effect".

[0091] S2: Construct a set of prior parameters;

[0092] In this embodiment, the prior set is composed of Composed of 1 sample, each parameter and Each sample was independently drawn from a uniform distribution. ,like Figure 2 As shown in (a) in the figure.

[0093] S3: Substitute each sample in the prior parameter set into the groundwater numerical model to obtain the corresponding model output;

[0094] Will Group parameter pair Substitute model You can then obtain the corresponding model output.

[0095] S4: Acquire observation data;

[0096] In this embodiment, the observed value is set as With additive Gaussian noise Under these conditions, the posterior parameter distribution approximates a unit circle.

[0097] S5: Determine the number of local sets and the local proportion in the local set update strategy;

[0098] For ILUES (K) and ILUES (DL) Considering the strong "different parameters with the same effect" characteristic in this embodiment, the local ratio is set to And in ILUES (DL) The number of local sets in the middle is specified as .

[0099] S6: Randomly select local center samples from the prior samples and calculate the similarity with the prior samples;

[0100] S7: For each local center sample, select prior samples with high similarity according to the local proportion to form a local prior sample set;

[0101] S8: Through a deep learning-based update process, transfer learning is used to update all local prior sample sets one by one to local posterior sample sets.

[0102] In this embodiment, ES (DL) and ILUES (DL) The deep learning-based updates in this example employ the same network architecture and hyperparameter settings. Specifically, the deep learning model used is a feedforward neural network containing three fully connected (FC) layers and two hyperbolic tangent (Tanh) activation layers, which is well-suited for the bounded posterior parameter space. The architecture can be represented as:

[0103]

[0104] The subscripts indicate the dimension of the corresponding network layer. This neural network contains approximately 626 trainable parameters. To ensure stable convergence of the model training, the entire training process is set to 50 training epochs, using a small learning rate and a mini-batch size of 256 samples.

[0105] S9: Uniformly and randomly draw posterior samples from all local posterior sample sets to obtain the posterior sample set of the population;

[0106] Specifically, since the total number of prior samples is set in this embodiment, And the total number of local sets is To maintain a constant set dimension, the algorithm randomly selects 10 samples in a proportional manner from each local posterior set (i.e., (Number). By merging the sampled subsets, a global posterior sample set is formed for the next iteration.

[0107] S10: Replace the prior sample set with the posterior sample set, and repeat the iterative steps until the maximum number of iterations is reached.

[0108] For all algorithms, i.e., ES (K) ES (DL) ILUES (K) and ILUES (DL) The number of iterations is fixed. .

[0109] Example Result Analysis:

[0110] The posterior distributions of the parameters obtained by the four algorithms are shown in Figure 2 In (b) to (e) of the text.

[0111] The results show that ILUES (K) and ILUES (DL) Both successfully recovered the circular posterior distribution, while ES... (K) and ES (DL) The failure to achieve this strongly demonstrates that local set updates are significantly more effective in addressing the "different parameters, same effect" problem. This embodiment further calculates the root-mean-square error (RMSE) between the model output and the observed values ​​of the estimated parameters, such as... Figure 2 As shown in (f) of ES. (K) ES (DL) ILUES (K) and ILUES (DL)The average RMSE values ​​between model predictions and observed values ​​were 1.6936, 1.3800, 0.0114, and 0.0215, respectively, clearly demonstrating that the two methods using local ensemble updates achieved significantly better prediction accuracy than the global method. (DL) RMSE is slightly higher than ILUES (K) This may be attributed to the fact that the neural network hyperparameters have not been fully optimized. Furthermore, limiting the number of local sets... This limits its ability to explore the characteristics of an infinitely multimodal posterior distribution, causing the posterior distribution to fail to resemble ILUES. (K) That would form a completely continuous unit circle. Nevertheless, it is clear that ILUES benefits from the local set update strategy. (DL) It can effectively solve the problem of "different parameters with the same effect" in problems, even those with infinitely many peaks in the posterior distribution.

[0112] Example 2:

[0113] In Example 1, the local set update strategy was shown to effectively address the non-uniqueness problem caused by multimodal posterior distributions. However, when non-Gaussianity stems from intrinsic properties of the parameters rather than "heterogeneous parameter equivalence," this strategy may no longer be sufficient to accurately characterize the posterior distribution. To test this limitation, this example uses ES... (K) ES (DL) ILUES (K) and ILUES (DL) These algorithms are applied to a non-Gaussian parameter field estimation problem. Specifically, they assimilate sparse head observations to estimate a channel-type permeability coefficient field.

[0114] Based on the above, this embodiment provides a groundwater model parameter estimation method combining deep learning and local set updates, the flowchart of which is as follows. Figure 1 As shown, the specific steps include the following:

[0115] S1: Establish a numerical model for groundwater;

[0116] The study area in this embodiment was designed as a two-dimensional channel-type aquifer, which... and The range in all directions is (L), and is uniformly discretized into The initial head for the transient groundwater flow simulation was uniformly set to 198 (L) throughout the region. Constant head boundaries of 202 (L) and 198 (L) were specified for the left and right boundaries, respectively, while the upper and lower boundaries were assumed to be impermeable. To further enhance groundwater flow and increase the nonlinearity of the problem, two wells were introduced into the region: one with a head of 150 (L). One well injects water at a rate of 1, while another well pumps water at the same rate. Their locations are... Figure 3 In (c), the triangles are marked with downward and upward triangles, respectively.

[0117] S2: Construct a set of prior parameters;

[0118] In this embodiment, ES (K) ES (DL) ILUES (K) and ILUES (DL) The algorithm was used to estimate a head from sparse transient head observations. Permeability field Channel-type penetration rate field parameters Composed of two contrasting media, it exhibits strong non-Gaussian distribution characteristics. The background matrix has a relatively low permeability coefficient. ( River channels, on the other hand, have a much higher permeability coefficient. ( This allows preferential flow to form in these highly permeable paths. This embodiment... The prior samples for the field were all generated using a multi-point geostatistical method, and the corresponding training images are as follows: Figure 3 As shown in (a). Total One sample is used to construct the prior set. Its mean field is as follows Figure 3 As shown in (d).

[0119] S3: Substitute each sample in the prior parameter set into the numerical model to calculate the corresponding model output;

[0120] Based on the above configuration, the MODFLOW groundwater flow model was used to analyze the channel field. Forward modeling is performed to generate transient head distribution within the simulation region. Its governing equations are:

[0121]

[0122] in ( () indicates the water storage rate; (L) represents the study area Spatial location on the grid; (T) represents the simulation time; ( The saturated groundwater flux is calculated using Darcy's law. For gradient operators; ( () represents the source / sink.

[0123] In order to obtain observation data, a total of The observation wells are evenly distributed throughout the area, such as Figure 3 The red circle in (c) indicates this. For each well, in the observation window... Head data were collected at ten evenly spaced time points within the time frame. The specific time points were: , produced Each observation value was analyzed, and additive noise was introduced. This represents measurement error.

[0124] Will By inputting a set of prior samples into the model, we can obtain the output set of the prior model. .

[0125] S4: Acquire observation data;

[0126] Generate reference field like Figure 3 As shown in (b), the observed data can be obtained by substituting it into the model. .

[0127] S5: Determine the number of local sets and the local proportion in the local set update strategy;

[0128] For ILUES (K) With ILUES (DL) In this embodiment, the following is consistently applied: The local scale, and set ILUES (DL) Number of local sets during the training and prediction phases .

[0129] S6: Randomly select local center samples from the prior samples and calculate the similarity with the prior samples;

[0130] S7: For each local center sample, select prior samples with high similarity according to the local proportion to form a local prior sample set;

[0131] S8: Through a deep learning-based update process, transfer learning is used to update all local prior sample sets one by one to local posterior sample sets.

[0132] In this embodiment, ES (DL) and ILUES (DL) The deep learning model used is built on top of the classic U-Net framework, which employs an encoder-decoder architecture (such as...). Figure 4 As shown in (a), from the information vector Information features are extracted and then mapped to the predicted parameter update vector. .enter It is a size of The tensor contains information arranged according to observation location and time step. The network outputs a tensor of size [size missing]. The tensor represents the permeability field. The corresponding updates. In the encoder, the network first reduces the spatial resolution of the input from [previous level] through transposed convolutions with ReLU activation and subsequent standard convolutional layers. Expand to Meanwhile, the number of feature channels is increased to 64. As the encoder deepens, the spatial dimension gradually decreases (e.g., , The channel depth is increased (up to 512), enabling the extraction of increasingly abstract and high-level features. To enhance robustness and reduce overfitting, dropout layers with a dropout rate of 30% are applied to selected layers. In this U-Net framework, the encoder performs a total of three downsampling operations. The decoder then upsamples the feature maps to restore spatial resolution while gradually reducing the number of channels, ultimately generating... The output. The skip connections between the corresponding encoder and decoder layers ( Figure 4 The green line in the diagram helps preserve information at a fine scale and significantly improves the accuracy of predicting permeability coefficient field updates.

[0133] Model training was performed using the DL toolbox in MATLAB R2024a. For ES... (DL) The network was trained using the Adam optimizer with a learning rate of 0.001, a mini-batch size of 512, and a maximum training epoch count of 50. All other training settings followed the toolbox's default configuration. For ILUES... (DL) A total of [number] were used in the training and prediction phases. For each local set, transfer learning techniques were employed to reduce computational costs. Specifically, the initial training used methods similar to ES... (DL) The same hyperparameters were used, but subsequent training (from the 2nd to the 50th iteration) used a reduced learning rate of 0.0005, a mini-batch size of 256, and 20 training epochs.

[0134] S9: Uniformly and randomly draw posterior samples from all local posterior sample sets to obtain the posterior sample set of the population;

[0135] Specifically, since the total number of prior samples is set in this embodiment, And the total number of local sets is To maintain a constant set dimension, the algorithm randomly selects 10 samples in a proportional manner from each local posterior set (i.e., By merging the sampled subsets, a global posterior sample set is formed for the next iteration.

[0136] S10: Replace the prior sample set with the posterior sample set, and repeat the iterative steps until the maximum number of iterations is reached.

[0137] Since the nonlinearity of this embodiment is relatively weak, all four algorithms in this embodiment only perform one iteration.

[0138] Example Result Analysis:

[0139] Figure 5 This case study demonstrates the ES (K) ES (DL) ILUES (K) and ILUES (DL) The estimation results. Figure 5 (a), (d), (g), and (j) show the average permeability fields inferred by the four algorithms. The results show that all four methods capture the non-Gaussian characteristics of the channel-type permeability coefficient field to some extent, but their accuracy varies. In particular, ES... (K) and ILUES (K) The obtained estimation results exhibit significant boundary blurring, structural discontinuities, and morphological distortions. In contrast, ES... (DL) and ILUES (DL) The obtained channel patterns exhibit higher clarity and spatial continuity, and better consistency with the reference field. This suggests that deep learning-based updates can extract more effectively. The field exhibits non-Gaussian characteristics while maintaining spatial correlation and structural integrity. Given the two inherent field properties of the channel-type permeability coefficient field, the posterior distribution is expected to... ( )and ( Two distinct peaks are observed at point ) . Figure 5 As shown in (b), (e), (h), and (k) (where the black dashed lines represent these reference values), the bimodal structure of the posterior distribution characteristics recovered using a deep learning-based update method is more pronounced. It is worth noting that the newly proposed ILUES... (DL) Generated more than ES (DL) Channels with clearer boundaries This more accurately characterizes the and The bimodal nature centered on this. Furthermore... Figure 5 The standard deviation fields in (c), (f), (i), and (l) show that ES (DL) and ILUES (DL)The resulting posterior uncertainty is lower than that of similar Kalman-based methods. This reduction in uncertainty further demonstrates the advantages of deep learning-based update schemes in capturing non-Gaussian features and improving inversion performance.

[0140] Example 3:

[0141] Examples 1 and 2 have demonstrated that both local set updates and deep learning-based update strategies are effective in addressing challenges arising from heterogeneity and non-Gaussianity. By integrating these two strategies, ILUES... (DL) Superior performance was achieved in Examples 1 and 2. This example will explore a more comprehensive and challenging scenario that involves high dimensionality, strong nonlinearity, parameter finality, and non-Gaussian parameter estimation problems. Specifically, this example focuses on the joint estimation of channel-type permeability coefficient fields and pollution source parameters, and systematically examines the sensitivity of the method of the present invention to different configurations in this case.

[0142] Based on the above, this embodiment provides a groundwater model parameter estimation algorithm that combines deep learning and local set updates, the flowchart of which is as follows: Figure 1 As shown, the specific steps include the following:

[0143] S1: Establish a numerical model for groundwater;

[0144] In this embodiment, a size of This paper investigates the coupling problem of steady-state groundwater flow and transient pollutant transport within the (L) region, and constructs a groundwater model to address this issue. The model uses impermeable boundaries at the top and bottom of the confined aquifer, with an initial hydraulic head of 11(L) for the entire region. To establish the hydraulic gradient, constant head boundaries of 12(L) and 11(L) are applied to the left and right boundaries, respectively. The aquifer is characterized as a non-Gaussian channel field system and discretized into... The grid, where each grid cell is assigned a permeability coefficient field. The values ​​are . The permeability coefficients of the aquifer matrix and the channel are set as . and In this embodiment, a pollution source with varying intensity over time is introduced into the steady-state groundwater flow field, and it is parameterized using eight variables. Specifically, the location of the source is determined by coordinates... (L) definition, in six consecutive time periods ( The mass loading rate within ) is then used ( This indicates that, based on the above configuration, this embodiment involves a total of 3,329 adjustable model parameters, namely... The remaining model parameters are determined by geological surveys or experimental data. Inputting these parameters into the numerical model allows for the simulation of steady-state groundwater flow fields and transient pollutant concentration fields.

[0145] S2: Construct a set of prior parameters;

[0146] Based on step S1, generate a process containing The set of prior parameters for each independent sample To characterize the prior uncertainty in the data assimilation framework. All penetration fields All were generated using the direct sampling algorithm, and their training images are as follows: Figure 3 Similar to (a), but the size of the region shown in the image is adjusted to... (L). Figure 6 Figure (d) shows the mean field of the permeability fields of all prior samples. The pollution source parameters are drawn from a uniform distribution within the specified boundaries, where... Figure 6 The blue area in (c) shows the prior distribution of pollution source intensity parameters, as detailed in Table 1.

[0147] Table 1. Prior distribution and reference values ​​of eight pollution source parameters

[0148]

[0149] S3: Substitute each sample in the prior parameter set into the numerical model to calculate the corresponding model output;

[0150] The set of prior parameters obtained in step S2 The 3,329 parameters of each sample were substituted into the groundwater numerical model constructed in step S1, and the forward simulation program was run to obtain the corresponding model output set. Specifically, by inputting pollution source parameters and permeability field parameters, and running the MODFLOW and MT3DMS numerical model software, the steady-state groundwater model and transient pollutant concentration distribution field are calculated.

[0151] Observational data collected from A uniformly distributed monitoring well, such as Figure 2 The data is shown in red circles in (a). Specifically, the dataset contains data at 9 time steps (a). A total of 32 concentration measurements were obtained from the data. 320 concentration values ​​and 32 head measurements were collected. Gaussian random noise was added to all 320 observations to account for measurement uncertainties. .

[0152] Forward simulation is implemented through the following process:

[0153] (1) The steady-state head is obtained by numerically solving the groundwater steady-flow equation. and pore water flow velocity ( ):

[0154]

[0155] as well as

[0156]

[0157] in In the Cartesian coordinate system, the first... Each spatial coordinate direction For along the first Permeability coefficient in direction (e.g., horizontal and vertical). (-) represents the porosity of the aquifer medium.

[0158] (2) The concentration of pollutants in the aquifer is calculated by numerically solving the convection-dispersion equation. ( ):

[0159]

[0160] in (T) represents time. ( () represents the volumetric flow rate per unit volume of the aquifer. ( () represents the solute concentration of the pollution source. In the Cartesian coordinate system, the first... Each has a spatial coordinate direction. The product of these two factors defines the pollutant mass loading rate. ( diffusion tensor ( The calculation formula for ) is as follows:

[0161]

[0162]

[0163]

[0164] in (L) and (L) represents longitudinal and lateral dispersion, respectively. and These represent the pore water velocity components along the longitudinal and transverse coordinate directions, respectively. This indicates that molecular diffusion is not considered.

[0165] After the above forward modeling calculation, the model output set is obtained, which consists of the state variables corresponding to the 3,329 parameters of all prior samples, and contains 320 observation data.

[0166] S4: Acquire observation data;

[0167] For the inversion experiment, a randomly selected sample is chosen as a reference case to generate observational data. Its parameters are considered as reference values ​​for evaluating the accuracy of the algorithm's parameter estimation. The reference permeability field is as follows: Figure 6 As shown in (b), the reference pollution source parameters are randomly selected from within the defined boundaries. Specifically, the reference location of the pollution source is determined by... Figure 6 The red triangle in (b) indicates that the reference value for the emission intensity of the pollution source is... Figure 6 The black dots in (c) represent [data points]. Detailed data can be found in Table 1. Substituting the reference parameters into the numerical model yields the observed data, denoted as [data point name]. .

[0168] S5: Determine the number of local sets and the local proportion in the local set update strategy;

[0169] Given the high complexity of the groundwater parameter inversion scenario set in this embodiment, this section systematically tests the impact of setting key hyperparameters (i.e., the number of local sets and the local proportion) in the local set update strategy on the overall performance of the algorithm. Specifically, sensitivity analysis experiments were conducted for different local proportions and local set number configurations. The relevant performance evolution patterns and detailed comparative test results will be fully described in the subsequent results analysis section.

[0170] S6: Randomly select local center samples from the prior samples and calculate the similarity with the prior samples;

[0171] S7: For each local center sample, select prior samples with high similarity according to the local proportion to form a local prior sample set;

[0172] S8: Through a deep learning-based update process, transfer learning is used to update all local prior sample sets one by one to local posterior sample sets.

[0173] Similar to Example 2, ES (DL) and ILUES (DL) They all use the same deep learning model architecture. For example... Figure 4 As shown in (b), the U-net structure is used to learn from the innovation vector. To update vector Nonlinear mapping. Input The dimension is The encoder gradually extracts deeper features from the observed data, increasing the number of channels to 64 at the expense of spatial resolution. The decoder then reconstructs the features into a final dimension of... of Meanwhile, skip connections between the corresponding encoder and decoder layers help preserve local structural information. The entire network contains 44 layers and approximately 33.1 million trainable parameters. The number of iterations is set to... For ES (DL) For each update step, the network is trained for 150 epochs with a learning rate of 0.0001, a mini-batch size of 512, and a learning rate of 0.001. Regularization factors are used to mitigate overfitting. However, for ILUES... (DL) The use of local sets significantly reduces the number of training samples in each iteration. To compensate for this limitation, this algorithm introduces a transfer learning mechanism during the update process. Specifically, during the first iteration of each update, the learning rate is set to 0.00005, the mini-batch size is fixed at 256, and the maximum number of training epochs is set to 100. After applying the transfer learning strategy, the maximum number of training epochs is further reduced to 30 to reduce computational costs.

[0174] S9: Uniformly and randomly draw posterior samples from all local posterior sample sets to obtain the posterior sample set of the population;

[0175] Specifically, since the total number of prior samples is set in this embodiment, And the total number of local sets is To maintain a constant set dimension, the algorithm randomly selects from each local posterior set. A set of samples. By merging the sampled subsets, a global posterior sample set is formed for the next iteration.

[0176] S10: Replace the prior sample set with the posterior sample set, and repeat the iterative steps until the maximum number of iterations is reached.

[0177] In this embodiment, the number of iterations is set to .

[0178] Example Result Analysis:

[0179] This embodiment focuses on a comparative analysis of ES based on traditional global updates. (K) With ES (DL) Algorithms, and ILUES based on local set updates (K) Compared with the ILUES proposed in this invention (DL) The algorithm's actual performance in solving the problem of multiple solutions ("different parameters with the same effect") and the coexistence of non-Gaussian parameters is analyzed, and the key hyperparameter (local scaling) is examined. Number of local sets Sensitivity tests were conducted.

[0180] Figure 7 Demonstrates ES-based (K) ES (DL) ILUES (K) and ILUES (DL) The obtained estimate The mean and its associated uncertainty (characterized by the standard deviation, where ILUES) (DL) based on and (Configuration). All four methods can restore the mean. The main channel structure in the field; however, due to the inherent complexity of the inversion problem and the limited quantity and quality of available observational data, no algorithm can perfectly reconstruct the complete channel distribution. Specifically, of the 320 observations, only 68 values ​​are greater than 1, and the average of the remaining observations is only 0.0497. This low-level observational data is highly susceptible to measurement noise, limiting the amount of information it provides. Figure 7 The posterior standard deviation fields (e) to (h) show that the uncertainty is significantly lower near the pollution source release area and along the main transport path, reflecting the availability of more informative observational data constraints in these areas. It is worth noting that ILUES... (DL) The results showed a lower overall standard deviation, indicating that the estimation results have higher consistency and stability in complex non-Gaussian scenarios.

[0181] Figure 8 This paper presents marginal posterior density estimates of eight pollution source parameters obtained after 10 consecutive replicate experiments using four methods. Compared to the prior range, the posterior density curves of all methods converge near the true values ​​(black vertical lines), significantly reducing parameter uncertainty. ILUES is one such method. (K) This yielded the most concentrated and nearly unbiased estimate, ES. (K) and ES (DL) It produced similar but slightly inferior results. In comparison, ILUES achieved the smallest root mean square error among all algorithms. (DL) The distribution exhibits a highly pronounced multimodal nature in its approximate posterior distribution (especially for pollution source location parameters). This multimodal behavior profoundly reflects the high-dimensional, strongly non-Gaussian nature of the joint inversion problem. The "different parameters, same effect" nature arising from the simultaneous estimation of field and pollution source parameters. Based on the Kalman linear update operator (ES... (K) and ILUES (K) Due to inherent statistical limitations, these potentially reasonable parameter combinations are obscured; while pure deep learning (ES) (DL)While it can alleviate the non-Gaussian effect, it still fails to fully characterize this multimodal posterior structure. Thanks to the organic combination of deep learning and local set updates, the ILUES of this invention... (DL) The algorithm not only successfully captured the non-Gaussianity, but also comprehensively revealed the multi-peak characteristics of the parameter distribution. This ability to retain multiple reasonable equivalent solutions is of great value for actual risk assessment and decision-making in groundwater research.

[0182] In ILUES (K) and ILUES (DL) In the middle, local proportions Controls the proportion of the local set relative to the global sample. Smaller It helps improve the exploration of multimodal posterior distributions; however, it also reduces the sample size of local ensembles, thus affecting ILUES. (K) Reliable estimation of set statistics and ILUES (DL) The training performance of deep learning models is negatively affected. To systematically evaluate this trade-off, 10 repeated experiments were conducted within the range of 0.10 to 0.40 with a step size of 0.05. Figure 9 As shown, with The increase, ILUES (DL) exist Both field estimation and data fitting showed slight but consistent performance improvements. Conversely, for ILUES... (K) ,when When the value exceeds 0.20, the root mean square error of the data matching gradually increases and exhibits significant volatility in repeated experiments. Overall comparison shows that across all tested... Under these conditions, the ILUES of the present invention (DL) The algorithm is always better than ILUES (K) Producing more accurate The field estimation demonstrated extremely strong parameter robustness.

[0183] Local set number This will also directly affect ILUES (DL) Assimilation performance and computational efficiency. Due to the large This means that more training iterations of the deep neural network are required, thus increasing computational costs. For example... Figure 10 As shown, for ILUES (DL) exist Tests were conducted and evaluated at different settings ranging from 10 to 100. The results showed that increasing... It will make ILUES (K) and ILUES (DL) The estimation accuracy was moderately improved due to the larger It allows for a more thorough exploration of the entire parameter space. However, when When the number of local sets is greater than or equal to 30, the overall performance improvement of both algorithms tends to level off and stabilize. This fully demonstrates that limiting the number of local sets to a recommended value (such as...) =50), which can significantly reduce ILUES with almost no sacrifice in assimilation accuracy. (DL) The training and computation costs are minimized, ensuring the efficiency and feasibility of the method in practical large-scale engineering applications.

Claims

1. A method for estimating parameters of a groundwater model combining deep learning and local set updates, characterized in that, Includes the following steps: S1: Based on the hydrogeological conditions of the study area, construct a groundwater numerical model and set the corresponding boundary conditions and physical parameters; S2: Based on the geostatistical characteristics and historical observation data of the study area, construct a sample set of prior parameters at the initial time of the iteration; S3: Substitute each parameter sample in the prior parameter sample set into the groundwater numerical model and run the forward simulation program to obtain the corresponding model output set; S4: Obtain observational data for the study area; S5: Determine the number of local sets and the local proportion in the local set update strategy; S6: Based on the number of local sets, randomly select local center samples from the prior parameter sample set and calculate the similarity with the prior samples; S7: For each local center sample, select the prior sample with the highest similarity according to the local proportion to form a local prior sample set; S8: Through a deep learning-based update process, transfer learning is used to update all local prior sample sets one by one to local posterior sample sets. S9: Randomly draw posterior samples from all local posterior sample sets to obtain the overall posterior sample set; S10: Replace the prior sample set with the posterior sample set, repeat the iterative steps until the maximum number of iterations is reached, and output the parameter estimation results.

2. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 1, characterized in that, The prior parameter sample set in step S2 is represented as follows: , This represents the physical parameters, and the superscript "(0)" indicates the initial time of the iteration process. This represents the total number of parameter samples contained in the set.

3. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 2, characterized in that, The forward simulation process in step S3 is as follows: A1: Obtain the steady-state hydraulic head by numerically solving the groundwater steady-flow equation. and pore water flow velocity : ； as well as ； in, In the Cartesian coordinate system, the first... Each spatial coordinate direction For along the first Permeability coefficient in direction Represents the porosity of the aquifer medium; A2: Calculate the pollutant concentration in the aquifer by numerically solving the convection-dispersion equation. : ； in, For time, The volumetric flow rate per unit volume of the aquifer. The solute concentration of the pollution source. In the Cartesian coordinate system, the first... Each coordinate direction; their product defines the pollutant release intensity. ;Diffusion tensor The calculation formula is as follows: ； ； ； in, and These represent longitudinal and lateral dispersion, respectively. and These represent the pore water velocity components along the longitudinal and transverse coordinate directions, respectively. The molecular diffusion coefficient; After the above forward modeling calculation, the model output set consisting of the state variables corresponding to each parameter sample is obtained.

4. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 3, characterized in that, The observation data in step S4 includes groundwater head data or solute concentration data obtained at preset observation well locations and at specific times. Based on the accuracy characteristics of the observation instruments, the corresponding observation error covariance matrix is ​​constructed. ;in, The diagonal elements are used to characterize the error variance of the measurement data from each observation well, while the off-diagonal elements are used to characterize the correlation between different observation data.

5. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 4, characterized in that, In step S5, the number of local sets in the local set update strategy is determined. With local proportions ; Local set number Defined as the total number of local sets generated from the prior sample set; Local proportions Defined as the ratio of the number of samples contained in a single local set to the total number of prior samples; the number of samples contained in a local set is given by the formula... The calculation shows that, among which This represents the number of samples within the local set. Indicates local proportions, This represents the total number of prior samples. This represents the floor operator.

6. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 5, characterized in that, Step S6 includes: Randomly draw from the prior sample set Each sample is used as a local center sample; For each local center sample Filter out those with the highest similarity 1 sample to construct the corresponding local prior sample set; Similarity is determined by a joint measure of the model output space bias and the parameter space distance: for the In the next iteration, the joint metric is obtained through the following composite cost function. Perform quantification calculations: ； in, Used to measure model output Compared with actual observation data The deviation between them, where the matrix This represents the observation error covariance matrix; while The quantization parameter vector is used to quantize local center samples. Compared with prior samples The distance between them, of which Represents the parametric covariance matrix; and They are respectively and The maximum value of is used as a normalization factor; These are weighting parameters used to adjust the relative weight contributions of the model output spatial distance and the parameter spatial distance in the composite cost function.

7. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 6, characterized in that, Step S7 includes: for each local center sample Its local set is obtained through in Selected from Minimum value The local set is constructed from samples and denoted as . .

8. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 7, characterized in that, In step S8, the parameter estimation process based on deep learning learns the nonlinear mapping from the innovation vector to the update vector by training a deep neural network model, and adopts a transfer learning strategy to update the local prior set to the local posterior set one by one. The definitions of the innovation vector and the update vector originate from the set smoother algorithm based on Kalman update, namely: ； in, For the iteration ordinal number, and They represent according to the set and The calculated sample cross-covariance matrix and autocovariance matrix; hyperparameters It is the first The observation error inflation coefficient of the next iteration satisfies the constraint condition. ,in This represents the total number of iterations. This is a random disturbance term that conforms to the distribution of observation error; From a functional perspective, an update step based on the Kalman formula is constructed from... arrive The explicit mapping, i.e.: ； in, Known as the innovation vector, This is called the update vector. For the first The next iteration uses the mapping operator from the innovation vector to the update vector constructed by the Kalman formula; For the first The iteration of the ... For each local set, a deep learning update process based on transfer learning strategy is used to update the local prior set. Update them one by one to the local posterior set .

9. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 8, characterized in that, The deep learning update process based on transfer learning strategy in step S8 includes: B1: Constructing the training dataset: by using local prior sets and Exhaustive pairing and bias calculation of all samples in the dataset generate a model without the need for additional forward modeling. The first set of training data, i.e., the second set of training data The iteration of the ... The training dataset generated by each local set is defined as follows: ； in A random sample representing the observation error; B2: Model Training: Employing a transfer learning strategy, build or load the initial deep neural network model, and then... The training setup shown learns the nonlinear mapping relationship from the observation data space to the parameter space in the training data, and obtains the nonlinear update operator. ; B3: Parameter Update: Calculate the observation innovation vector Inputting this into the trained neural network model yields the parameter update vector: and with the prior local set The update is completed by addition, resulting in the posterior local set. Then, by substituting the model parameters into the numerical model, the local posterior output set can be obtained. .

10. The groundwater model parameter estimation method combining deep learning and local set updating according to claim 9, characterized in that, In step B2, the transfer learning strategy refers to using a blank deep neural network model for training only during the first update of the local sample set. To be continued in the next update When using a local sample set, directly load the model trained on the previous local set. As the initial model, and based on the current training set Hyperparameter fine-tuning and weight updates are performed to achieve knowledge transfer across local sets.

Images