Underground water movement substitution model construction method and system based on deep learning

Abstract

The invention provides an underground water movement substitution model construction method and system based on deep learning, and the method comprises the steps: solving a parameterized underground water movement partial differential equation in a heterogeneous aquifer based on a physical information neural network, and constructing a substitution model; constructing a loss function fusing physical information, and enabling the output of the substitution model to meet initial conditions and boundary conditions through a hard constraint method; parameters of the substitution model are optimized through an iterative training process; groundwater head distribution under different hydrogeological conditions is predicted based on the optimized substitution model, and uncertainty quantitative analysis is completed. According to the method, the prediction accuracy and reliability of the substitution model under the condition of no data are greatly improved, and the calculation time for completing uncertainty quantification is greatly shortened.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of water environment simulation, and particularly relates to a groundwater movement surrogate model construction method based on deep learning. BACKGROUND

[0002] Building a groundwater flow model is of great significance for water resource management, environmental protection, and achieving sustainable development of the environment. An accurate groundwater flow model not only helps us to deeply understand the dynamic change law of the groundwater system, but also provides strong support for formulating scientific and reasonable groundwater management and protection strategies, so as to ensure the rational use and long-term stability of groundwater resources.

[0003] Natural soil layers are usually a composite body of different types of soil interlaced together, so the aquifer hydrogeological parameters such as the permeability coefficient K have strong variability in space. Due to the inhomogeneity of the porous medium distribution and the incomplete information in the distribution, the groundwater movement model has a certain degree of uncertainty. Therefore, it is necessary to quantify the uncertainty of groundwater movement caused by inhomogeneous aquifer medium. The Monte Carlo method is the most commonly used numerical method for uncertainty quantification, which generates a large number of realizations of the random distribution (also known as random field) of hydrogeological parameters. The numerical calculation method can be used to solve the groundwater water level distribution corresponding to the random field.

[0004] The common numerical method for establishing a groundwater movement model is to solve the groundwater movement equation (partial differential equation), mainly including the finite difference method, the finite element method and the boundary element method. They discretize the model space and boundary into nodes in space and time to solve the partial differential equation. Due to its solid theoretical foundation and robustness, these methods have been widely used. However, in problems such as uncertainty quantification that require repeated simulation, the traditional numerical method quantifies the uncertainty of groundwater flow by sampling a large number of parameters, repeatedly solving the parameterized partial differential equation, and then constructing a fitting function to efficiently approximate the relationship between the input and output using interpolation methods. However, this method often faces great difficulties in terms of computational resources and time costs when dealing with time-dependent or high-dimensional problems, and the method of discretizing the space-time domain in the process has a high mathematical knowledge threshold and high requirements for users. Therefore, it is necessary to construct a surrogate model to replace the solving process of the partial differential equation in the numerical simulator.

[0005] With the rapid development of artificial intelligence, deep learning methods provide new techniques and methods for developing surrogate models. Data-constrained surrogate models mainly rely on the exact solution of partial differential equations, also known as labeled data, and then use a loss function to make the results close to the labeled data with the expected accuracy, so as to obtain a neural network model that can represent the partial differential equation. However, when the amount or quality of data cannot be guaranteed, the learning ability of data-driven models decreases significantly, and at the same time, the generalization ability is insufficient when making long-term predictions, and there is a lack of physical connection with the real world. SUMMARY

[0006] The present application proposes an unsupervised deep learning method based on Karhunen-Loève (K-L) expansion to solve the parameterized groundwater flow equation in heterogeneous aquifers, and the model training process does not rely on any labeled data. The surrogate model constructed by this method is based on the physical information neural network (PINN) framework, and the hard physical constraints (such as control equations, boundary conditions and initial conditions) are directly incorporated into the loss function during the training process. The input of the model includes the configuration points of space and time and the parameterized variables in the groundwater flow equation, such as pumping rate, and random variables used to describe the uncertainty of groundwater flow in heterogeneous aquifers. Here, two cases of heterogeneous aquifers with pumping wells are used to compare the prediction results of the surrogate model with the results of MODFLOW to test the performance of the surrogate model.

[0007] To solve the above technical problems, the present application provides the following technical solutions: A deep learning-based groundwater flow surrogate model construction method, comprising the following steps: Solving the parameterized groundwater flow partial differential equation in a heterogeneous aquifer based on a physical information neural network to construct a surrogate model; Constructing a loss function that integrates physical information, and using a hard constraint method to make the output of the surrogate model satisfy the initial conditions and boundary conditions; optimizing the surrogate model parameters through an iterative training process; Predicting the groundwater head distribution under different hydrogeological conditions based on the optimized surrogate model, and completing the uncertainty quantification analysis.

[0008] Further, the specific steps of constructing the surrogate model include: Establishing a conceptual model of groundwater flow area, setting the area range and physical parameters related to groundwater flow, source and sink conditions, initial conditions and boundary conditions; Selecting the method of sampling the spatio-temporal boundary and the internal region according to the specific circumstances of the regional conceptual model; A physical information neural network framework is constructed, and the number of hidden layers and the number of neurons in each hidden layer are set. The inputs are spatiotemporal configuration points and parameters representing different pumping flows or random fields with different permeability coefficients.

[0009] Furthermore, the source and sink terms are represented by a two-dimensional normal distribution function instead of the Dirac function to indicate the location of the pumping well, expressed as:

[0010] in, It refers to the coordinates of points within the study area. Indicates that it is located at Source and sink items at the location, These are the coordinates of the pumping well. The number of wells ( i =1, 2, ...), For located The pumping capacity of the pumping well at that location; It is a two-dimensional normal distribution function;

[0011] Among them, parameters s The standard deviation of the distribution.

[0012] Furthermore, the permeability coefficient random field is parameterized using the KL expansion method:

[0013] in, It is the permeability coefficient, in m / d; For random variables, For coordinates within the spatial domain, and Characterize the eigenvalues ​​and eigenfunctions to be solved, respectively. This represents the average value of the permeability field; based on the covariance function... Solving the equation yields its eigenvalues ​​and eigenfunctions. and Represents the coordinates of any two points in space. Represents the x and y coordinates of a point in a random field. σ 2 lnK It is the variance of the random field. and It is the correlation length of the random field; The KL expansion of a random field contains an infinite number of terms; it becomes a finite number of terms:

[0014] in, denotes the truncation coefficient.

[0015] Further, the hard constraint method is constructed by the following formula:

[0016]

[0017] wherein, the groundwater level output by the surrogate model after hard constraint; is a function satisfying the initial condition and the first type of boundary condition; is the output of the surrogate model; is a distance function, and the value on the initial time and the first type of boundary is 0; denotes the time t =0; , denotes the value of the initial condition, denotes the value of the first type of condition; and denote the horizontal and vertical coordinates of the spatial point; denotes the time; denotes the parameter of the neural network.

[0018] Further, the loss function is constructed based on the groundwater motion control equation, and the expression is:

[0019] wherein,

[0020]

[0021] wherein, is the loss function, , respectively denote the residual of the control equation and the residual of the second type of boundary condition; is a weight parameter for balancing and ; and respectively denote the control equation term and the second type of boundary condition term in the loss function; is the number of internal collocation points; is the number of points on the second type of boundary.

[0022] Further, the residual of the control equation is calculated by substituting the water head value predicted by the surrogate model into the parameterized groundwater motion equation:

[0023] second-type boundary condition residual f 2 is represented as:

[0024] wherein, is the surrogate model predicted water head; in the phreatic aquifer represents the water supply degree, is the second-type boundary condition.

[0025] Further, the iterative training process comprises: inputting the spatiotemporal configuration points and the parameterized variables into the surrogate model to obtain an original output; converting the original output into a water head prediction value satisfying the initial condition and the first-type boundary condition through a hard constraint method; substituting the water head prediction value into the control equation to calculate a control equation residual, and calculating a second-type boundary condition residual on the second-type boundary; calculating a total loss function value based on the control equation residual and the second-type boundary condition residual; calculating the gradient of the total loss function with respect to the surrogate model parameters, and updating the surrogate model parameters using an optimization algorithm; repeating the above steps until the loss function converges or a preset number of iterations is reached.

[0026] Further, the uncertainty quantification analysis comprises: generating multiple sets of KL expansion parameters representing different permeability coefficient field realizations based on the statistical characteristics of the permeability coefficient field; inputting each set of parameters into the trained surrogate model to obtain the corresponding water head distribution prediction; statistically analyzing the prediction results to calculate the statistical characteristic quantities of the water head to quantify the uncertainty.

[0027] In another aspect, the present application provides a deep learning-based groundwater movement surrogate model construction system, comprising: a surrogate model construction module: which is used to solve the parameterized groundwater movement partial differential equation in the heterogeneous aquifer based on the physical information neural network, and construct a surrogate model; a surrogate model optimization module: which is used to construct a loss function fused with physical information, and make the surrogate model output satisfy the initial condition and the boundary condition through a hard constraint method; and optimize the surrogate model parameters through an iterative training process; a prediction analysis module: which is used to predict the groundwater water head distribution under different hydrogeological conditions based on the optimized surrogate model, and complete the uncertainty quantification analysis.

[0028] Compared with the prior art, the present application has the following beneficial effects: The present application aims at the problem that the existing numerical calculation model is time-consuming in solving parameterized equations and quantifying uncertainty, and the performance of the alternative model based on data constraints is limited when the labeled data is scarce, and the abnormal values are difficult to explain, and proposes an improved method of fusing physical constraints. By introducing the hard constraint as a physical constraint, the residual of the groundwater movement equation with Karhunen-Loeve expansion is regarded as part of the loss function, which not only significantly improves the prediction accuracy and reliability of the alternative model without data, but also greatly shortens the calculation time of quantifying uncertainty. It only takes 94 seconds to process 40,000 random fields, which is more than 100 times more efficient than the traditional numerical model. In addition, since the input of the neural network includes different pumping flows or Gaussian random variables representing the permeability coefficient random field, the alternative model can directly predict the groundwater movement under different pumping flows or different random fields without retraining the model. At the same time, the addition of physical constraints makes the model results easier to explain with physical laws, enhancing the model's explainability and robustness, reducing the occurrence of abnormal values, and providing a more efficient and reliable solution for related research and application, effectively balancing the calculation efficiency and prediction accuracy, and expanding the application range of alternative models in complex tasks of groundwater movement simulation. BRIEF DESCRIPTION OF DRAWINGS

[0029] In order to more clearly illustrate the technical solutions in the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description are some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor.

[0030] Figure 1 is the overall framework of the alternative model of the embodiment of the present application; Figure 2 is a schematic diagram of the specific problem setting of the embodiment of the present application. (a) shows the problem setting schematic diagram of embodiment one, and (b) shows the problem setting schematic diagram of embodiment two; Figure 3 is a schematic diagram of different sampling strategies of the embodiment of the present application. (a) shows a schematic diagram of three methods of sampling space points, and (b) shows a schematic diagram of three methods of sampling time points; Figure 4 is a schematic diagram of the change of different evaluation indexes with pumping flow of the embodiment of the present application. Figures (a), (b), and (c) respectively show the change of the prediction results of the embodiment with pumping flow; MAE, RRMSE and R 2 with pumping flow; Figure 5Figures (a) and (b) show the mean and variance of the groundwater level predicted by the alternative model and MODFLOW simulation at different times in the embodiments of the present invention. Figure (c) shows the probability density function (PDF) of the groundwater level predicted by the alternative model under 40,000 random fields. Figure (d) shows the probability density function (PDF) of the groundwater level predicted by the alternative model under 40,000 random fields when the spatiotemporal location is (100m, 100m, 10d). Detailed Implementation

[0031] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0032] In specific implementation, the method proposed in the technical solution of this invention can be automatically executed by those skilled in the art using computer software technology. System devices for implementing the method, such as computer-readable storage media storing the corresponding computer program of the technical solution of this invention and computer equipment including the computer program running the corresponding computer program, should also be within the protection scope of this invention.

[0033] like Figure 1 As shown, this embodiment provides a method for constructing a groundwater movement alternative model based on deep learning, including the following steps: Step 1. Solve the parametric partial differential equations of groundwater movement in heterogeneous aquifers based on physical information neural networks, and construct an alternative model; Step 2. Construct a loss function that integrates physical information, and use a hard constraint method to ensure that the output of the alternative model satisfies the initial and boundary conditions; optimize the parameters of the alternative model through an iterative training process; Step 3. Based on the optimized alternative model, predict the distribution of groundwater head under different hydrogeological conditions and complete the uncertainty quantification analysis.

[0034] In one specific embodiment, step 1 specifically includes: S1. Establish a conceptual model of the groundwater movement area, and set the area range as well as the physical parameters, source and sink conditions, initial conditions and boundary conditions related to groundwater movement; S2, according to the specific situation of the regional conceptual model, the method of selecting the sampling space-time boundary and the interior of the region, using the physical information of the groundwater movement control equation, generating a loss function term including the initial condition, boundary condition and the residual of the control equation in the region, physically constraining the physical information neural network, and using a hard constraint method to make the residual of the space-time configuration point conforming to the initial condition and the first type of boundary condition be 0; S3, construct a physical information neural network framework, set the number of hidden layers in the network and the number of neurons in each hidden layer. The input is the space-time configuration point and different pumping flow or parameters representing different permeability coefficient random fields, and the output after hard constraint is the groundwater water head.

[0035] Preferably, the physical information of the groundwater movement control equation generates a loss function term including the initial condition, boundary condition and the residual of the control equation in the region. S21, take the groundwater movement equation with non-homogeneous source and sink term as the physical information, and the equation is expressed as: , , ,

[0036] wherein, h is the groundwater water head, m; represents the value of the initial condition, represents the value of the first type of condition, represents the value of the second type of condition; represents the unit normal vector on the second type of boundary; is the specific yield (phreatic aquifer) or the release coefficient (confined aquifer, m -1 ); is the hydraulic conductivity m 2 / d, and the specific expression form for different aquifers is as follows, wherein K is the permeability coefficient, m / d; represents the coordinate position, represents the groundwater water head at a point at a time, m; represents the groundwater water head at a point at the initial time, m; , , represents the spatial domain of the initial condition, the first type of boundary condition and the second type of boundary condition.

[0037]

[0038] wherein, is the aquifer thickness, m.

[0039] In the governing equations It is a source / sink term, which can also be represented as , for( x , y The set of ) is expressed as follows:

[0040] in, It refers to the coordinates of points within the study area. These are the coordinates of the pumping well. The number of wells, For located The pumping capacity of the pumping well at that location. Let be the Dirac function. The Dirac function, which approximates the well-source problem, is not differentiable everywhere and can only be used to represent the location of wells. Therefore, this invention replaces the Dirac function with a two-dimensional normal distribution function in the network model, as shown in the following expression:

[0041] Among them, parameters s The standard deviation of the distribution.

[0042] Hydrogeological parameters of aquifers, such as permeability coefficient K Due to the strong spatial variability, this invention treats the spatial distribution of aquifer permeability coefficient as a random field, with the specific aquifer being a realization of this random field. The permeability coefficient is described by a log-normal distribution, and the KL (Karhunen-Loève) expansion is used to simulate the random field. For a second-order stationary random field, the permeability coefficient field can be expressed as:

[0043] in, It is the permeability coefficient, in m / d; ξ i For random variables, For coordinates within the spatial domain, and Characterize the eigenvalues ​​and eigenfunctions to be solved, respectively. This represents the average value of the permeability field; based on the covariance function... Solving the equation yields its eigenvalues ​​and eigenfunctions. and Represents the coordinates of any two points in space. Represents the x and y coordinates of a point in a random field. σ 2 lnK It is the variance of the random field. and is the correlation length of the random field.

[0044] The KL expansion expression of the random field contains infinite terms. For the convenience of calculation and programming, we use the truncated coefficients M to make it finite terms:

[0045] The expression of the random field representing the permeability coefficient is brought into the groundwater flow equation to obtain:

[0046] S22, according to the groundwater flow equation and boundary conditions, add the residuals related to the physical information in the loss function. The output of the neural network is automatically satisfied by the hard constraint of the initial conditions and the first type of boundary conditions, so the two residuals are 0, and finally generate the loss function term containing the second type of boundary conditions and the physical formula inside the region.

[0047] The preferred hard constraint is expressed as follows:

[0048]

[0049]

[0050] wherein, is the function satisfying the initial conditions and the first type of boundary conditions; is the output of the surrogate model; is the distance function, whose value at the initial time and on the first type of boundary is 0; respectively represent the horizontal and vertical coordinates and time. If the spatial domain is rectangular, the distance function expression is as follows:

[0051] wherein, represents the vertical coordinate; represents time; and respectively represent the minimum and maximum values of the spatial domain on the axis; represents the simulation duration. This formula makes the value of the coordinate axis equal to or the distance function value of the point on the first type of boundary condition is 0.

[0052] Preferably, the loss function term includes:

[0053]

[0054]

[0055]

[0056]

[0057] wherein, is a loss function, , respectively represent the residual of the control equation and the residual of the second type of boundary condition; is a weight parameter for balancing and between the size relationship; and respectively represent the control equation term and the second type of boundary condition term in the loss function; is the water head predicted by the surrogate model; is the number of internal collocation points; is the number of points on the second type of boundary.

[0058] The specific steps of constructing the physical information neural network framework to train the surrogate model are preferably expressed as follows: S31, the overall framework of the surrogate model for solving the groundwater flow equation is as shown in Figure 1 The model input includes the combination of collocation points of space-time coordinates and parameterized variables, represented as , and the output is a scalar . This scalar is then mapped to the groundwater level h by a hard constraint to ensure that the solution satisfies the specified boundary conditions. By iteratively calculating the training loss and updating the network parameters, the model gradually improves the prediction accuracy of the groundwater level. The symbol represents different physical parameters in each case. In the case involving a pumping well, represents the pumping flow rate Q varying within the interval. For a surrogate model containing a random field of permeability coefficients, represents a set of values of a Gaussian random variable ξ i (i=1,…,M) in this deep neural network, there is no labeled data as input.

[0059] S32, the present application constructs a surrogate model with PINN as the skeleton to complete the solution of the groundwater motion equation, and the neural network is composed of l +1 layers, wherein the 0th layer is the input layer, the l layer is the output layer.

[0060] Preferably, the neural network is expressed by a mathematical expression as follows:

[0061]

[0062] in, and They are the first l Layer weights and biases; Indicates the first l The output of layer neurons; the network uses a non-linear activation function, expressed as... For ease of representation, let ; operator " " indicates the composition of operations; Θ represents the parameters that need to be calculated iteratively during network training.

[0063] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0064] To verify the effectiveness of the method of this invention, a physical information neural network based on KL expansion was used to construct a substitute model for groundwater movement in heterogeneous aquifers to solve two problems. The first problem is the groundwater movement in heterogeneous aquifers under different pumping rates, and the second problem is the groundwater movement in a random field with different randomly distributed permeability coefficients. Python was used as the programming language, and the PyTorch package was called to build a deep learning framework. The specific problem settings are as follows: Figure 2 As shown, the specific situation is as follows.

[0065] Example 1 This embodiment addresses the problem of single-well pumping in a two-dimensional unconfined aquifer, where the pumping well is located in the spatial domain Ω = [ 500,500]×[ At the center of 500,500m, the flow rate of the pumping well is Q i ( N w =1); the northern and southern boundaries are designated as Class I boundaries with constant head, i.e., at Γ 1,D ={(x,y):∣y∣=500m} (above) g 1( x , y , t =80m; the western and eastern boundaries are designated as impermeable boundaries, i.e., at Γ 2,D ={(x,y):∣x∣=500m} (above) g 2( x , y , t )=0; the initial condition is set to g 0( x, y , t )=80m; the simulation time is T=10d; the remaining hydrogeological parameters are as follows: specific yield μ =0.05. Specifically, the following is included: The surrogate model can predict the groundwater movement under different pumping rates. The heterogeneous permeability coefficient field is generated by one implementation of K-L expansion. The pumping rate ranges from 1000 to 3000 m 3 / d, and the model input includes the corresponding flow and space-time configuration points. The number of flows generated by the uniform sampling strategy is 50, the space-time points in the internal region are sampled using the local encryption sampling strategy, and the space-time points on the boundary are sampled using the uniform sampling strategy, and a total of 1445*50 space-time configuration points are obtained.

[0066] The schematic diagram of different sampling strategies is shown in Figure 3 This embodiment uses a local encryption sampling strategy near the pumping well, that is, the number of spatial sampling points is increased in the local spatial range near the pumping well in order to better capture the trend of groundwater level change.

[0067] According to the calculation formula of the loss function term at each sampling point constructed, the mean square error form is used for calculation; The network architecture of this embodiment is 5 hidden layers, and each hidden layer has 90 neurons; each surrogate model is optimized by 2000 times of adaptive moment estimation (Adam) algorithm to minimize the loss function, and the loss function λ 2 is 100; the initial learning rate is set to 10 -2 , and the decay rate is 0.7 (the learning rate is adjusted every 1000 times of training).

[0068] The actual prediction model is verified by using MODFLOW to construct a case under the same conditions, and the conditions are set as follows: The simulation time is 10 days, a total of 10 stress periods, each stress period is 1 day, and the time step of each stress period is 20, x direction and y The number of discrete grids in the direction is 50, and a total of 2500 grids are generated. The other parameter settings in the two cases are as described above.

[0069] The error between the groundwater level predicted by the surrogate model and the numerical solution of MODFLOW is described by three indicators, and the calculation formula is as follows:

[0070]

[0071]

[0072] wherein,y i is the numerical solution of MODFLOW; i is the predicted groundwater level by the surrogate model; is y i is the average value of N is the subscript i number.

[0073] The input pumping rates are 1220, 1550, 2000, 2350 and 2810 m 3 / d into the surrogate model trained in Example One and predict the spatial distribution of its groundwater level. The values of three evaluation indicators (R MAE , RRMSE and R ²) at the 2.5th, 5th, 7.5th and 10th days vary with the pumping rate as shown in Figure 4 . In all pumping rate settings MAE and RRMSE are quite close to 0, while R ² is always close to 1. The surrogate model shows strong prediction ability and robustness in replacing the pumping rate, and can accurately predict the water head even when close to the boundary of the training set.

[0074] The present application generates a random distribution of the permeability coefficient by K-L expansion corresponding to a random seed. A random seed corresponds to an implementation of a random field.

[0075] Example Two This example is the same as Example One, which is a single-well pumping problem in a two-dimensional phreatic aquifer. Unlike Example One, the mean and variance of the random field of permeability coefficient are lnK=2.5 and σ 2 lnK =1.0; x and y the correlation length in the two directions is η x = η x =408m; the stage coefficient of K-L expansion is M =20, so the input of the surrogate model is the spatiotemporal configuration point and a set of 20 random variables ξ ={ ξ 1, ξ 2,..., ξ 20}; the pumping rate is set to 2000 m 3The same way and number of spatio-temporal configuration points as in Case 1 were sampled, and 20 sets of random variables were used as inputs (20 sets of random field realizations) through a Latin hypercube sampling strategy.

[0076] Six realizations with random seeds of 5, 10, 38, 100, 125, and 500 in Example 2 were selected for testing. Table 1 shows the prediction results of the surrogate model and the MODFLOW simulation results at different realizations and times MAE , RRMSE and R ². All of the MAE are below 0.02 m, RRMSE are below 0.08%, and the R ² in most of the results are above 0.97, which indicates that the surrogate model has strong ability to predict the groundwater level and establish the relationship from the random variables (inputs) to the water head (output).

[0077] Table 1 Error table of the surrogate model predicting groundwater level at different random seeds in Example 2

[0078] In this example, the surrogate model is used to quantify uncertainty by the Monte Carlo method, in which 40,000 realizations of the permeability field are generated by K-L expansion. Figure 5 The mean, variance, and probability density function (PDF) of the groundwater level at different times are shown for the surrogate model and the MODFLOW simulation under 40,000 realizations. Except for the well center position, the absolute error values of the mean and variance are not greater than 0.05 throughout the region, and the PDF curve predicted by the surrogate model is almost coincident with the reference PDF curve of MODFLOW. The surrogate model only takes 94 seconds to process 40,000 realizations of the random field, which is less than 1% of the time required by the numerical model (194 minutes, i.e., 11640 seconds).

[0079] Example Three The present example provides a deep learning-based groundwater movement surrogate model construction system, comprising: a surrogate model construction module: which is used to solve the parametric groundwater movement partial differential equation in a heterogeneous aquifer based on a physical information neural network, and construct a surrogate model; a surrogate model optimization module: which is used to construct a loss function fused with physical information, and make the output of the surrogate model satisfy the initial conditions and boundary conditions through a hard constraint method; and optimize the surrogate model parameters through an iterative training process; a prediction analysis module: which is used to predict the groundwater head distribution under different hydrogeological conditions based on the optimized surrogate model, and complete uncertainty quantification analysis.

[0080] It should be understood that parts of the specification not specifically described are part of the prior art.

[0081] It should be understood that the above description of the preferred embodiments is merely illustrative and not restrictive of the patent protection scope of the present application, and it is not necessary or possible to enumerate all the embodiments. Those skilled in the art can make substitutions or modifications under the inspiration of the present application without departing from the scope of the patent protection of the present application, and all fall within the scope of the patent protection of the present application. The patent protection scope of the present application should be subject to the appended claims.

Claims

1. A method for constructing a groundwater movement alternative model based on deep learning, characterized in that, Includes the following steps: A physical information neural network is used to solve the parameterized partial differential equations of groundwater movement in heterogeneous aquifers, and an alternative model is constructed. A loss function that integrates physical information is constructed, and the output of the alternative model is made to satisfy the initial and boundary conditions through a hard constraint method. Optimize alternative model parameters through an iterative training process; Based on the optimized alternative model, the distribution of groundwater head under different hydrogeological conditions is predicted, and the uncertainty is quantitatively analyzed.

2. The method for constructing a groundwater movement substitution model based on deep learning according to claim 1, characterized in that, The specific steps for constructing the alternative model include: Establish a conceptual model of the groundwater movement area, and define the area's scope as well as the physical parameters, source and sink conditions, initial conditions, and boundary conditions related to groundwater movement. The sampling spatiotemporal boundary and the method for sampling the interior of the region are selected according to the specific circumstances of the region concept model; A physical information neural network framework is constructed, and the number of hidden layers and the number of neurons in each hidden layer are set. The inputs are spatiotemporal configuration points and parameters representing different pumping flows or random fields with different permeability coefficients.

3. The method for constructing a groundwater movement substitution model based on deep learning according to claim 2, characterized in that, The source and sink terms use a two-dimensional normal distribution function to replace the Dirac function to represent the location of the pumping well, and the expression is: in, It refers to the coordinates of points within the study area. Indicates that it is located at Source and sink items at the location, These are the coordinates of the pumping well. The number of wells, =1, 2, ..., For located The pumping capacity of the pumping well at that location; It is a two-dimensional normal distribution function; Among them, parameters s The standard deviation of the distribution.

4. The method for constructing a groundwater movement substitution model based on deep learning according to claim 2, characterized in that, The permeability coefficient random field is parameterized using the KL expansion method: in, K It is the permeability coefficient, in m / d; For random variables, For coordinates within the spatial domain, and Characterize the eigenvalues ​​and eigenfunctions to be solved, respectively. This represents the average value of the permeability field; based on the covariance function... Solving the equation yields its eigenvalues ​​and eigenfunctions. and Represents the coordinates of any two points in space. Represents the x and y coordinates of a point in a random field. σ 2 lnK It is the variance of the random field. and It is the correlation length of the random field; The KL expansion of a random field contains an infinite number of terms; it becomes a finite number of terms: in, This represents the cutoff coefficient.

5. The method for constructing a groundwater movement substitution model based on deep learning according to claim 1, characterized in that, The hard constraint method is implemented by the following formula: in, The groundwater level output by the replacement model after applying hard constraints; It is a function that satisfies the initial conditions and the first kind of boundary conditions; It is the output of the alternative model; It is a distance function, with a value of 0 at the initial moment and on the first-class boundary; Indicates time t =0 , Indicates the value of the initial condition, This represents the value of the first type of condition; and Represents the x and y coordinates of a point in space; Indicates time; This represents the parameters of the neural network.

6. The method for constructing a groundwater movement substitution model based on deep learning according to claim 3, characterized in that, The loss function is constructed based on the groundwater movement control equation, and its expression is: in, in, For loss function, , These represent the residuals of the governing equations and the residuals of the second type of boundary conditions, respectively. It is used for balance and Weight parameters for the size relationship between them; and These represent the governing equation terms and the second-type boundary condition terms in the loss function, respectively. It refers to the number of internal coordinate points; It is the number of points on the second type of boundary.

7. The method for constructing a groundwater movement substitution model based on deep learning according to claim 6, characterized in that, The residuals of the governing equations are calculated by substituting the hydraulic head values ​​predicted by the alternative model into the parameterized groundwater movement equations: Second type of boundary condition residual f 2 is represented as: in, It is an alternative model for predicting water head; in unconfined aquifers Indicates water supply degree, It is a second type of boundary condition.

8. The method for constructing a groundwater movement substitution model based on deep learning according to claim 1, characterized in that, The iterative training process includes: The spatiotemporal configuration points and parameterized variables are input into the substitution model to calculate the original output; The original output is converted into a head prediction value that satisfies the initial conditions and the first type of boundary conditions using a hard constraint method. Substitute the predicted head value into the control equation to calculate the control equation residual, and calculate the residual of the second type of boundary condition on the second type of boundary; The total loss function value is calculated based on the residuals of the governing equations and the residuals of the second type of boundary conditions. Calculate the gradient of the total loss function with respect to the surrogate model parameters, and update the surrogate model parameters using an optimization algorithm; Repeat the above steps until the loss function converges or the preset number of iterations is reached.

9. The method for constructing a groundwater movement substitution model based on deep learning according to claim 1, characterized in that, The uncertainty quantification analysis includes: Based on the statistical characteristics of the permeability coefficient field, multiple sets of KL expansion parameters are generated to characterize different permeability coefficient field implementations; Each set of parameters is input into the trained alternative model to obtain the corresponding head distribution prediction. Statistical analysis is performed on the prediction results to calculate the statistical characteristics of the water head in order to quantify the uncertainty.

10. A system for constructing a groundwater movement alternative model based on deep learning, characterized in that, include: Alternative Model Construction Module: It is used to solve the parametric partial differential equations of groundwater movement in heterogeneous aquifers based on physical information neural networks and construct alternative models; Alternative model optimization module: It is used to construct a loss function that integrates physical information, and to make the output of the alternative model satisfy the initial and boundary conditions through hard constraint methods; it optimizes the parameters of the alternative model through an iterative training process; Predictive analysis module: It is used to predict the distribution of groundwater head under different hydrogeological conditions based on optimized alternative models and complete the uncertainty quantification analysis; The deep learning-based groundwater movement alternative model construction system is used to perform the steps in the deep learning-based groundwater movement alternative model construction method according to any one of claims 1-9.

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