Underground water pollution thermodynamic diagram simulation and prediction method, equipment and medium

Abstract

The invention relates to a groundwater pollution thermodynamic diagram simulation and prediction method, equipment and a medium. The method comprises the following steps: constructing a convection diffusion equation for describing a groundwater pollution physical phenomenon; constructing a blocking residual connection neural network model, embedding a convective diffusion equation into a loss function, optimizing model parameters to minimize the difference between a test solution output by the model and a real solution, and training the model; wherein in the blocking residual error connection neural network model, from a second hidden layer, the neurons are subjected to block processing, the neurons in different blocks are not connected, the neurons in the same block are completely connected with the neurons in the previous layer, and the residual error is added to each block; and performing groundwater pollution diffusion prediction by adopting the trained blocking residual error connection neural network model, and generating a groundwater pollution thermodynamic diagram simulation prediction result. Compared with the prior art, the method has the advantage of high prediction accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of groundwater pollution prediction, and in particular to a method, device and medium for simulating and predicting a groundwater pollution heat map. Background Art

[0002] The sources of groundwater pollution are complex and diverse, including industrial wastewater discharge, leakage of chemical fertilizers and pesticides in agricultural activities, leakage of urban domestic sewage, etc. With the acceleration of the industrialization and urbanization processes, the scope and degree of groundwater pollution are continuously expanding, bringing huge challenges to water resource protection and pollution control. Accurately simulating and predicting the migration and diffusion of groundwater pollutants is of crucial significance for formulating effective pollution control strategies, conducting risk assessments, and protecting limited water resources.

[0003] The simulation of groundwater pollution involves complex physical and chemical processes, mainly including the convection, diffusion, and adsorption of pollutants, etc. These processes are affected by multiple factors such as the flow characteristics of groundwater, the heterogeneity of geological media, and the properties of pollutants themselves. Traditional numerical simulation methods, such as the finite difference method, finite element method, and finite volume method, although can describe the migration process of pollutants to a certain extent, have many limitations. For example, these methods usually require complex grid division, with high computational costs, and are extremely sensitive to initial conditions and boundary conditions. In addition, traditional numerical methods often struggle when dealing with high-dimensional problems and complex non-linear relationships, and are difficult to accurately capture the complex dynamic behaviors in the pollutant migration process. Summary of the Invention

[0004] The purpose of the present invention is to overcome the defects existing in the above-mentioned prior art, and provide a method, device and medium for simulating and predicting a groundwater pollution heat map. By embedding the convection-diffusion equation of the groundwater pollution physical process into the loss function of the neural network, the model can more accurately simulate the migration and diffusion process of pollutants. The blocked residual connection neural network processes neurons in blocks and adds the output of each block to the residual of the input, which can better capture complex non-linear relationships, thereby improving the simulation accuracy, and is especially suitable for complex groundwater pollution migration and diffusion problems.

[0005] The purpose of the present invention can be achieved by the following technical solutions:

[0006] According to the first aspect of the present invention, there is provided a method for simulating and predicting a groundwater pollution heat map, including:

[0007] Construct a convection-diffusion equation for describing the physical phenomena of groundwater pollution;

[0008] Construct a blocked residual connection neural network model, embed the convection-diffusion equation into the loss function, and train the model by optimizing the model parameters to minimize the difference between the experimental solution and the true solution of the model output; among them, in the blocked residual connection neural network model, starting from the second hidden layer, the neurons are processed in blocks, the neurons in different blocks are not connected, the neurons in the same block are fully connected to the neurons in the previous layer, and the input residual is added to each block;

[0009] Use the trained blocked residual connection neural network model to predict the diffusion of groundwater pollution and generate a simulation prediction result of the groundwater pollution heat map.

[0010] Preferably, the constructed convection-diffusion equation for describing the groundwater pollution phenomenon has the expression:

[0011]

[0012] g(t,x) = -e -t (sin(πx) - π 2 sin(πx)), x ∈ [-1,1], t ∈ [0,1] (2)

[0013] Among them, the initial condition is:

[0014] u(0,x) = sin(πx); (3)

[0015] The boundary condition is:

[0016] u(t,-1) = u(t,1) = 0; (4)

[0017] True solution:

[0018] u(t,x) = e -t sin(πx) (5)

[0019] In the formula: u(t,x) represents the displacement of the groundwater pollution source at time t and position x, and is used to describe the evolution of the path of the underground polluted water.

[0020] Preferably, the blocked residual connection neural network model includes an input layer, k hidden layers and an output layer, specifically:

[0021] For the d-dimensional residual input x = (x1,x2,…,x d ), after full connection through the input layer, the output Output a 0 Enters the first hidden layer to obtain the output of the first hidden layer

[0022]

[0023] Wherein: is the weight from the i-th neuron in the input layer to the j-th neuron in the first hidden layer; is the bias of the j-th neuron in the first hidden layer;

[0024] Starting from the second hidden layer, each layer is divided into different blocks; among them, the neurons in different blocks are not connected, the neurons in the same block are fully connected to the neurons in the previous layer, and the d-dimensional residual input x is added to each block;

[0025] The output a of the second hidden layer 2 :

[0026]

[0027] Wherein: N is the number of neurons in the first hidden layer; is the weight from the i-th neuron in the first hidden layer to the j-th neuron in the n-th block of the second hidden layer; is the bias of the j-th neuron in the n-th block of the second hidden layer; is the weight from the i-th neuron in the input layer to the j-th neuron in the n-th block of the second hidden layer;

[0028] The output a of the k-th hidden layer k :

[0029]

[0030] Wherein: N is the number of neurons in the k-th hidden layer; is the weight from the i-th neuron in the k-1-th hidden layer to the j-th neuron in the n-th block of the k-th hidden layer; is the bias of the j-th neuron in the n-th block of the k-th hidden layer; is the weight from the i-th neuron in the input layer to the j-th neuron in the n-th block of the k-th hidden layer;

[0031] The output a of the last hidden layer is fully connected to the output layer, then the output N(θ, x) of the output layer: k N(θ, x) = ω

[0032] N(θ, x) = ω o a k +(b) T (11)

[0033] Wherein: ω o is the weight from the k-th hidden layer to the output layer; b is the bias of the output layer; θ is the parameter loss.

[0034] Preferably, the trial solution output by the blocked residual connection neural network model has the following mathematical expression:

[0035] Φ t = A(x) + B(x)N(x; θ) (12)

[0036] Where: N(θ,x) is the output of the output layer, x is the model input, and θ is the parameter loss; A and B are constructed through the boundary conditions of the convection-diffusion equation.

[0037] Preferably, the parameter loss θ is optimized by minimizing the loss function L(θ) of the numerical solution of the partial differential equation corresponding to the convection-diffusion equation. The expression of the loss function L(θ) is:

[0038]

[0039] Where: Ω is the solution domain; κ(x) is the weight coefficient related to the position x, such as the thermal conductivity, elastic modulus, etc.; is the trial solution of the blocked residual connection neural network model, satisfying is the gradient of; f(x) is the parameter corresponding to the boundary condition of the convection-diffusion equation.

[0040] Preferably, the Monte Carlo method is used to randomly select N1 distribution points P in the solution domain (x,y)∈Ω to discretely estimate the loss function L(θ), and the estimated loss function has the following expression:

[0041]

[0042] Where: is the trial solution of the blocked residual connection neural network model corresponding to the discrete point P(x i ,y i ), is the gradient of, and f(x i ,y i ) is the parameter corresponding to the boundary condition of the convection-diffusion equation at the discrete point P(x i ,y i ).

[0043] Preferably, the adaptive moment estimation algorithm is used to process the network parameter gradient. The update expression of the parameter loss θ is:

[0044]

[0045] Where: is the estimated loss function; represents the gradient of the estimated loss function; α is the learning rate decay value.

[0046] Preferably, based on the deep Ritz method, initialize the blocked residual connection neural network, including setting the solution domain Ω, the learning rate decay value α, and the trial solution of the convection-diffusion equation Use the blocked residual connection neural network model to approximate the true solution of the convection-diffusion equation. The training process specifically includes:

[0047] Take N1 points in the solution domain Ω;

[0048] Calculate the loss function estimate

[0049] Calculate the loss function estimate of the gradient and the corresponding parameter loss θ;

[0050] Through the gradient and the learning rate decay value α, update the parameter loss θ;

[0051] Generate test points and calculate the residuals.

[0052] According to the second aspect of the present invention, there is provided an electronic device, including a memory and a processor, where a computer program is stored on the memory, and when the processor executes the program, it implements the method according to any one of the above.

[0053] According to the third aspect of the present invention, there is provided a computer-readable storage medium, on which a computer program is stored, and when the program is executed by a processor, it implements the method according to any one of the above.

[0054] Compared with the prior art, the present invention has the following beneficial effects:

[0055] 1) By embedding the convection-diffusion equation of the physical process of groundwater pollution into the loss function of the neural network, the model can more accurately simulate the migration and diffusion process of pollutants. The blocked residual connection neural network can better capture complex non-linear relationships by processing neurons in blocks and adding the output of each block to the residual of the input, thereby improving the simulation accuracy, especially suitable for complex groundwater pollution migration and diffusion problems.

[0056] 2) The blocked residual connection neural network structure of the present invention can effectively extract the features of high-dimensional data, improve the adaptability of the model to different geological conditions and pollutant types. Through the blocked residual connection, the model has stronger generalization ability in different scenarios and can more reliably predict the pollution distribution under unknown conditions.

[0057] 3) The adaptive moment estimation algorithm is used to process the gradient of network parameters, which improves the approximation of the test solution and the true solution in the groundwater pollution advection-diffusion equation by the blocked residual connection neural network model. It can adaptively adjust the learning rate according to the gradient changes of different parameters, enabling the model to have stronger generalization ability in different groundwater pollution scenarios and being able to more reliably predict the pollution distribution under unknown conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 is the flowchart of the method of the present invention;

[0059] Figure 2 is the neural network structure diagram;

[0060] Figure 3 is the blocked residual connection neural network structure diagram;

[0061] Figure 4 is the image of the predicted values and the exact solutions of the three neural network methods of LNN, PINN, and BRCNN;

[0062] Figure 5 is the error iteration diagram of the three neural network methods of LNN, PINN, and BRCNN;

[0063] Figure 6 is the comparison diagram of the predicted values and the exact values at different x when t = 0.25;

[0064] Figure 7 is the comparison diagram of the predicted values and the exact values at different x when t = 0.5. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0065] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0066] Embodiment

[0067] As Figure 1 shown, this embodiment provides a method for simulating and predicting the thermal map of groundwater pollution, and the method includes:

[0068] S1. Construct an advection-diffusion equation for describing the physical phenomenon of groundwater pollution, specifically:

[0069]

[0070] g(t,x) = -e -t (sin(πx) - π2 sin(πx)), x ∈ [-1, 1], t ∈ [0, 1] (2)

[0071] Among them, the initial condition is:

[0072] u(0, x) = sin(πx); (3)

[0073] The boundary conditions are:

[0074] u(t, -1) = u(t, 1) = 0; (4)

[0075] True solution:

[0076] u(t, x) = e -t sin(πx) (5)

[0077] In the formula: u(t, x) represents the displacement of the groundwater pollution source at time t and position x, and is used to describe the evolution of the underground polluted water path.

[0078] S2. Construct a blocked residual connection neural network model, embed the convection-diffusion equation into the loss function, and minimize the difference between the experimental solution and the true solution of the model output by optimizing the model parameters to train the model; among them, in the blocked residual connection neural network model, starting from the second hidden layer, the neurons are processed in blocks, the neurons in different blocks are not connected, the neurons in the same block are fully connected to the neurons in the previous layer, and the input residual is added to each block.

[0079] For a multi-layer perceptron (MLP) model, function approximation is realized by constructing a multi-layer fully connected neural network, as Figure 2 shown. For the input x = (x1, x2, L, x d ), d represents the dimension, and the output y has only one dimension. Suppose there are N hidden layers, then N0 = d (neurons in the input layer), N n+1 = 1 (neuron in the output layer), a i is the input value of the i-th layer, is the i-th neuron from the k-th layer, and its output is to the weight of the j-th neuron in the k + 1-th layer, and its output is And is the bias for each neuron in the k-th layer (k = 1, 2,..., n), and the activation function is represented as σ.

[0080] For the input x = (x1, x2, …, x d ), d represents the dimension, and the output y is one dimension.

[0081] Suppose there are N hidden layers, then N0 = d (neurons in the input layer), Nn+1 = 1 (neuron in the output layer), a i is the input value of the i-th layer, is the i-th neuron from the k-th layer, whose output is to the weight of the j-th neuron in the k + 1-th layer, whose output is and is the bias for each neuron in the k-th layer (k = 1, 2,..., n), and the activation function is denoted as σ.

[0082] For d-dimensional input: x = (x1, x2, …, x d )

[0083]

[0084] For a j (j = 1, 2,...) in a 1 in,

[0085]

[0086] Substitute into Equation (6) and rewrite it as:

[0087]

[0088] Considering j = 1, 2,..., N1, we have:

[0089]

[0090] Meanwhile, the weight matrix in Equation (8) can be written in the form of ω 0 of, Then the above formula can be expressed as:

[0091] (a 1 ) T = ω 0 (a 0 ) T + (b 1 ) T (9)

[0092] In this way, the neural network transfers data from the input layer to the first hidden layer. Considering the mapping effect of the activation function σ in the hidden layer, Equation (9) is expressed as:

[0093] (a k+1 ) T = σ(ω k (a k ) T + (b k+1 ) T), k = 0, 1, 2, ..., N - 1 (10)

[0094] Determine the activation function σ through the training loss function, and obtain the optimal network parameters ω and b.

[0095] As Figure 3 shown, in this embodiment, the blocked residual connection neural network model includes an input layer, k hidden layers, and an output layer, specifically:

[0096] For the d-dimensional residual input x = (x1, x2, …, x d ), after full connection through the input layer, the output output a 0 enters the first hidden layer to obtain the output of the first hidden layer

[0097]

[0098] wherein:[[]]END]] is the weight from the i-th neuron of the input layer to the j-th neuron of the first hidden layer; is the bias of the j-th neuron in the first hidden layer;

[0099] Starting from the second hidden layer, each layer is divided into different blocks; among them, the neurons in different blocks are not connected, the neurons in the same block are fully connected to the neurons in the previous layer, and the d-dimensional residual input x is added to each block;

[0100] The output a 2 of the second hidden layer:[[]]END]]

[0101]

[0102] wherein: N is the number of neurons in the first hidden layer; is the weight from the i-th neuron of the first hidden layer to the j-th neuron in the n-th block of the second hidden layer; is the bias of the j-th neuron in the n-th block of the second hidden layer; is the weight from the i-th neuron of the input layer to the j-th neuron in the n-th block of the second hidden layer;

[0103] The output a k of the k-th hidden layer:[[]]END]]

[0104]

[0105] wherein: N is the number of neurons in the k-th hidden layer; is the weight from the i-th neuron of the k - 1-th hidden layer to the j-th neuron in the n-th block of the k-th hidden layer; is the bias of the j-th neuron in the n-th block of the k-th hidden layer; is the weight from the i-th neuron in the input layer to the j-th neuron in the n-th block of the k-th hidden layer;

[0106] The output a of the last hidden layer k is fully connected to the output layer, then the output N(θ,x) of the output layer:

[0107] N(θ,x) = ω o a k + (b) T (16)

[0108] where: ω o is the weight from the k-th hidden layer to the output layer; b is the bias of the output layer; θ is the parameter loss.

[0109] The trial solution of the output of the blocked residual connection neural network model, the mathematical expression is:

[0110] Φ t = A(x) + B(x)N(x;θ) (17)

[0111] where: N(θ,x) is the output of the output layer, x is the model input, θ is the parameter loss; A and B are constructed through the boundary conditions of the convection-diffusion equation.

[0112] The general form of the Poisson equation with homogeneous boundary conditions is known:

[0113]

[0114] Based on the variational principle, construct the energy functional:

[0115]

[0116] Use Green's first formula to rewrite equation (19) as:

[0117]

[0118] The solution of equation (17) is equivalent to the solution u * ∈H 1 and to minimize equation (19), that is:

[0119]

[0120] Discretize equation (20) by the Monte Carlo method;

[0121] According to the deep Ritz method, the parameter loss θ is optimized by minimizing the loss function L(θ) of the numerical solution of the partial differential equation corresponding to the convection-diffusion equation. That is, the expression of the loss function L(θ) corresponding to the numerical solution is:

[0122]

[0123] where: Ω is the solution domain; κ(x) is the weight coefficient related to the position x, such as thermal conductivity, elastic modulus, etc.; is the trial solution of the blocked residual connection neural network model, satisfying is the gradient of; f(x) is the parameter corresponding to the boundary condition of the convection-diffusion equation.

[0124] The parameter θ is optimized by reducing L(θ). The gradient descent method based on Monte Carlo integration is used. That is, in the t-th parameter update, the stochastic gradient descent method is used to randomly select N1 points with a uniform distribution P in (x, y) ∈ Ω, and is used to estimate L in the Monte Carlo integration method:

[0125]

[0126] where: is the trial solution of the blocked residual connection neural network model corresponding to the discrete point P(x i , y i ), satisfying is the gradient of, f(x i , y i ) is the parameter corresponding to the boundary condition of the convection-diffusion equation at the discrete point P(x i , y i ).

[0127] The minimum value of the loss function corresponding to Equation (23) is obtained along the gradient descent direction. The iteration expression is:

[0128]

[0129] where: represents the current momentum at the current time t, is the gradient of the network parameters, and α is the learning rate decay value; the adaptive moment estimation Adam algorithm is used to process the gradient, and then the gradient descent is performed according to the current learning rate.

[0130] In this embodiment, based on the deep Ritz method, the blocked residual connection neural network is initialized, including setting the solution domain Ω, the learning rate decay value α, and the trial solution of the convection-diffusion equation A blocked residual connection neural network model is used to approximate the true solution of the convection-diffusion equation, and the training process is shown in Table 1 below.

[0131] Table 1

[0132]

[0133]

[0134] S3. Use the trained blocked residual connection neural network model to predict the groundwater pollution diffusion, and generate a heat map of groundwater pollution to simulate the prediction results.

[0135] In this embodiment, uniformly distributed sample points are selected within the solution domain. The uniformly distributed sample points can ensure that the performance of the model in different regions can be evaluated, and avoid misjudgment of performance caused by uneven distribution of test points. A total of 2601 (51×51) sample points are selected for training, with a sample spacing of 0.02 in the t direction and 0.04 in the x direction. Among them, 51 points are located on the initial conditions, and 101 points are located on the boundary conditions.

[0136] The error evaluation indicators include:

[0137] Mean Squared Error (MSE): MSE is used to measure the average squared error between the actual observed values (true values) and the predicted values. MSE is more sensitive to larger errors because the square of the error will amplify the impact of the error. It is usually used to evaluate the overall fitting effect of the model.

[0138] Mean Absolute Error (MAE): MAE is used to measure the maximum absolute error between the actual observed values and the predicted values. Feature: MAE directly reflects the error of the model in the worst case and can help evaluate the performance of the model in extreme cases.

[0139] The calculation formulas for MSE and MAE are as follows:

[0140]

[0141] where N represents the total number of data points, y i represents the actual observed values (true values), represents the corresponding predicted values. Through the MSE and MAE indicators, the prediction error of the model is quantified, and the accuracy and reliability of the model are evaluated.

[0142] Numerical experiments

[0143] In this part, numerical experiments demonstrate the effectiveness of the proposed BRCNN method. All experiments were run on a computer with an Intel(R) Xeon(R) CPU E5-2630 V4 @ 2.20GHz, and the code was written in Python 3.6.5. The experiment was the convective diffusion equation described above, and the mean square error (MSE) and maximum absolute error (MAE) were used to measure the error magnitude.

[0144] Table 2 Error comparison of different methods

[0145]

[0146]

[0147] To illustrate the performance of the method, first, classical methods such as the finite element method were compared, and it was found that the BRCNN method could obtain smaller maximum error and mean square error; second, when compared with general DNN, RESNN, and PINN, it can be clearly seen from Table 2 that the solution of the BRCNN approximation equation (1) is better than that of ordinary neural networks. In this sharp contrast, the superiority of the BRCNN method was verified.

[0148] The MSE and MAE between the solution results of BRCNN with different numbers of layers and the exact solution are shown in Table 3. It can be clearly seen from Table 2 that, while keeping the number of training samples unchanged, as the number of layers increases, the error gradually decreases. It is worth noting that when comparing the two-layer structure and the one-layer structure, both the MSE and MAE are significantly reduced by one order of magnitude. However, it is observed that the rate of decrease of the MSE and MAE starts to decline with the introduction of the five-layer structure. Therefore, for comparison with other methods, a four-layer neural network model was selected in this embodiment.

[0149] Table 3 Error comparison of neural networks with different numbers of layers

[0150] Number of layers MSE MAE 1 <![CDATA[6.72×10 -6 > <![CDATA[8.65×10 -4 > 2 <![CDATA[4.19×10 -7 > <![CDATA[8.41×10 -5 > 3 <![CDATA[8.63×10 -8 > <![CDATA[6.05×10 -5 > 4 <![CDATA[3.26×10 -8 > <![CDATA[4.93×10 -5 > 5 <![CDATA[2.84×10 -8 > <![CDATA[4.65×10 -5 > 6 <![CDATA[2.52×10 -8 > <![CDATA[4.41×10 -5 > 7 <![CDATA[2.45×10 -8 > <![CDATA[4.28×10 -5 >

[0151] Table 4 gives the mean square errors obtained by three different neural network methods for solving the diffusion equation at different time points (t = 0.3, t = 0.6, t = 1.0) and spatial positions (x = -0.5, x = 0, x = 0.3, x = 0.6, x = 1.0) after 6000 iterations. It can be clearly seen from Table 4 that in the LNN method, the error magnitude is generally between 10 -3 and 10 -4 with occasional slight variations. Compared with LNN, the PINN method has lower errors. It is worth noting that compared with the first two methods, the BRCNN method achieves a significant reduction in errors, and the error magnitude at most time steps and spatial points is generally between 10 -7 to 10-8 indicates the minimum error.

[0152] Table 4 Mean Square Error of the Diffusion Equation at Different t and x

[0153]

[0154]

[0155] Figure 4 are the predicted values and exact values of the convection-diffusion equation obtained by three different methods. Figure 5 shows the MSE and MAE of the three methods at different iteration times. As Figure 5 shown, PINN performs well at a relatively small number of iteration times (less than 2000 times), but its loss term stabilizes as the number of iteration times increases. In contrast, PINN performs better at a higher number of iteration times. Overall, BRCNN is superior to other algorithms in terms of both convergence speed and prediction accuracy, with an MSE of 6.03×10 -9 , and an MAE of 2.96×10 -6 .

[0156] Figure 6 and Figure 7 respectively describe the comparison between the predicted values and the exact values of BRCNN at different x positions when t = 0.25 and t = 0.5. These data indicate that when the neural network fully captures the smooth region, its performance decreases at the inflection point. To address this limitation, additional training points can be strategically assigned to these key regions during the neural network training process.

[0157] Figure 4 Images of the predicted values and exact solutions of the three neural network methods of LNN, PINN, and BRCNN Figure 5 Error iteration graphs of the three neural network methods of LNN, PINN, and BRCNN Figure 6 Comparison graph of the predicted values and exact values at different x when t = 0.25 Figure 7 Comparison graph of the predicted values and exact values at different x when t = 0.5.

[0158] The present invention proposes a method for simulating and predicting the groundwater pollution heat map based on a Block Residual Connection Neural Network (BRCNN for short). This method aims to solve the limitations existing in traditional numerical simulation methods when dealing with groundwater pollution problems, such as high computational cost, sensitivity to initial and boundary conditions, and difficulty in dealing with complex non-linear relationships. BRCNN enhances the model's feature extraction ability and non-linear approximation ability for complex data by processing neurons in blocks and adding the output of each block to the residual of the input.

[0159] The present invention proposes a deep learning method based on the Block Residual Connection Neural Network (BRCNN) for the method of simulating and predicting the groundwater pollution heat map. Compared with the Finite Difference Method (FDM), the Finite Element Method (FEM), and the Laguerre Neural Network (LNN), BRCNN can avoid discretization and improve the efficiency of the solution. In addition, BRCNN has stronger representation ability, which leads to a higher-precision solution than the Radial Basis Function (RBF) neural network. Similarly, compared with the Deep Neural Network (DNN), the Residual Neural Network (RESNN), and the Physics-Informed Neural Network (PINN), BRCNN can obtain better results in solving oscillatory differential equations and is easier to handle the groundwater pollution simulation equations with complex variations. Therefore, the proposed BRCNN method has significant advantages over existing methods in solving oscillatory groundwater pollution simulation equations.

[0160] The present invention applies the Block Residual Connection Neural Network BRCNN to the simulation and prediction of the groundwater pollution heat map by embedding the physical process of groundwater pollution (such as the convection-diffusion equation) as a constraint. For the first time, the Block Residual Connection Neural Network (BRCNN) is applied to the field of groundwater pollution simulation. BRCNN can effectively process high-dimensional data by processing neurons in blocks and adding the output of each block to the residual of the input, and improve the accuracy of the model by fitting the residual layer by layer.

[0161] The electronic device of the present invention includes a Central Processing Unit (CPU), which can execute various appropriate actions and processes according to the computer program instructions stored in the Read-Only Memory (ROM) or the computer program instructions loaded from the storage unit into the Random Access Memory (RAM). In the RAM, various programs and data required for device operation can also be stored. The CPU, ROM, and RAM are connected to each other through a bus. The Input / Output (I / O) interface is also connected to the bus.

[0162] Multiple components in the device are connected to the I / O interface, including: an input unit, such as a keyboard, a mouse, etc.; an output unit, such as various types of displays, speakers, etc.; a storage unit, such as a disk, an optical disc, etc.; and a communication unit, such as a network card, a modem, a wireless communication transceiver, etc. The communication unit allows the device to exchange information / data with other devices through a computer network such as the Internet and / or various telecommunication networks.

[0163] The processing unit executes the various methods and processes described above, such as methods S1 to S3. For example, in some embodiments, methods S1 to S3 may be implemented as a computer software program that is tangibly contained in a machine-readable medium, such as the storage unit. In some embodiments, part or all of the computer program may be loaded and / or installed onto the device via the ROM and / or the communication unit. When the computer program is loaded into the RAM and executed by the CPU, one or more steps of methods S1 to S3 described above can be executed. Alternatively, in other embodiments, the CPU may be configured to execute methods S1 to S3 by any other suitable means (e.g., by means of firmware).

[0164] The functions described above herein can be performed at least in part by one or more hardware logic components. For example, without limitation, exemplary types of hardware logic components that can be used include: field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), application specific standard products (ASSPs), systems on a chip (SOCs), complex programmable logic devices (CPLDs), and so on.

[0165] The program code for implementing the method of the present invention can be written in any combination of one or more programming languages. These program codes can be provided to a processor or controller of a general-purpose computer, a special-purpose computer, or other programmable data processing devices, such that when the program codes are executed by the processor or controller, the functions / operations specified in the flowcharts and / or block diagrams are implemented. The program codes can be executed entirely on the machine, partially on the machine, as an independent software package partially on the machine and partially on a remote machine, or entirely on a remote machine or server.

[0166] In the context of the present invention, a machine-readable medium can be a tangible medium that can contain or store a program for use by or in connection with an instruction execution system, apparatus, or device. The machine-readable medium can be a machine-readable signal medium or a machine-readable storage medium. The machine-readable medium can include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of the machine-readable storage medium would include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0167] As described above, the above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention can easily think of various equivalent modifications or substitutions, and these modifications or substitutions should all be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims

1. A method for simulating and predicting groundwater pollution heat map, characterized in that: include: Construct the convection-diffusion equations used to describe the physics of groundwater contamination; A blocked residual connection neural network model is constructed, the convection-diffusion equation is embedded into the loss function, and the model is trained by optimizing the model parameters to minimize the difference between the experimental solution and the true solution output by the model; wherein, in the blocked residual connection neural network model, starting from the second hidden layer, the neurons are processed in blocks, the neurons in different blocks are not connected, the neurons in the same block are fully connected to the neurons in the previous layer, and the input residual is added to each block; The trained blocked residual connection neural network model is used to predict the diffusion of groundwater pollution and generate the simulation prediction results of groundwater pollution heat map.

2. A method for simulating and predicting groundwater pollution heat map according to claim 1, characterized in that: The construction is used to describe the convection-diffusion equation of groundwater pollution, and the expression is: g(t,x)=-e -t (sin(πx)-π 2 sin(πx)),x∈[-1,1],t∈[0,1](2) The initial conditions are: u(0,x)=sin(πx); (3) The boundary conditions are: u(t,-1)=u(t,1)=0;(4) Real solution: u(t,x)=e -t sin(πx)(5) Where: u(t,x) represents the displacement of the groundwater pollution source at time t and position x, which is used to describe the evolution of the underground contaminated water path.

3. A method for simulating and predicting groundwater pollution heat map according to claim 1, characterized in that: The blocked residual connection neural network model includes an input layer, k hidden layers and an output layer, specifically: For the d-dimensional residual input x=(x1,x2,…,x d ), and the output is obtained after the full connection of the input layer Output a 0 Enter the first hidden layer and get the output of the first hidden layer Where: is the weight from the i-th neuron in the input layer to the j-th neuron in the first hidden layer; is the bias of the jth neuron in the first hidden layer; Starting from the second hidden layer, each layer is divided into different blocks; neurons in different blocks are not connected, neurons in the same block are fully connected to neurons in the previous layer, and the d-dimensional residual input x is added to each block; The output of the second hidden layer is a 2 : Where: N is the number of neurons in the first hidden layer; is the weight from the i-th neuron in the first hidden layer to the j-th neuron in the n-th block of the second hidden layer; is the bias of the jth neuron in the nth block of the second hidden layer; is the weight from the i-th neuron in the input layer to the j-th neuron in the n-th block of the second hidden layer; The output of the kth hidden layer is a k : Where: N is the number of neurons in the kth hidden layer; is the weight from the i-th neuron in the k-1-th hidden layer to the j-th neuron in the n-th block of the k-th hidden layer; is the bias of the jth neuron in the nth block of the kth hidden layer; is the weight from the i-th neuron in the input layer to the j-th neuron in the n-th block of the k-th hidden layer; The output of the last hidden layer is a k Fully connected to the output layer, the output of the output layer is N(θ,x): N(θ,x)=ω o a k +(b) T (11) Where: o is the weight from the kth hidden layer to the output layer; b is the bias of the output layer; θ is the parameter loss.

4. A method for simulating and predicting groundwater pollution heat map according to claim 1, characterized in that: The experimental solution of the output of the blocked residual connection neural network model is expressed as follows: F t =A(x)+B(x)N(x;θ)(12) Where: N(θ,x) is the output of the output layer, x is the model input, and θ is the parameter loss; A and B are constructed through the boundary conditions of the convection-diffusion equation.

5. A method for simulating and predicting groundwater pollution heat map according to claim 4, characterized in that: The parameter loss θ is optimized by minimizing the loss function L(θ) of the numerical solution of the partial differential equation corresponding to the convection-diffusion equation. The expression of the loss function L(θ) is: Where: Ω is the solution domain; κ(x) is the weight coefficient related to the residual input x; is the experimental solution of the blocked residual connection neural network model, satisfying for The gradient of ; f(x) is the parameter corresponding to the boundary condition of the convection-diffusion equation.

6. A method for simulating and predicting groundwater pollution heat map according to claim 5, characterized in that: Using the Monte Carlo method, N1 distribution points P are randomly selected in the solution domain (x, y)∈Ω to perform discrete estimation of the loss function L(θ), and the loss function estimate is obtained. The expression is: Where: For discrete points P(x i ,y i ) is the experimental solution of the blocked residual connection neural network model corresponding to for The gradient of f(x i ,y i ) is a discrete point P(x i ,y i ) is the parameter corresponding to the boundary conditions of the convection-diffusion equation.

7. A method for simulating and predicting groundwater pollution thermodynamic maps according to claim 6, characterized in that: The adaptive moment estimation algorithm is used to process the network parameter gradient, and the parameter loss θ update expression is: Where: is the loss function estimate; Represents the gradient of the loss function estimate; α is the learning rate decay value.

8. A method for simulating and predicting groundwater pollution heat map according to claim 7, characterized in that: Based on the deep Ritz method, the blocked residual connection neural network is initialized, including setting the solution domain Ω, the learning rate decay value α and the experimental solution of the convection diffusion equation A blocked residual connection neural network model is used to approximate the true solution of the convection-diffusion equation. The training process specifically includes: Take N1 points in the solution domain Ω; Compute loss function estimate Compute loss function estimate Gradient And the corresponding parameter loss θ; By gradient Update the parameter loss θ with the learning rate decay value α; Generate test points and compute residuals.

9. An electronic device comprising a memory and a processor, wherein a computer program is stored in the memory, wherein: When the processor executes the program, the method according to any one of claims 1 to 8 is implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the method according to any one of claims 1 to 8 is implemented.

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