A multi-front multi-target grey wolf optimization method for groundwater pollution monitoring

Abstract

This application provides a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring, relating to the field of wastewater well monitoring. The method includes: acquiring pollutant concentration data for each monitoring well at different times and performing preprocessing and parameter initialization to obtain initial individuals; based on the initial individuals, executing the multi-frontier, multi-objective gray wolf optimization algorithm to output a Pareto optimal solution set; and selecting a monitoring well layout scheme that meets engineering requirements based on the output Pareto optimal solution set. The technical solution of this application improves the stability and solution quality of multi-objective optimization solutions.

Description

Technical Field

[0001] This application relates to the field of wastewater well monitoring, and in particular to a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring. Background Technology

[0002] Groundwater is a vital basic water resource, widely serving industry, agriculture, and residential life. With the continuous expansion of industrial activities and urban construction, groundwater pollution problems are becoming increasingly prominent due to chemical production, solid waste storage, and leaks from underground facilities. Groundwater pollution is characterized by its high degree of concealment, complex diffusion processes, and long remediation cycles. If it cannot be identified and effectively monitored in a timely manner, it can easily have long-term impacts on ecological environment safety and public health.

[0003] Deploying monitoring wells to monitor groundwater pollution is a primary technical means of obtaining groundwater quality information, identifying the scope of pollution, and judging the evolution trend of pollution. The deployment plan directly affects the representativeness and effectiveness of the monitoring data. Under limited construction and operating costs, how to rationally deploy monitoring wells to improve the efficiency of pollution identification and monitoring has become a key technical issue in the field of groundwater environmental management and pollution prevention.

[0004] In existing technologies, the problem of groundwater pollution monitoring well deployment is typically abstracted as a multi-objective optimization problem. Its objective functions are usually maximizing monitoring information and minimizing monitoring costs, with decision variables primarily including the number of monitoring wells and their spatial location. To address this type of problem, researchers have proposed and applied various intelligent optimization algorithms, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Non-Dominated Sorting Genetic Algorithm (NSGA-II), and Multi-Objective Grey Wolf Optimization Algorithm (MOGWO).

[0005] Among them, the gray wolf optimization algorithm, by simulating the social hierarchy and hunting behavior of gray wolf packs, has the advantages of few parameters, simple implementation, and strong global search capabilities. The multi-objective gray wolf optimization algorithm, by introducing mechanisms such as non-dominated sorting and crowding distance, can simultaneously handle multiple conflicting optimization objectives and has been used to solve problems such as engineering site selection, resource allocation, and monitoring network optimization.

[0006] However, existing multi-objective gray wolf optimization algorithms still have the following shortcomings in high-dimensional and complex constraint application scenarios such as the deployment of groundwater pollution monitoring wells: (1) The Pareto solution set is unevenly distributed. Existing algorithms usually rely on a single Pareto front or a limited number of leader individuals to guide the search process, which easily leads to the solution set being concentrated in a local area, making it difficult to fully cover the Pareto front and reducing the decision reference value of the optimization results; (2) The search process is prone to premature convergence. When the solution space dimension is high and the objective conflict is significant, the search ability of traditional multi-objective gray wolf optimization algorithms is insufficient, and it is easy to converge to the local optimum solution too early, affecting the overall performance of the final deployment scheme; (3) The adaptability to complex monitoring well deployment problems is limited. Groundwater pollution monitoring well deployment problems often involve multiple decision factors such as the number of wells, spatial location, and operation strategy. The optimization problem presents high-dimensional, nonlinear, and strongly constrained characteristics. When dealing with such problems, the search efficiency and solution set quality of existing algorithms still need to be improved. Summary of the Invention

[0007] The purpose of this invention is to address the problems of uneven Pareto solution set distribution, easy getting trapped in local optima, and insufficient adaptability to complex decision variables in existing multi-objective gray wolf optimization algorithms when applied to high-dimensional and complex constraint scenarios such as groundwater monitoring well deployment. This invention provides a multi-frontier multi-objective gray wolf optimization method for groundwater pollution monitoring.

[0008] The above-mentioned objective of this application is achieved through the following technical solution: Step S1: Obtain pollutant concentration data for each monitoring well at different times, perform preprocessing and parameter initialization to obtain initial data; Step S2: Based on the initial individuals, execute the multi-front, multi-objective gray wolf optimization algorithm to output the Pareto optimal solution set; Step S3: Based on the output Pareto optimal solution set, select a monitoring well deployment scheme that meets the engineering requirements.

[0009] Optionally, step S1 includes: S11: Determine the feasible domain for hydrogeological parameters and pollution source parameters, including: aquifer permeability coefficient, pollutant discharge rate, and water discharge rate; S12: The input sample set is generated using Latin hypercube sampling and divided into a training set and a validation set; S13: Input the input sample into the numerical simulation model of groundwater solute transport to obtain pollutant concentration data of each monitoring well at different times; S14: Set the core parameters of the multi-objective gray wolf optimization algorithm, including population size, maximum number of iterations, archive capacity, and number of grid cells in the objective dimension; S15: Define three-dimensional decision variables, including monitoring well location, number of new wells, and monitoring frequency, and use binary encoding; S16: Randomly generate the initial population within the feasible region of the decision variables, thus obtaining the initial individuals.

[0010] Optionally, step S2 includes: S21: Calculate the fitness value of each initial individual, including the objectives of maximizing monitoring information and minimizing monitoring costs; S22: Perform non-dominated sorting of all individuals in the population based on fitness values, and establish a dual archiving mechanism to store the first non-dominated Pareto front solution and the second non-dominated Pareto front solution respectively. S23: The leader wolf is selected from two archives using a roulette wheel selection strategy. The α wolf comes from the first non-dominated Pareto front solution, and the β and δ wolves come from the second non-dominated Pareto front solution. S24: Design a nonlinear convergence factor to dynamically adjust the algorithm's search strategy; S25: Update the positions of other individuals in the population based on the position of the leader wolf; S26: A resolution-based solution archiving mechanism filters archived solutions to ensure a uniform distribution of the solution set; S27: Determine convergence based on the number of iterations or the IGD value, and output the Pareto optimal solution set.

[0011] Optionally, step S21 includes: The objective function for maximizing the amount of monitoring information is:

[0012] in, The number of monitoring wells; In order to monitor frequency Downhole monitoring The pollutant concentration sequence; Concentration sequence The marginal entropy represents The amount of information it contains; Concentration sequence and Mutual information represents the amount of information transmitted between preferred and non-preferred wells; This represents a 0-1 control variable, calculating only the mutual information between preferred and non-preferred wells; This indicates taking the maximum value; Represents a three-dimensional combination vector of decision variables; The objective function for minimizing the monitoring cost is:

[0013] in, This indicates taking the minimum value; The cost of deploying each monitoring well. For sampling and monitoring costs; For monitoring frequency The corresponding number of monitoring sessions; Indicates the first Potential monitoring well.

[0014] Optionally, step S24 includes: The nonlinear convergence factor The calculation formula is:

[0015] in, This represents the current iteration number. It is the square of the maximum number of iterations.

[0016] Optionally, step S26 includes: The resolution-based dearchive mechanism includes: The range of values ​​for the objective function is divided into a fixed grid, with each objective dimension divided into 10 grids; the objective function values ​​include: the amount of monitoring information. and monitoring costs ; For each non-dominated Pareto front solution, map it to the lower left reference point of the corresponding grid and calculate the Euclidean distance; Based on Euclidean distance, only the solution with the smallest distance is retained within the same grid, and the rest are discarded.

[0017] An electronic device includes a processor, a memory, a user interface, and a network interface. The memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory to cause the electronic device to perform a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring.

[0018] A computer-readable storage medium storing instructions that, when executed, perform a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring.

[0019] The beneficial effects of the technical solution provided in this application are: High uniformity of solution set distribution: The innovative dual-archiving and multi-frontier guidance mechanism takes the α wolf from the globally optimal solution library and the β and δ wolves from the suboptimal solution library. Combined with the resolution-based solution archiving strategy, the Pareto solution set fully covers the target space and has higher decision reference value.

[0020] Superior global search performance: The algorithm enhances its exploration capabilities in the early stages of the search through a nonlinear convergence strategy and strengthens its local exploration capabilities in the later stages, reducing the risk of the algorithm getting trapped in local optima.

[0021] Adaptable to complex deployment problems: Supports three-dimensional decision variables of "quantity-location-frequency", which can effectively handle multi-dimensional decision variables and complex constraints involved in the deployment of groundwater pollution monitoring wells, and is suitable for high-dimensional and nonlinear monitoring well optimization problems. Attached Figure Description

[0022] The present application will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a flowchart of an embodiment of this application; Figure 2 This is a diagram of the MPMOOGWO algorithm in the embodiments of this application; Figure 3 This is a curve showing the change of the convergence factor in the embodiments of this application; Figure 4 This is a schematic diagram of dearchiving in the embodiments of this application; Figure 5 This is a graph showing the change of HV index of all algorithms in the embodiments of this application with the number of iterations; Figure 6 This is a graph showing the change of IGD metrics of all algorithms in the embodiments of this application with the number of iterations; Figure 7 This is a schematic diagram of the electronic device structure in the embodiments of this application. Detailed Implementation

[0023] To provide a clearer understanding of the technical features, objectives, and effects of this application, the specific embodiments of this application will now be described in detail with reference to the accompanying drawings.

[0024] The embodiments of this application provide a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring.

[0025] Please refer to Figure 1 , Figure 1 This is a flowchart of a multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring, as described in an embodiment of this application, including: Step S1: Obtain pollutant concentration data for each monitoring well at different times, perform preprocessing and parameter initialization to obtain initial data; Step S2: Based on the initial individuals, execute the multi-front, multi-objective gray wolf optimization algorithm to output the Pareto optimal solution set; Step S3: Based on the output Pareto optimal solution set, select a monitoring well deployment scheme that meets the engineering requirements.

[0026] Step S1 includes: S11: Determine the feasible domain for hydrogeological parameters and pollution source parameters, including: aquifer permeability coefficient, pollutant discharge rate, and water discharge rate; S12: The input sample set is generated using Latin hypercube sampling and divided into a training set and a validation set; S13: Input the input sample into the numerical simulation model of groundwater solute transport to obtain pollutant concentration data of each monitoring well at different times; S14: Set the core parameters of the multi-objective gray wolf optimization algorithm, including population size, maximum number of iterations, archive capacity, and number of grid cells in the objective dimension; S15: Define three-dimensional decision variables, including monitoring well location, number of new wells, and monitoring frequency, and use binary encoding; S16: Randomly generate the initial population within the feasible region of the decision variables, thus obtaining the initial individuals.

[0027] As one example, the core parameters are set as follows: population size of 50, maximum number of iterations. =100, archive capacity is 30, leader selection pressure parameter c=4, target dimension grid number is 10. Population encoding: Each individual corresponds to the three-dimensional decision variable combination of "monitoring well location - number of new additions - monitoring frequency", and the monitoring well location is represented by binary encoding ( , =1 indicates the deployment of the first... Potential monitoring wells), the number of new wells meets the requirements. ≤10, with a monitoring frequency ranging from 7 to 12 times per year (compliant with the requirements of the "Technical Specification for Groundwater Environmental Monitoring"). Among these: The number of monitoring wells includes existing and newly added monitoring wells; initial population generation: 50 individuals are randomly generated within the feasible region of the decision variables to ensure that the initial population covers the key areas of the solution space, laying the foundation for subsequent searches.

[0028] Step S2 includes: S21: Calculate the fitness value of each initial individual, including the objectives of maximizing monitoring information and minimizing monitoring costs; S22: Perform non-dominated sorting of all individuals in the population based on fitness values, and establish a dual archiving mechanism to store the first non-dominated Pareto front solution and the second non-dominated Pareto front solution respectively. As one example, non-dominated ranking: all individuals in the population are ranked according to their bi-objective fitness values ​​to identify the dominance relationship between individuals—if individual A is better than individual B in both objectives, then A dominates B; if the two individuals each have an advantage in different objectives, then they are in a non-dominant relationship.

[0029] As one example, the dual archive partitioning is implemented: the first non-dominated Pareto front (the set of optimal non-dominated solutions) obtained after sorting is stored in Archive1 as a global optimal solution candidate library; after clearing the first front solution, the second non-dominated Pareto front (the set of suboptimal non-dominated solutions) obtained by re-sorting the remaining individuals is stored in Archive2 as a differentiated guided solution candidate library, thereby realizing the independent storage and management of the two types of front solutions.

[0030] As one example, the archive initialization verification is as follows: if the number of solutions in the initial archive exceeds the capacity limit (30), the solution archiving mechanism is immediately triggered to filter and ensure the validity of the archived solutions.

[0031] S23: The leader wolf is selected from two archives using a roulette wheel selection strategy. The α wolf comes from the first non-dominated Pareto front solution, and the β and δ wolves come from the second non-dominated Pareto front solution. S24: Design a nonlinear convergence factor to dynamically adjust the algorithm's search strategy; S25: Update the positions of other individuals in the population based on the position of the leader wolf; S26: A resolution-based solution archiving mechanism filters archived solutions to ensure a uniform distribution of the solution set; S27: Determine convergence based on the number of iterations or the IGD value, and output the Pareto optimal solution set.

[0032] As one example, the convergence condition is: when the number of iterations reaches... The algorithm stops iterating when the change in the IGD value (back-generation distance) is less than 1e-4 over 10 consecutive iterations. Output: Outputs the Pareto optimal solution set from Archive1 and Archive2, containing optimization schemes with different combinations of "number of monitoring wells-location-frequency" for selection in practical engineering.

[0033] Step S21 includes: The objective function for maximizing the amount of monitoring information is:

[0034] in, The number of monitoring wells; In order to monitor frequency Downhole monitoring The pollutant concentration sequence; Concentration sequence The marginal entropy represents The amount of information it contains; Concentration sequence and Mutual information represents the amount of information transmitted between preferred and non-preferred wells; This represents a 0-1 control variable, calculating only the mutual information between preferred and non-preferred wells; This indicates taking the maximum value; Represents a three-dimensional combination vector of decision variables; The objective function for minimizing the monitoring cost is:

[0035] in, This indicates taking the minimum value; The cost of deploying each monitoring well. For sampling and monitoring costs; For monitoring frequency The corresponding number of monitoring sessions; Indicates the first Potential monitoring well.

[0036] Step S24 includes: The nonlinear convergence factor The calculation formula is:

[0037] in, This represents the current iteration number. It is the square of the maximum number of iterations.

[0038] As one example, a nonlinear convergence factor is designed. It replaces the traditional linear decay factor.

[0039] Step S26 includes: The resolution-based dearchive mechanism includes: The range of values ​​for the objective function is divided into a fixed grid, with each objective dimension divided into 10 grids; the objective function values ​​include: the amount of monitoring information. and monitoring costs ; For each non-dominated Pareto front solution, map it to the lower left reference point of the corresponding grid and calculate the Euclidean distance; Based on Euclidean distance, only the solution with the smallest distance is retained within the same grid, and the rest are discarded.

[0040] As one example, the objective function (monitoring information content) Monitoring costs The target value range is divided into grids, with 10 grids per target dimension, forming a fixed-resolution target space grid. The target value of each non-dominated solution is mapped to the lower-left reference point of the corresponding grid, and the Euclidean distance between the solution and the reference point is calculated. Only the solution closest to the reference point is retained within the same grid, ensuring uniqueness within each grid, effectively eliminating redundant non-dominated solutions, and controlling the sparsity of the solution set distribution. Compared to traditional crowding ranking, this mechanism controls the distribution density of solutions through resolution quantization, avoiding concentration or redundancy in the solution set, and improving the selectivity and practicality of the optimal solution.

[0041] As one embodiment, step S1: acquire pollutant concentration data of each monitoring well at different times and perform preprocessing and parameter initialization to obtain initial individuals; 1.1 First, determine the feasible domain of parameters: clarify the range of values ​​for hydrogeological parameters (aquifer permeability coefficient, assuming that the permeability coefficients of the 8 aquifers in the study area are uniformly distributed between [0, 5.0]) and pollution source parameters (pollutant emission rate, water discharge rate) to ensure the possible range of parameter changes in the actual covered site. The emission rates of pollutants from pollution sources all follow a normal distribution with a mean of 1.0 and a standard deviation of 0.5, and the feasible region is defined as [0, 2.0]. The water discharge rates all follow a normal distribution with a mean of 100.0 and a standard deviation of 50.0, and the feasible region is defined as [0, 200.0].

[0042] 1.2 Sampling to generate input samples: Latin hypercube sampling method is used to draw a sufficient number of input samples within the feasible region of each parameter. Each sample group contains all input variables to be optimized. 2500 sets of input samples were generated using Latin hypercube sampling. Each set of samples contained 12 input variables (8 permeability coefficients + 2 pollutant discharge rates + 2 water discharge rates) and was divided into 2000 training sets and 500 validation sets. (1) Stratification: The range of values ​​of each group of 12 input variables is divided into 10 non-overlapping sub-ranges with the same probability, as shown in Table 1.

[0043] (2) Screening: Take 12 random permutations of (1, 2, ... 10)T to form 2000 x 12-dimensional training samples and 500 x 12-dimensional validation sample matrices respectively. The following is an example of a training sample matrix.

[0044]

[0045] (3) Sampling: According to the sample matrix obtained in (2), the 12 values ​​in each row represent the stratified interval number of each group of input samples in their respective feasible domains. Then, a value is randomly selected from the corresponding interval as the sample of the corresponding dimension input of this group. Taking the above training samples as an example, the labels extracted in the first row of the matrix are 8, 9, 7, 4, 5, 6, 8, 6, 3, 4, 5, 1. These labels will correspond to the small interval range of the stratified result table. Random sampling is performed within this range. "8" corresponds to the 8th small interval. The permeability coefficient of the first aquifer is extracted from the interval (3.5, 4.0). If the random sampling result is 3.7, then the training sample result table 2 can be obtained by analogy.

[0046] 1.3 Obtaining Output Data: Substitute all input samples one by one into the numerical simulation model of groundwater solute transport, run the model to obtain pollutant concentration data of each monitoring well at different times, form an "input-output" dataset, which serves as the core basis for subsequent fitness calculation; The simulation was performed using MODFLOW6 with a time span of one year. After the simulation was completed, the water level and pollutant concentration of the six wells were read. The results are shown in Table 3.

[0047] 1.4 Data Preprocessing: Remove outliers and missing values ​​from the dataset to ensure data integrity and reliability. At the same time, divide the training set and validation set proportionally to provide data support for subsequent optimization.

[0048] 1.5 Set the core parameters of the algorithm: population size is set to 50 (50 monitoring schemes are optimized at the same time in each iteration), maximum number of iterations is 100 (avoid invalid iterations), archive capacity is 30 (a maximum of 30 optimal non-dominated solutions are retained), leader selection pressure parameter c=4, number of grids in the target dimension is 10 (for subsequent solution archiving); 1.6 Define three-dimensional decision variables: ① Monitoring well location: using binary encoding, ,in =1 indicates that the iwth potential monitoring well is deployed. =0 indicates no deployment, and the code length is consistent with the total number of potential monitoring wells; ② New additions: The number of "1"s in the code must satisfy the constraint formula. ( The total number of monitoring wells (including existing and new ones) should be used to avoid excessive costs due to too many wells or insufficient monitoring due to too few wells. ③ Monitoring frequency: Select a specific value from 7-12 times / year, which must meet the constraint formula. This corresponds to different monitoring time intervals; 1.7 Generate the initial population: Randomly generate 50 initial individuals (i.e. 50 initial monitoring schemes) within the feasible region of the decision variables, ensuring that each individual satisfies all constraints and that the initial population can cover the key regions of the solution space, providing a diverse starting point for subsequent iterative searches.

[0049] Step S2: Based on the initial individuals, execute the multi-front, multi-objective gray wolf optimization algorithm to output the Pareto optimal solution set; 2.1 Calculate fitness value The "monitoring information content" and "monitoring cost" of each initial individual (monitoring plan) are transformed into quantifiable and comparable values, providing a basis for subsequent ranking and selection.

[0050] 2.1.1 Calculate the amount of monitoring information ① Extract the concentration sequence of all wells (including existing and newly added monitoring wells) for the current individual at the corresponding monitoring frequency. (e.g., a certain scheme for well layout) If the monitoring frequency is 8 times per year, then extract the concentration data from 8 monitoring sessions of this well. ② Calculate the marginal entropy

[0051] Calculate the probability of each concentration value occurring in the concentration sequence of each deployed well. (The number of times a certain concentration value appears divided by the total number of monitoring times) is substituted into the formula to calculate the information contained in a single well. Then, the marginal entropy of all deployed wells is summed to obtain...

[0052] ③ Calculate mutual information

[0053] in To obtain the joint entropy of the concentration sequences of the two wells, calculate the mutual information between all deployed wells, and sum them up.

[0054] in The variables are 0-1 control variables, and only the mutual information between preferred wells and non-preferred wells is calculated.

[0055] ④ Substitute into the formula to obtain the current monitoring information amount for the individual.

[0056]

[0057] 2.1.2 Calculate monitoring costs ① Deployment costs

[0058] Number of newly added monitoring wells × ( =1, single-well deployment cost constant), starting from the 4th item, are newly added monitoring wells, and the first 3 items are existing monitoring wells.

[0059] ② Sampling and monitoring fees

[0060] (Number of existing monitoring wells + Number of newly added monitoring wells) × Monitoring frequency × ( =0.01 (cost per sampling monitoring)

[0061] Substituting into the formula, we obtain the monitoring cost for the current individual. The goal is to minimize

[0062] 2.1.3 Constraint Verification Each individual is checked to see if the number of new monitoring wells, monitoring frequency, and well spacing meet the formula. Unqualified individuals are directly removed and do not participate in subsequent iterations.

[0063] ① Constraints on the number of new monitoring wells

[0064] ② Monitoring frequency constraints

[0065] ③ Monitoring well density constraints

[0066] The distance between the two monitoring wells shall not be less than 10 meters. , The difference between the horizontal and vertical coordinates of the two monitoring wells. 2.2 Non-dominated sorting and dual archive establishment "High-quality solutions" are selected and stored in layers to avoid the omission of high-quality solutions due to the single archiving of traditional algorithms, and to provide a differentiated candidate pool for the selection of multiple cutting-edge leaders in the future.

[0067] 2.2.1 Non-dominated sorting The first and second non-dominated Pareto fronts are selected based on dominance relationships (the solution set where no other individual can dominate after removing the first front).

[0068] 2.2.2 Dual Archive Division The first non-dominated Pareto front solution is stored in Archive1 (the global optimal candidate library) as the selection pool for the subsequent α wolf (global optimal leader); the second non-dominated Pareto front solution is stored in Archive2 (the suboptimal guidance library) as the selection pool for the subsequent β wolf and δ wolf (suboptimal leader). 2.2.3 Archive capacity verification If the number of solutions in Archive1 or Archive2 exceeds the preset archive capacity (30), the subsequent resolution-based solution archiving mechanism will be immediately triggered to remove redundant solutions.

[0069] 2.3 Multiple Leading Leaders' Choices Selecting a representative leader wolf guides the population to search for better regions, avoiding the problems of insufficient population diversity and susceptibility to local optima caused by a single frontier in traditional algorithms. By selecting β and δ wolves based on the second Pareto front, when there are better solutions around the leader wolf, ω wolves need to boldly move closer to them to avoid missing better solutions, thus improving MOGWO performance. The specific leader selection process is detailed in [link to relevant documentation]. Figure 2 2.3.1 Leader Selection Strategy Using the roulette wheel selection method, based on Calculate the selection probability, where c=4 (leader selection pressure parameter). This represents the number of Pareto optimal solutions in the corresponding archive. 2.3.2 Specific Selection One solution is selected from Archive1 by selection probability as α wolf to ensure that the search process always moves towards the global optimum. Two solutions are extracted from Archive2 according to the selection probability as β wolf and δ wolf respectively, and differentiated suboptimal guidance directions are introduced to enrich the population search path; 2.3.3 Output Leader Combination Alpha wolf (balanced optimal solution), β wolf (high information content suboptimal solution), and δ wolf (low cost suboptimal solution) provide multi-dimensional guidance for population position updates.

[0070] 2.4 Adjustment of Nonlinear Convergence Factor The algorithm's search strategy is dynamically adjusted according to the iteration stage. In the early stage, the search scope is expanded (global exploration), and in the later stage, the focus is on fine optimization (local development), which solves the problem of insufficient adaptability of traditional linear convergence factors.

[0071] 2.4.1 Calculate the convergence factor Substitute into the formula

[0072] Where t is the current iteration number. =100 is the maximum number of iterations; ① Early stage of iteration (t∈[0,40]) The value decreases slowly and remains above 1.2; at this point, the coefficient... ( The probability of (being a random number) increases significantly, the population mainly explores globally, and the solution space coverage is expanded; ② Mid-cycle (t∈[40,80]) The value dropped rapidly from 1.2 to 0.4. Fluctuating around 1, it achieves a smooth transition between global exploration and local development, balancing the breadth and accuracy of the search; ③ Later stage of iteration (t∈[80,100]) The value approaches 0. The probability of <1 increases significantly, the population focuses on local development, performs fine search on high fitness regions, and quickly converges to the optimal solution set; The convergence factor variation curve is shown in the figure. Figure 3 .

[0073] 2.4.2 Calculate the auxiliary coefficient (Control the search step size) ( ∈[0,1] is a random number, which controls the randomness of the search direction and provides parameter support for population position updates.

[0074] 2.5 Population Location Update By leveraging the superior characteristics of α, β, and δ wolves, the decision variables of other individuals (ω wolves) are adjusted to gradually optimize the solution towards "high information content and low cost".

[0075] 2.5.1 Calculate Distance Calculate the spatial distance between wolf ω and wolves α, β, and δ:

[0076] in , , Let X be the combination of decision variables for α, β, and δ wolves, respectively, X be the current combination of decision variables for ω wolf, and C be the auxiliary coefficient; 2.5.2 Calculate the intermediate position: Substituting into the formula, we obtain the three potential directions of movement for ω wolf:

[0077] Where A is an auxiliary coefficient that controls the size of the movement step; 2.5.3 Determine the new location The average of the three potential directions is taken as the new decision variable combination for ω wolf, i.e.

[0078] 2.5.4 Constraint Verification The updated decision variables are adjusted to ensure that the number of new monitoring wells, monitoring frequency, and well spacing all meet the constraints, thus obtaining effective new individuals.

[0079] 2.6 Resolution-Based Dearchiving Redundant non-dominated solutions are eliminated to ensure that the solution set is evenly distributed in the target space of "monitoring information volume - monitoring cost", thereby improving the selectivity of actual engineering decisions. A schematic diagram of solution archiving is shown below. Figure 4 'a' represents the bottom left corner of the grid, and 'b', 'c', and 'd' represent different solutions. It can be seen that solution 'd' is closest to point 'a', therefore, only solution 'd' will be retained within this grid.

[0080] 2.6.1 Target Space Meshing right (Monitoring information volume) and The range of values ​​for (monitoring cost) is divided into 10 grids, forming a fixed resolution grid of 10×10. 2.6.2 Calculate Distance For each non-dominated solution ( , The value is mapped to the lower left reference point of the corresponding grid, and substituted into the Euclidean distance formula.

[0081] Calculate the distance between the solution and the reference point; 2.6.3 Filtering Rules Within the same grid, only the solution with the smallest distance is retained, and other redundant solutions are eliminated to ensure that only one optimal solution exists in each grid. This results in a uniformly distributed set of non-dominated solutions, which are then stored in Archive1 and Archive2.

[0082] 2.7 Convergence Judgment and Result Output To avoid wasting computational resources on ineffective iterations, ensure that the output solution has reached a stable optimal state.

[0083] 2.7.1 Convergence Judgment ① Iteration count check: If the current iteration count reaches the maximum iteration count of 100, the algorithm stops iterating; ② IGD value determination: Calculate the IGD value (back-generation distance) of the current solution set and substitute it into the formula.

[0084] in, The ideal Pareto front is represented by Q, which is the solution set obtained by the current algorithm. If the change in IGD value is less than 1e-4 over 10 consecutive iterations, the algorithm stops iterating. 2.7.2 Result Output Output the Pareto optimal solution set in Archive1 and Archive2, which includes optimization schemes for different "number of new monitoring wells-location-monitoring frequency", covering the entire range from "low cost-low information content" to "high cost-high information content", for selection in actual engineering scenarios.

[0085] Step S3: Based on the output Pareto optimal solution set, select a monitoring well deployment scheme that meets the engineering requirements.

[0086] The changes in HV metric for all algorithms with iteration number and the changes in IGD metric for all algorithms with iteration number are shown in [link to documentation]. Figure 5 , Figure 6 As shown in the figure, MOPSO's HV metric initially exhibits a high value, indicating rapid convergence. However, its convergence falls below that of MPMOGWO and NSGA-II in the later stages of evolution. MPMOGWO's convergence is slightly inferior to NSGA-II in the early stages but outperforms all other algorithms in the later stages. The IGD metric shows a similar trend to the HV metric. Therefore, based on the convergence performance of the algorithms, MPMOGWO can achieve better convergence results in the later stages.

[0087] In one embodiment, Table 1 shows the sampling stratification results, Table 2 shows the training sample dataset, and Table 3 shows the simulated data results of each monitoring well over one year.

[0088] Table 1

[0089] Table 2

[0090] Table 3

[0091] This application also discloses an electronic device. (See reference...) Figure 7 , Figure 7 This is a schematic diagram of the structure of an electronic device disclosed in an embodiment of this application. The electronic device 500 may include: at least one processor 501, at least one network interface 504, a user interface 503, a memory 505, and at least one communication bus 502.

[0092] The communication bus 502 is used to enable communication between these components.

[0093] The user interface 503 may include a display screen, and optionally, the user interface 503 may also include a standard wired interface or a wireless interface.

[0094] The network interface 504 may optionally include a standard wired interface or a wireless interface (such as a Wi-Fi interface).

[0095] This application also discloses a computer-readable storage medium storing multiple instructions adapted for loading by a processor to execute the above-described multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring.

[0096] The above are merely exemplary embodiments of this disclosure and should not be construed as limiting the scope of this disclosure. Any equivalent changes and modifications made in accordance with the teachings of this disclosure shall still fall within the scope of this disclosure.

[0097] This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not described in this disclosure. The specification and embodiments are to be considered exemplary only, and the scope and spirit of this disclosure are defined by the claims.

Claims

1. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring, characterized in that, The method comprises the following steps: Step S1: obtaining the pollutant concentration data of each monitoring well at different time and preprocessing and parameter initialization to obtain initial individuals; Step S2: based on the initial individuals, performing a multi-front multi-objective grey wolf optimization algorithm, and outputting a Pareto optimal solution set; Step S3: selecting a monitoring well layout scheme meeting the engineering requirements according to the output Pareto optimal solution set.

2. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring as claimed in claim 1 wherein, Step S1 comprises: S11: determining the feasible region of hydrogeological parameters and pollution source parameters, including the aquifer permeability coefficient, the pollutant discharge rate and the water discharge rate; S12: generating an input sample set by using a Latin hypercube sampling method, and dividing it into a training set and a validation set; S13: inputting the input sample into a groundwater solute transport numerical simulation model to obtain the pollutant concentration data of each monitoring well at different time; S14: setting the core parameters of the multi-objective grey wolf optimization algorithm, including the population size, the maximum number of iterations, the archive capacity and the target dimension grid number; S15: defining three-dimensional decision variables, including the monitoring well position, the number of new wells and the monitoring frequency, and using binary coding; S16: randomly generating an initial population in the feasible region of the decision variables, that is, obtaining the initial individuals.

3. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring as claimed in claim 1 wherein, Step S2 comprises: S21: calculating the fitness value of each initial individual, including the monitoring information maximization objective and the monitoring cost minimization objective; S22: performing non-dominated sorting on all individuals in the population based on the fitness value, establishing a double-archiving mechanism, and storing the first non-dominated Pareto front solution and the second non-dominated Pareto front solution respectively; S23: selecting leader wolves from the double archive by using a roulette wheel selection strategy, the alpha wolf comes from the first non-dominated Pareto front solution, and the beta wolf and the delta wolf come from the second non-dominated Pareto front solution; S24: designing a nonlinear convergence factor for dynamically adjusting the search strategy of the algorithm; S25: updating the positions of other individuals in the population according to the positions of the leader wolves; S26: screening the archived solutions based on a resolution-based solution archiving mechanism to ensure uniform distribution of the solution set; S27: judging convergence according to the number of iterations or the IGD value, and outputting the Pareto optimal solution set.

4. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring as claimed in claim 3 wherein, Step S21 comprises: The monitoring information maximization objective function is: in, The number of monitoring wells; In order to monitor frequency Downhole monitoring The pollutant concentration sequence; Concentration sequence The marginal entropy represents The amount of information it contains; Concentration sequence and Mutual information represents the amount of information transmitted between preferred and non-preferred wells; This represents a 0-1 control variable, calculating only the mutual information between preferred and non-preferred wells; This indicates taking the maximum value; Represents a three-dimensional combination vector of decision variables; The monitoring cost minimization objective function is: wherein, represents taking the minimum value; is the cost of the layout for each monitoring well, is the cost of sampling monitoring; is the monitoring frequency corresponds to the number of monitoring times; represents the potential monitoring well.

5. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring as claimed in claim 3 wherein, Step S24 comprises: The non-linear convergence factor The calculation formula is: wherein, is the current iteration number, is the square of the maximum iteration number.

6. A multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring as claimed in claim 4 wherein, Step S26 comprises: The resolution-based solution archiving mechanism comprises: The value range of the objective function is divided into a fixed grid, and each objective dimension is divided into 10 grids; the value of the objective function includes: monitoring information quantity and monitoring cost ; Mapping each non-dominated Pareto front solution to the left lower corner reference point of the corresponding grid, and calculating the Euclidean distance; Based on the Euclidean distance, only the solution with the smallest distance in the same grid is retained, and the rest is removed.

7. An electronic device, comprising: The electronic device comprises a processor, a memory, a user interface and a network interface, the memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory to enable the electronic device to perform the multi-front multi-objective grey wolf optimization method for groundwater pollution monitoring according to any one of claims 1-6.

8. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores instructions that, when executed by a computer, perform the multi-frontier, multi-objective gray wolf optimization method for groundwater pollution monitoring as described in any one of claims 1-6.

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