Groundwater seepage flow prediction method and system based on stochastic maximum posterior variational inference

Abstract

This invention discloses a groundwater seepage flow prediction method and system based on stochastic maximum apost-variational inference, which belongs to the field of groundwater management and prediction. [Solution] The groundwater seepage flow prediction method disclosed in the present invention introduces a stochastic maximum posterior variational inference method based on the steady-state Darcy flow equation and directly obtains parameter samples from the posterior distribution by stochastic sampling, thereby solving the nonlinear problem in groundwater seepage flow and providing more accurate parameter posterior estimation. This significantly improves the accuracy of groundwater seepage flow prediction. Compared to the prior art, the method of the present invention not only estimates parameters but also quantifies the uncertainty of the prediction results, providing decision-makers with a comprehensive risk assessment and enhancing the scientific and robustness of decision-making.

Description

[Technical Field]

[0001] This invention belongs to the field of groundwater management and prediction, and more specifically, relates to a groundwater seepage flow prediction method and system based on stochastic maximum posterior variational inference. [Background technology]

[0002] The rational development and scientific management of groundwater resources are crucial for regional water security, supporting agricultural development, and maintaining ecosystem balance. Accurate simulation of groundwater flow is essential for tasks such as assessing water resource conditions, designing rational extraction plans, and preventing and managing groundwater pollution. The steady-state Darcy flow equation, a fundamental mathematical model describing groundwater flow, becomes complex to solve directly due to its nonlinear properties, particularly when considering geological heterogeneity and uncertainty. Conventional numerical methods often require significant computational resources to address such problems, and effectively quantifying the uncertainty of model parameters is difficult. Therefore, developing new and efficient groundwater seepage flow prediction methods is a critical need in the field of hydrogeology.

[0003] In the field of groundwater seepage flow prediction, Bayesian methods are attracting attention because they naturally incorporate prior knowledge and quantify uncertainty. However, in practical applications, Bayesian methods face a major challenge: how to efficiently extract information from posterior distributions, especially when the dimensionality of model parameters is high and the calculations are complex. Variational inference, as an approximate Bayesian inference method, reduces computational cost by approximating complex posterior distributions by finding a single simple distribution. However, conventional variational inference methods often require linearization of the model when dealing with nonlinear problems, resulting in the loss of important information and impaired prediction accuracy. Therefore, developing a variational inference method that can handle nonlinear problems while maintaining high computational efficiency is an urgent issue in the current field of groundwater seepage flow prediction. [Overview of the project] [Problems that the invention aims to solve]

[0004] The present invention aims to provide a groundwater seepage flow prediction method and system based on probabilistic maximum posterior variational inference to solve the technical problem of low prediction accuracy in conventional prediction methods, which are required to handle nonlinear problems. [Means for solving the problem]

[0005] To achieve the above objective, the present invention is realized by employing the following technical solutions.

[0006] The present invention S1: A step to construct a groundwater seepage flow model based on the steady-state Darcy flow equation and determine the unknown parameters of the groundwater seepage flow model based on the steady-state Darcy flow equation, S2: The steps involve acquiring observational data and constructing a mathematical model of groundwater seepage flow observation data based on the observational data and unknown parameters. S3: The steps include setting a prior distribution for the target parameters of a mathematical model of groundwater seepage flow observation data, obtaining the prior distribution of the target parameters, finding the optimal solution by substituting the prior distribution of the target parameters and the observation data based on a stochastic cost function, and obtaining a posterior approximation sample of the target parameters in the optimal solution. S4: A step of calculating the approximate post-mortem measure and uncertainty quantification of the target parameter according to the post-mortem approximation sample of the target parameter, S5: Discloses a groundwater seepage flow prediction method based on probabilistic maximum posterior variational inference, which includes the steps of applying the approximation of the posterior measure and uncertainty of the target parameters to a groundwater seepage flow model based on the steady-state Darcy flow equation using a probabilistic maximum posterior variational inference method, performing groundwater seepage flow prediction, and obtaining prediction results.

[0007] Furthermore, in S1, the equation for the groundwater seepage flow model based on the steady-state Darcy flow equation is as follows:

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[0008] Furthermore, in S2, the equation of the mathematical model of the groundwater infiltration flow observation data is as follows.

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[0009] Furthermore, the following requirements are satisfied.

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[0010] Furthermore, in S3, the target parameters of the mathematical model of the groundwater infiltration flow observation data include the hydraulic conductivity u, further include λ and τ, and λ is a hyperparameter that controls the prior distribution of the hydraulic conductivity u.

[0011] Furthermore, in S3, the equation of the probabilistic cost function is as follows.

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[0012] Furthermore, in S4 and S5, according to the posterior approximate samples of the target parameters, the approximate posterior measure and the quantification of uncertainty of the target parameters are calculated, and the adopted equations are as follows,

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[0013] The present invention A model building module for constructing a groundwater seepage flow model based on the steady-state Darcy flow equation, determining unknown parameters of the groundwater seepage flow model based on the steady-state Darcy flow equation, acquiring observational data, and constructing a mathematical model of groundwater seepage flow observational data according to the observational data and unknown parameters. A prior assumption and inference module for setting prior distributions for target parameters of a mathematical model of groundwater seepage flow observation data, obtaining prior distributions for target parameters, finding the optimal solution by substituting the prior distributions and observation data for the target parameters based on a stochastic cost function, and obtaining a posterior approximation sample of the target parameters in the optimal solution. A post-hoc estimation and uncertainty quantification module for calculating approximate post-hoc measures and uncertainty quantifications of target parameters, based on post-hoc approximation samples of target parameters, Further disclosure relates to a groundwater infiltration flow prediction system based on stochastic maximum posterior variational inference, which includes a numerical experiment module for obtaining prediction results, by applying the approximation of the posterior measure and uncertainty of target parameters to a groundwater infiltration flow model based on the steady-state Darcy flow equation using a stochastic maximum posterior variational inference method to predict groundwater infiltration flow.

[0014] Compared to the prior art, the present invention has the following beneficial effects.

[0015] This invention discloses a groundwater seepage flow prediction method based on stochastic maximum posterior variational inference. By introducing a stochastic maximum posterior variational inference method based on the steady-state Darcy flow equation, convergence can be achieved within a few steps, significantly reducing the number of iterations and the need for computational resources. By directly obtaining parameter samples from the posterior distribution through stochastic sampling, computational cost and time are greatly reduced, complex posterior distribution analysis calculations are avoided, excellent mesh independence is demonstrated, stability and reliability of prediction results are ensured, and the prediction method is not affected by discretization accuracy. This prediction method can solve nonlinear problems in groundwater seepage flow and provide more accurate parameter posterior estimation, thereby significantly improving the accuracy of seepage flow prediction. Compared to the prior art, the method of this invention can not only estimate parameters but also quantify the uncertainty of prediction results, providing decision-makers with a comprehensive risk assessment and enhancing the scientific and robustness of decision-making.

[0016] Furthermore, when dealing with large-scale groundwater seepage flow problems, especially when the amount of data is large and the parameter dimensions are high, the prediction method of the present invention is particularly efficient, can be applied to various complex geological conditions and different groundwater seepage flow scenes, and has broad applicability.

[0017] Furthermore, the prediction method of the present invention contributes to avoiding environmental problems caused by unreasonable water resource development, improves the efficiency of water resource utilization, and brings about significant environmental and economic benefits. In short, the present invention provides a new tool for the scientific management of groundwater resources and has significant practical value and broad applicability. [Brief explanation of the drawing]

[0018] [Figure 1] This is a flowchart of the groundwater seepage flow prediction method based on stochastic maximum posterior variational inference in the present invention. [Figure 2] This figure shows a comparison of the posterior mean function of the hydraulic conductivity u and schematic diagrams of the point dispersion fields in Embodiment 1 of the present invention, where (a) is the posterior mean function of the hydraulic conductivity u, (b) is the true function of the hydraulic conductivity u, and (c) is the point dispersion field of the posterior measure. [Figure 3]This is a comparison of the posterior mean function and the true function at different points in a 50 × 50 discrete mesh in Embodiment 1 of the present invention, where (a) is a comparison of the posterior mean and the true function in confidence interval I, (b) is a comparison of the posterior mean and the true function in confidence interval II, and (c) is a comparison of the posterior mean and the true function in confidence interval III. [Figure 4] The relative error of the hydraulic conductivity u, the posterior mean of λ, and the posterior mean of τ in Embodiment 1 of the present invention are shown, where (a) is the change in the relative error of the hydraulic conductivity u with respect to the number of iteration steps, (b) is the change in the posterior mean of λ (λ* N) with respect to the number of iteration steps, and (c) is the change in the posterior mean of τ (τ* N) with respect to the number of iteration steps. [Modes for carrying out the invention]

[0019] To enable those skilled in the art to better understand the technical concept of the present invention, the technical concept of embodiments of the present invention will be described below clearly and completely with reference to the drawings of embodiments of the present invention. It will be clear that the embodiments described are only some embodiments of the present invention, not all embodiments. Any other embodiments that those skilled in the art can obtain without creative ingenuity based on the embodiments of the present invention are all within the scope of protection of the present invention.

[0020] Furthermore, terms such as "First," "Second," etc., in the specification, claims, and drawings of the present invention are used to distinguish similar objects and do not necessarily describe a specific order or sequence. It should be understood that the data used in this manner is interchangeable where appropriate, so that the embodiments of the present invention described herein may be carried out in an order other than that illustrated or described herein. In addition, the terms "includes," "has," and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or apparatus that includes a series of steps or units may include other steps or units that are not explicitly listed or that are specific to these processes, methods, products, or apparatus, and are not limited to those steps or units that are explicitly listed.

[0021] The present invention will be described in more detail below with reference to the drawings.

[0022] As shown in Figure 1, the present invention discloses a method for predicting groundwater seepage flow based on stochastic maximum apost-variational inference, which includes the following steps.

[0023] Step 1: Model Building A groundwater seepage flow model based on the steady-state Darcy flow equation was constructed to describe the partial differential equations of groundwater flow, particularly steady-state flow in porous media. The equation is as follows:

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[0024] Step 2: Parameter Definition We determined the unknown parameter, namely the hydraulic conductivity u, of the groundwater seepage flow model based on the steady-state Darcy flow equation. In groundwater prediction, an important parameter is porosity e u and supply rate, porosity e u This affects the permeability of the medium, i.e., the ability of water or a fluid to pass through the medium, and in the steady-state Darcy flow equation, the porosity e u The hydraulic conductivity (u) directly affects the flow and distribution of groundwater, while the porosity (e) is a parameter that describes the permeability of the medium. u Closely related to this, the replenishment rate describes the rate at which a groundwater system obtains water from surface water bodies or other sources. In the steady-state Darcy flow equation, the replenishment rate appears directly as the source function p, representing the rate of groundwater replenishment or discharge per unit volume. The source function p describes the source terms in the groundwater system, which may be factors influencing groundwater flow, such as rainwater infiltration, spring water seepage, or well pumping.

[0025] Step 3: Building a mathematical model of groundwater seepage flow observation data Observational data was acquired, and a mathematical model of groundwater seepage flow observation data was constructed based on the observational data and unknown parameters. The formula is as follows:

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[0026] Step 4: Prior Assumptions A prior distribution is set for the target parameters of the mathematical model of groundwater seepage flow observation data, and the prior distribution of the target parameters is obtained. With respect to the hydraulic conductivity u, The prior distribution of the hydraulic conductivity u is u~N(0,λ 2 Let C0 be an operator, where C0 is (I-0.05Δ) -2 It is defined as follows, where I is the identity matrix, Δ is the Laplace operator, and we show that the Neumann boundary conditions are satisfied in the study region Ω (i.e., the derivative at the boundary is zero), N is a normal distribution, and C0 satisfies the square of the reciprocal of the Laplace operator. The Neumann boundary conditions mean that there is no fluid flow at the boundary, which is the most common assumption in groundwater models. In the stochastic maximum posterior variational inference method, the prior assumptions for the hydraulic conductivity u are as follows.

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[0027] Step 5: Stochastic Maximum Aposterior Variational Inference A stochastic cost function is used to transform a nonlinear problem into a stochastic programming problem, and the optimal solution of the stochastic cost function is obtained as a posterior approximation sample of the target parameters. The formula for the stochastic cost function is as follows:

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[0028] Step 6: Ex-post estimation and quantification of uncertainty Depending on the posterior approximation sample of the target parameter, the approximate posterior measure and uncertainty quantification of the target parameter are calculated. The approximate post-hoc measure of the permeability coefficient u is calculated using the following formula.

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[0029] Step 7: Numerical Experiment The approximate posterior measure and uncertainty quantification of target parameters are applied to a groundwater seepage flow model of the steady-state Darcy flow equation using a stochastic maximum posterior variational inference method to predict groundwater seepage flow, obtain prediction results, and verify the accuracy and reliability of the prediction results by comparing them with observational data. In numerical experiments, the effectiveness of the method is verified by applying the stochastic maximum posterior variational inference method to a nonlinear inverse problem controlled by the steady-state Darcy flow equation. In numerical experiments of groundwater seepage flow prediction, the settings adopted by the present invention to avoid inverse crime are shown in Example 1.

[0030] Example 1 Observation data acquisition: Observation data is acquired using a 500x500 discrete mesh. This is to avoid information loss caused by using a discrete mesh that is too coarse in numerical experiments, and to prevent further impact on the accuracy of the inverse analysis results.

[0031] Research area Ω:[0,1] 2 This sets the region to one unit square area.

[0032] The observation points are fixed at (i / 20, j / 20), where both i and j take integer values from 1 to 20, and the number of mesh points of the discrete mesh is the number of observation points for obtaining the observation data. That is, N d = 500×500, which means that 400 observation points are evenly distributed within the research area Ω, and i and j are the i-th or j-th observation points in the row index and column index, respectively.

[0033] <00002�1>True value of the permeability coefficient u: Assume a non-linear function, and the expression is as follows.

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[0034] Hyperparameter τ for controlling the noise level of the observation data: The intensity of the noise term ε can be adjusted by τ, which affects the variance of the noise term ε in the observation data.

[0035] Inverse analysis method: Perform inverse analysis of the probabilistic maximum a posteriori variational inference method in a 50×50 discrete mesh.

[0036] In the present invention, the posterior mean function is the expected value of the parameter considering the observation data. That is, the expected value of the posterior distribution is the posterior mean function, which represents the average prediction value of parameter estimation. The true function refers to the true value or true distribution of the parameter in the natural world or actual application, that is, the actual true distribution of the target parameter, which is the target that the model attempts to accurately estimate or predict. The point variance field of the posterior measure describes the degree of uncertainty at each point of parameter estimation after the observation data is given, and is usually used to quantify the reliability of prediction. The quantification of uncertainty is performed by the point variance field of the posterior measure.

[0037] Figures 2 to 4 show numerical results in the resolution of a nonlinear inverse problem controlled by the steady-state Darcy flow equation using the stochastic maximum posterior variational inference method (rMAP-VI), displaying the posterior estimation results for u, λ, and τ, respectively.

[0038] Figure 2 shows a comparison of the posterior mean functions of the hydraulic conductivity u in Example 1 and a schematic diagram of the point-by-point dispersion field. The horizontal and vertical coordinates indicate the inverse analysis interval, and the scale on the right indicates the function value. Figure 2(a) displays the posterior mean function of the hydraulic conductivity u obtained by the stochastic maximum posterior variational inference method. The change in color in the figure indicates hydraulic conductivity estimates in different regions, with the yellow region indicating a high value of the hydraulic conductivity u. Figure 2(b) displays the true function of the hydraulic conductivity u, i.e., the actual hydraulic conductivity distribution. This is used as a criterion for comparison to evaluate the accuracy of the model. Comparing Figure 2(a) and Figure 2(b), it can be seen that the shape of the posterior mean function of the hydraulic conductivity u is similar to the true function of the hydraulic conductivity u, particularly at the positions of the two peaks, indicating that the distribution of the hydraulic conductivity u predicted by the model closely matches the actual situation. Figure 2(c) displays the point-by-point dispersion field of the ex-post measures, quantifying the uncertainty of the prediction and contributing to understanding the reliability of predictions in different regions in groundwater forecasting. Darker colors indicate greater uncertainty in the prediction. This may be due to the complexity of the geological structure or the sparsity of the data.

[0039] Figure 3 compares the performance of the posterior mean function and the true function at different points in a 50x50 discrete mesh. In Figure 3, the horizontal axis represents different discrete mesh points, the vertical axis represents the corresponding function value, the black dashed line represents the posterior mean function, the red solid line represents the true function, and the two green lines represent the upper and lower bounds of the 95% confidence region of the estimated posterior mean function, i.e., the confidence intervals. The width of the confidence intervals reflects the uncertainty of the prediction, with wider intervals indicating greater uncertainty.

[0040]

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[0041] Figure 4 shows the relative error of the hydraulic conductivity u, the posterior mean of λ, and the posterior mean of τ. Figure 4(a) shows the change in the relative error of the hydraulic conductivity u with respect to the number of iteration steps, where the horizontal coordinate represents the number of iteration steps and the vertical coordinate represents the percentage of the relative error. It can be seen that the relative error rapidly converged to approximately 1.6% within a few steps and then stabilized. This indicates that the posterior mean function of the hydraulic conductivity u is quantitatively close to the true mean function. Figure 4(b) shows the posterior mean of λ. * N The change with the number of iteration steps is displayed, and the final result is λ * N Approximately 33.5 is displayed, which is the posterior mean λ of the algorithm. * N This allows for rapid and accurate estimation. Figure 4(c) shows the posterior mean τ. * N The change with the number of iteration steps is displayed, and the final result is τ * N ≈9.3 × 10 6 This is displayed. This similarly proves the efficiency and accuracy of the algorithm in estimating τ.

[0042] The technical concept of the present invention has been described above, but this does not limit the scope of protection of the present invention. Any modifications made based on this technical proposal without departing from the technical concept proposed in the present invention are also within the scope of protection of the present invention.

Claims

1. A groundwater seepage flow prediction method based on stochastic maximum posterior variational inference, S1: A groundwater seepage flow model based on the steady-state Darcy flow equation is constructed according to the following equation, and the target parameters of the groundwater seepage flow model based on the steady-state Darcy flow equation are determined. [Math 1] ω = 0 is a boundary condition at the boundary ∂Ω of the research area. S2: A step of acquiring observational data and constructing a mathematical model of groundwater seepage flow observation data according to the observational data and target parameters, S3: A step of setting a prior distribution for the target parameters of a mathematical model of groundwater seepage flow observation data, obtaining the prior distribution of the target parameters, substituting the prior distribution of the target parameters and the observation data based on a stochastic cost function to find the optimal solution, and obtaining a posterior approximation sample of the target parameters in the optimal solution. S4: A step of calculating the approximate post-hoc measure and uncertainty quantification of the target parameter according to the post-hoc approximation sample of the target parameter, S5: This includes the steps of applying the approximation of the posterior measure and uncertainty of the target parameters to a groundwater seepage flow model based on the steady-state Darcy flow equation using a stochastic maximum posterior variational inference method, performing groundwater seepage flow prediction, and obtaining prediction results. In S2, the mathematical model equation for the groundwater seepage flow observation data is as follows: [Math 2] [Math 3] In S3, the target parameters of the mathematical model of the groundwater seepage flow observation data include the hydraulic conductivity u, and further include λ and τ, wherein λ is a hyperparameter that controls the prior distribution of the hydraulic conductivity u, characterized in that this is a groundwater seepage flow prediction method based on stochastic maximum posterior variational inference.

2. In S3, the formula for the stochastic cost function is as follows, characterized in that the groundwater seepage flow prediction method based on stochastic maximum apost-variational inference according to claim 1. [Math 4]

3. In S4 and S5, the approximate posterior measure and uncertainty quantification of the target parameter are calculated according to the posterior approximation sample of the target parameter, and the formulas used are as follows: [Math 5] The groundwater seepage flow prediction method based on probabilistic maximum posterior variational inference according to claim 2, characterized in that the stopping criteria used when employing a probabilistic maximum posterior variational inference method and applying it to a groundwater seepage flow model based on the steady-state Darcy flow equation are as follows. [Math 6]

4. A groundwater seepage flow prediction system based on stochastic maximum posterior variational inference, A model building module is provided to construct a groundwater seepage flow model based on the steady-state Darcy flow equation according to the following equation, determine the target parameters of the groundwater seepage flow model based on the steady-state Darcy flow equation, acquire observational data, and construct a mathematical model of the groundwater seepage flow observation data according to the observational data and target parameters. [Math 1] ω = 0 is a boundary condition at the boundary ∂Ω of the research area. A prior assumption and inference module for setting prior distributions for target parameters of a mathematical model of groundwater seepage flow observation data, obtaining prior distributions for target parameters, finding the optimal solution by substituting the prior distributions and observation data for the target parameters based on a stochastic cost function, and obtaining a posterior approximation sample of the target parameters in the optimal solution. A post-hoc estimation and uncertainty quantification module for calculating approximate post-hoc measures and uncertainty quantifications of target parameters, based on post-hoc approximation samples of target parameters, This includes a numerical experiment module for obtaining prediction results by applying the approximation of the posterior measure and uncertainty of the target parameter to a groundwater seepage flow model of the steady-state Darcy flow equation using a stochastic maximum posterior variational inference method, thereby performing groundwater seepage flow prediction. The aforementioned model construction module constructs a mathematical model of the groundwater seepage flow observation data according to the following formula: [Math 2] [Math 3] A groundwater seepage flow prediction system based on stochastic maximum posterior variational inference, characterized in that the target parameters of the mathematical model of the groundwater seepage flow observation data, in which the prior assumptions and inference modules set prior distributions, include a hydraulic conductivity u, and further include λ and τ, wherein λ is a hyperparameter that controls the prior distribution of the hydraulic conductivity u.

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