A heterogeneous aquifer permeability inversion method based on PINN and CGAN network

Abstract

The application provides a heterogeneous aquifer permeability coefficient inversion method based on a PINN and CGAN network, and relates to the technical field of heterogeneous aquifer permeability coefficient inversion and identification. The application comprises the following steps: constructing an initial model, obtaining small sample data of a to-be-measured region and inputting the small sample data into the initial model to obtain a corresponding permeability coefficient field and a water head value at a flow field sampling point; constructing a physical constraint condition; performing loss calculation according to the physical constraint condition and the corresponding permeability coefficient field and the water head value at the flow field sampling point to obtain a generator loss function and a discriminator loss function; updating the initial model by using the generator loss function and the discriminator loss function to obtain a final model; and determining an optimal permeability coefficient according to the final model. The application solves the problem that the hydrogeological parameter inversion method in the prior art is limited in terms of sample scarcity, spatial heterogeneity and calculation stability, and it is difficult to efficiently and accurately obtain a continuous permeability coefficient field at a regional scale.

Description

Technical Field

[0001] This invention relates to the field of permeability coefficient inversion and identification technology for heterogeneous aquifers, and in particular to a method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks. Background Technology

[0002] The estimation and identification of hydrogeological parameters has always been a key focus and challenge in groundwater research, especially in areas with limited reference data. The inversion of global hydrogeological parameters poses a significant challenge to researchers in the field. Currently, common methods for hydrogeological parameter inversion include experimental methods, numerical simulation methods, data assimilation methods, and deep learning methods. Experimental methods require quantitative parameters such as the skin factor, which are inherently difficult to obtain and are not suitable for regional studies. Numerical simulation methods are computationally intensive and limited in their ability to address complex spatial heterogeneity issues. Data assimilation methods mostly require a batch of continuous real-world background parameters. Deep learning models achieve the conversion from linear to nonlinear parameters, but the resulting models are computationally intensive and their stability is difficult to guarantee. It can be seen that data assimilation and deep learning methods show potential in solving the problem of parameter field heterogeneity, focusing on simulating the generation of the parameter field. Given that the continuous distribution of regional parameter fields is difficult to determine in advance in practical work, the inversion problem needs to focus more on deep learning methods, which have a lower dependence on such distributions. Summary of the Invention

[0003] To overcome the shortcomings of existing technologies, the purpose of this invention is to provide a method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks. This invention solves the problem that existing hydrogeological parameter inversion methods are limited in terms of sample scarcity, spatial heterogeneity, and computational stability, making it difficult to efficiently and accurately obtain continuous permeability coefficients at the regional scale.

[0004] To achieve the above objectives, the present invention provides the following solution:

[0005] A method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks includes:

[0006] Construct an initial PICGAN model, which includes an initial generator and an initial discriminator;

[0007] Small sample data of the area to be tested are obtained and input into the initial PICGAN model to obtain the corresponding permeability field and hydraulic head value at the flow field sampling point;

[0008] Construct physical constraints;

[0009] Based on the physical constraints and the corresponding permeability coefficient field and the head value at the flow field sampling point, the loss is calculated to obtain the generator loss function and the discriminator loss function.

[0010] The initial PICGAN model is updated using the generator loss function and the discriminator loss function until the initial generator and the initial discriminator reach a stable state, so as to obtain the final PICGAN model.

[0011] The optimal permeability coefficient is determined based on the final PICGAN model.

[0012] Preferably, the initial generator uses a residual structure to link the Dense layer.

[0013] Preferably, the physical constraints include:

[0014] The loss includes the governing equation, boundary condition loss, and initial condition loss.

[0015] Preferably, the expression for the loss of the governing equation is:

[0016] ;

[0017] Among them, S s The water storage rate of a confined aquifer, K is the permeability coefficient of the confined aquifer, and H is the hydraulic head of the confined aquifer.

[0018] Preferably, the expression for the boundary condition loss is:

[0019] ;

[0020] Where h is the head value output by the sampling point discriminator; This represents the actual boundary head value at the corresponding sampling point.

[0021] Preferably, the expression for the initial conditional loss is:

[0022] ;

[0023] in, This represents the actual head value at the initial moment of the corresponding sampling point.

[0024] Preferably, the expression for the discriminator loss function is:

[0025] Loss dis =L physical +L data ;

[0026] Among them, L physical Generate value K for the generator gen The discriminator generates the value h.p The calculation error in the physical governing equations; L data The discriminator uses a generator to generate the value K. gen Simulated head value h generated with spatiotemporal information p Compared with the actual head value h r The error;

[0027] L data = ;

[0028] in, The spatial discrete point coordinate pairs sampled by the discriminator; These are the discrete time points sampled by the discriminator; The permeability coefficient value generated by the generator for the corresponding sampling point; This represents the head value at the sampling point.

[0029] Preferably, the generator loss function is expressed as follows:

[0030] Loss gen =L k_real +L physical ;

[0031] Among them, L physical Generate value K for the generator gen The discriminator generates the value h. p The calculation error in the physical governing equations; L k_real K is the true value of spatial discrete sampling real Generate value K at the corresponding position of the generator gen The error;

[0032] L k_real = ;

[0033] in, Spatial discrete point coordinate pairs sampled by the generator; This corresponds to the actual permeability coefficient value.

[0034] The present invention discloses the following technical effects:

[0035] This invention provides a method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks, comprising: constructing an initial PICGAN model, wherein the initial PICGAN model includes an initial generator and an initial discriminator; acquiring small sample data of the area to be measured and inputting it into the initial PICGAN model to obtain the corresponding permeability coefficient field and hydraulic head values ​​at the flow field sampling points; constructing physical constraints; performing loss calculations based on the physical constraints and the corresponding permeability coefficient field and flow field sampling points to obtain generator loss functions and discriminator loss functions; updating the initial PICGAN model using the generator loss functions and discriminator loss functions until the initial generator and initial discriminator reach a stable state to obtain the final PICGAN model; and determining the optimal permeability coefficient based on the final PICGAN model. This invention is based on small sample hydrogeological parameters and hydraulic head data. It uses a residual network to construct a PICGAN model for hydrogeological parameter inversion and identification. While ensuring accuracy, this invention can fully explore and utilize a small amount of known hydrogeological parameter information and its relationship with hydraulic head changes, thereby using a generative model to invert and generate a global-scale heterogeneous aquifer permeability coefficient field. Attached Figure Description

[0036] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0037] Figure 1 A flowchart illustrating a method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks, provided for embodiments of the present invention.

[0038] Figure 2 The flowchart of the PICGAN model provided in the embodiments of the present invention;

[0039] Figure 3 A technical flowchart provided for embodiments of the present invention;

[0040] Figure 4 A schematic diagram of the residual network structure of the present invention provided in an embodiment of the present invention;

[0041] Figure 5 A schematic diagram of the activation function used in the embodiments of the present invention.

[0042] Figure 6 A schematic diagram of a calculation example provided in the embodiments of the present invention.

[0043] Figure 7Loss variation of generator and discriminator provided in embodiments of the present invention

[0044] Figure 8 This invention provides a generated permeability coefficient field and its error distribution for heterogeneous confined aquifers in embodiments of the present invention. Detailed Implementation

[0045] 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.

[0046] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0047] like Figure 1 As shown, this invention provides a method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks, comprising:

[0048] Step 100: Construct the initial PICGAN model, which includes an initial generator and an initial discriminator;

[0049] Step 200: Obtain small sample data of the area to be tested. In this study, small sample data refers to the actual permeability coefficient value set in the calculation example (in addition, different types of hydrogeological parameters such as infiltration coefficient, water supply coefficient and water storage coefficient in hydrogeological research are all small sample data). Input them into the initial PICGAN model to obtain the corresponding permeability coefficient field and hydraulic head value at the flow field sampling point.

[0050] Step 300: Construct physical constraints;

[0051] Step 400: Calculate the loss based on the physical constraints and the corresponding permeability coefficient field and head value at the flow field sampling point to obtain the generator loss function and the discriminator loss function;

[0052] Step 500: Update the initial PICGAN model using the generator loss function and the discriminator loss function until the initial generator and the initial discriminator reach a stable state to obtain the final PICGAN model.

[0053] Step 600: Determine the optimal permeability coefficient based on the final PICGAN model.

[0054] Furthermore, the initial generator employs a residual structure to link the Dense layer.

[0055] Specifically, the construction of the permeability coefficient generator includes: obtaining real permeability coefficient samples and their spatial location information; constructing a Dense layer based on Linear and ELU functions; using a residual structure to link the Dense layer to form a generator; and substituting the spatial information of the parameter samples into the generator to generate a global permeability coefficient field.

[0056] The construction of the flow field discriminator includes: acquiring real flow field samples and their spatial location information; constructing a Dense layer based on Linear and ELU functions; using a residual structure to link the Dense layer to form a generator; and substituting the generated global permeability coefficient field and the spatial information of the flow field samples into the discriminator to generate the sample point head.

[0057] Furthermore, such as Figure 2-3 As shown, a numerical model for the embodiment is established and solved. The embodiment is set as a two-dimensional unsteady flow in a confined aquifer, see [link to example]. Figure 4 , Figure 4 a shows the mesh subdivision, sampling point distribution, and boundary type of the embodiment. Figure 4 b shows the actual permeability coefficient distribution in the example. The example area is defined as a rectangular region of 500m × 500m, discretized into a 10m × 10m grid. The north and south sides are set as impermeable boundaries, the western boundary is set as a given head boundary with a head value of 10m, and the eastern side is set as an outflow boundary with a flow rate of -10m. 3 / d. Initially, the hydraulic head in the area was 0m, and water was replenished at the western boundary until the flow field reached stability. The aquifer consisted of heterogeneous fine sand with a high permeability coefficient. Logarithmic parameters were set with a mean of 2m / d, a standard deviation of 0.5m / d, and a range of 20m. The total simulation period was 50 days, with 1 day as one stress period, uniformly discretized into 50 stress periods. Numerical methods were used to solve the model, resulting in 50 stress periods, each with 2500 data points, for a total of 125,000 hydraulic head data points.

[0058] The model is constructed using a generator and a discriminator. The generator receives the spatial location information of the parameter field to be generated, passes it through two dense layers, activates it through a Layer_after layer, and then enters the output layer. The output layer directly performs a linear transformation using a Linear function and outputs the result without further activation. The logarithmic transformation of the output result yields the generated permeability coefficient field. The discriminator also consists of two dense layers, one Layer_after layer, and one Layer_out layer. The discriminator receives the coordinate information, time information, and permeability coefficients generated and transformed by the generator from the flow field samples. After calculation, it outputs the simulated head values ​​for the flow field sample points.

[0059] Furthermore, the physical constraints include:

[0060] The loss includes the governing equation, boundary condition loss, and initial condition loss.

[0061] Specifically, physical constraints are incorporated into the neural network. The discriminator generates simulated head values ​​and their corresponding spatiotemporal information, which are then input into the physical constraints, i.e., substituted into the hydrodynamic control equations, boundary control equations, and initial condition control equations to calculate the loss.

[0062] The physical constraint loss and the permeability coefficient sampling point loss are superimposed to form the generator's total loss, while the physical constraint loss and the flow field sampling point loss are superimposed to form the discriminator's total loss. The hyperparameters of the generator and discriminator are continuously updated using backpropagation until the loss reaches the convergence criterion. At this point, the permeability coefficient field generated by the generator becomes the final inversion identification result of the model.

[0063] Using global spatial information as input, the output is the global permeability coefficient value:

[0064] Configure the number of layers, neurons per layer, and activation function of the generator network, and input spatial information data into the generator residual network, such as... Figure 5 The trained generator shown makes:

[0065] GM F ( );

[0066] Where GM represents the GRU neural network model, and F represents the mapping relationship between the feature variables that the generator can learn and the permeation coefficient.

[0067] The discriminator generates simulated head values ​​including:

[0068] Using the spatiotemporal information of the flow field sampling points and the corresponding permeability coefficients generated by the generator as input, the output is the simulated head value of the flow field sampling points:

[0069] Set the number of layers, the number of neurons per layer, and the activation function of the discriminator network. Substitute the input data into the discriminator network and train the discriminator so that:

[0070] DM F' ( );

[0071] Where GM represents the GRU neural network model, and F' represents the mapping relationship between the discriminator's learnable feature variables and the head value.

[0072] Accuracy evaluation of the generator and discriminator. The root mean square error and Nash efficiency coefficient are selected to evaluate the PICGAN model.

[0073] The root mean square error can be expressed as:

[0074] E RMSE= ;

[0075] In the formula: x iT The actual permeability coefficient at grid i in the example is shown; x iS The value of the permeability coefficient generated at grid i in the example is shown; n is the grid involved in the calculation.

[0076] The Nash efficiency coefficient can be expressed as:

[0077] N=1- ;

[0078] In the formula: p i This is the simulated value of the permeability coefficient; i This represents the true value of the permeability coefficient; is the mean of the true values; n is the number of samples involved in the calculation.

[0079] Figure 6 The activation function used by the generator and discriminator—the ELU function—can be expressed as follows:

[0080] ELU(x) ;

[0081] In the formula: x is the data that has undergone linear layer transformation; α is a preset parameter used to control the saturation of the output when x is negative.

[0082] The ELU function is a non-linear differentiable function that can learn complex data relationships within a network and helps improve the stability of gradient descent. Compared to the commonly used ReLU function, the ELU function can accept negative inputs and output negative values, thereby mitigating the vanishing gradient problem and facilitating data centralization.

[0083] Furthermore, the expression for the loss of the governing equation is:

[0084] ;

[0085] Among them, S s The water storage rate of a confined aquifer, K is the permeability coefficient of the confined aquifer, and H is the hydraulic head of the confined aquifer.

[0086] Furthermore, the expression for the boundary condition loss is:

[0087] ;

[0088] Where h is the head value output by the sampling point discriminator; This represents the actual boundary head value at the corresponding sampling point.

[0089] Furthermore, the expression for the initial conditional loss is:

[0090] ;

[0091] in, This represents the actual head value at the initial moment of the corresponding sampling point.

[0092] Furthermore, the expression for the discriminator loss function is as follows:

[0093] Loss dis =L physical +L data ;

[0094] Among them, L physical Generate value K for the generator gen The discriminator generates the value h. p The calculation error in the physical governing equations; L data The discriminator uses a generator to generate the value K. gen Simulated head value h generated with spatiotemporal information p Compared with the actual head value h r The error;

[0095] L data = ;

[0096] in, The spatial discrete point coordinate pairs sampled by the discriminator; These are the discrete time points sampled by the discriminator; The permeability coefficient value generated by the generator for the corresponding sampling point; This represents the head value at the sampling point.

[0097] Furthermore, the expression for the generator loss function is:

[0098] Loss gen =L k_real +L physical ;

[0099] Among them, L physical Generate value K for the generator gen The discriminator generates the value h. p The calculation error in the physical governing equations; L k_real K is the true value of spatial discrete sampling real Generate value K at the corresponding position of the generator gen The error;

[0100] L k_real = ;

[0101] in, Spatial discrete point coordinate pairs sampled by the generator; This corresponds to the actual permeability coefficient value.

[0102] Furthermore, at the start of training, the weights of each layer are randomly assigned. The network is trained using the Adam optimizer with a learning rate of 0.0001. The initial number of nodes (neurons) is set to 512, and the number of layers is set to 2. The model is trained for 500 epochs. The loss changes of the generator and discriminator during training are shown in [link to documentation]. Figure 7 ,in Figure 7 'a' shows the changes in the generator's loss. Figure 7 b shows the changes in the discriminator's loss.

[0103] As shown in the figure, both the generator and the discriminator decreased simultaneously, gradually stabilizing after 100 training iterations.

[0104] Figure 8 This shows the global permeability coefficient field finally generated by the generator and its error distribution compared with the true value, where Figure 8 'a' represents the global continuous permeability coefficient field ultimately generated by the generator. Figure 8 b shows the error distribution after subtracting the generated permeability field from the actual permeability field. It can be seen that the generated and actual values ​​have a high degree of consistency in both numerical values ​​and trends. After 500 training iterations, the generator's global root mean square error is 0.69 m / d, and its Nash efficiency coefficient is 0.95. The discriminator's global root mean square error is 0.94 m, and its Nash efficiency coefficient is 0.89. This indicates that the established model can effectively invert and identify the global permeability field using a small amount of sample data, and can also simulate and predict head changes.

[0105] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0106] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks, characterized in that, include: Construct an initial PICGAN model, which includes an initial generator and an initial discriminator; Inputting small sample data into the initial PICGAN model yields the corresponding global permeability field and head values ​​at the flow field sampling points; Construct physical constraints; Based on the physical constraints and the corresponding permeability coefficient field and the head value at the flow field sampling point, the loss is calculated to obtain the generator loss function and the discriminator loss function. The initial PICGAN model is updated using the generator loss function and the discriminator loss function until the initial generator and the initial discriminator reach a stable state, so as to obtain the final PICGAN model. The optimal permeability coefficient is determined based on the final PICGAN model.

2. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 1, characterized in that, The initial generator uses a residual structure to link the Dense layer.

3. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 1, characterized in that, The physical constraints include: The loss includes the governing equation, boundary condition loss, and initial condition loss.

4. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 3, characterized in that, The expression for the loss of the governing equation is: ； Among them, S s The water storage rate of a confined aquifer, K is the permeability coefficient of the confined aquifer, and H is the hydraulic head of the confined aquifer.

5. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 4, characterized in that, The expression for the boundary condition loss is: ； Where h is the head value output by the sampling point discriminator; This represents the actual boundary head value at the corresponding sampling point.

6. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 5, characterized in that, The expression for the initial conditional loss is: ； in, This represents the actual head value at the initial moment of the corresponding sampling point.

7. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 6, characterized in that, The expression for the discriminator loss function is: Loss dis =L physical +L data ; Among them, L physical Generate value K for the generator gen The discriminator generates the value h. p Calculation error in the physical governing equations; L data The discriminator uses a generator to generate the value K. gen Simulated head value h generated with spatiotemporal information p Compared with the actual head value h r The error; L data = ； in, The spatial discrete point coordinate pairs sampled by the discriminator; These are the discrete time points sampled by the discriminator; The permeability coefficient value generated by the generator for the corresponding sampling point; This represents the head value at the sampling point.

8. The method for inverting the permeability coefficient of heterogeneous aquifers based on PINN and CGAN networks according to claim 7, characterized in that, The generator loss function is expressed as follows: Loss gen =L k_real +L physical ; Among them, L physical Generate value K for the generator gen The discriminator generates the value h. p Calculation error in the physical governing equations; L k_real K is the true value of spatial discrete sampling real Generate value K at the corresponding position of the generator gen The error; L k_real = ； in, Spatial discrete point coordinate pairs sampled by the generator; This corresponds to the actual permeability coefficient value.

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