A new simulation method for characterizing the spatiotemporal evolution of groundwater levels

Abstract

The present invention discloses a novel simulation method for characterizing the spatiotemporal evolution of groundwater levels, belonging to the field of computer machine learning technology. The novel simulation method for characterizing the spatiotemporal evolution of groundwater levels obtains long-term groundwater level data based on long-term groundwater observation wells. Based on the groundwater level information, multiple hydrogeological parameter information is superimposed, and a two-dimensional groundwater seepage equation is used as a physical mechanism to construct a regional neural network model with a physical mechanism, thereby simulating the temporal and spatial evolution of the regional groundwater level. The present invention can strengthen the rational use of groundwater resources and implement scientific and effective supervision, providing a scientific basis for protecting regional groundwater safety and promoting the sustainable development of groundwater resources.

Description

Technical Field

[0001] The present invention relates to the technical field of groundwater simulation, and in particular to a novel simulation method for characterizing the temporal and spatial evolution of groundwater levels. Background Art

[0002] As one of Earth's most important freshwater resources, groundwater not only impacts the ecological environment but also controls socioeconomic development. Especially in areas lacking surface water, groundwater is often the only reliable source. Globally, over one-third of all water used by humans comes from groundwater. However, groundwater shortages are exacerbated by factors such as human activities and climate change, leading to numerous geological and environmental problems such as land subsidence, groundwater pollution, and ecosystem degradation. To effectively manage and protect groundwater resources, groundwater simulation and prediction are urgently needed.

[0003] Traditional groundwater simulation methods require clear discretization equations and the construction of networks based on the geometric structure and boundary conditions of the problem. For complex geometric shapes or multi-physics field coupling problems, modeling and solving may be very complex and time-consuming. Traditional groundwater simulation methods are mostly based on control equations, so there are obvious limitations in dealing with nonlinear problems. Compared with traditional numerical simulation methods, machine learning and deep learning algorithms have obvious advantages in solving nonlinear problems and improve computational efficiency. However, such methods are all driven by data, and the model lacks physical constraints, which makes the internal mechanism of the model difficult to explain and the reliability of the model results is questioned. In order to solve the above problems, the present invention proposes a new simulation method for characterizing the spatiotemporal evolution of groundwater levels. Summary of the Invention

[0004] The purpose of this invention is to obtain the changing information of groundwater in time series and quantitatively describe the evolution law, thereby proposing a new simulation method for characterizing the spatiotemporal evolution of groundwater level and performing efficient groundwater level simulation.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A new simulation method for characterizing the spatiotemporal evolution of groundwater levels includes the following steps:

[0007] S1. Obtain monthly precipitation data covering the study area through long-term satellite remote sensing precipitation products; obtain monthly groundwater level information through long-term groundwater observation wells within the study area.

[0008] S2. Obtain basic hydrogeological data within the study area and, combined with the groundwater control equation, construct a two-dimensional groundwater seepage equation for the study area.

[0009] S3. Based on the two-dimensional groundwater seepage equation in the study area and combined with the neural network model, a deep learning neural network model that incorporates physical knowledge - the PINNs model - was created.

[0010] S4, using the precipitation data and groundwater level data obtained in S1 as input to train the deep learning neural network model integrating physical knowledge - PINNs model, and obtain the data error term between the network output and the training data;

[0011] S5, bringing the network output result obtained from the model in S4 into the physical equation, solving the physical equation through automatic differentiation to obtain the physical residual term;

[0012] S6. Superimpose the data error term obtained in S4 and the physical residual term obtained in S5 to obtain the overall loss of the PINNs model, which is back-propagated back to the neural network model. Learning is performed through model iteration to achieve accurate simulation of the groundwater level.

[0013] Preferably, when S1 obtains long-term satellite remote sensing precipitation products, it uses geoscience-related software for processing. The geoscience-related processing software is ArcGIS, QGIS or GeoScence .

[0014] Preferably, it is characterized in that said S2 constructing the two-dimensional groundwater seepage equation of the study area specifically comprises the following steps:

[0015] S2.1. Based on the groundwater governing equation, determine the physical laws used for the neural network constraints. Under heterogeneous isotropic conditions, the basic differential equation for the unstable motion of groundwater can be expressed as:

[0016]

[0017] Where h represents the groundwater level, Represents the water storage rate of the water-containing medium; and are the permeability coefficients of the aquifer in the x and y directions, respectively. Under isotropic conditions, for any point in space, the value of K is the same in all directions ( ), but in space, the value of K will vary - heterogeneity; W is the source and sink term, which represents the impact of external recharge or pumping;

[0018] S2.2, the equation satisfies the following conditions under the initial conditions:

[0019]

[0020] Where, represents the groundwater level at the initial moment of the model; t represents time;

[0021] Preferably, the S3 constructs a deep learning neural network model that incorporates physical knowledge, and its construction specifically includes the following steps:

[0022] S3.1. Construct the model input layer to receive groundwater level information including: water level height, water point location; physical parameter information including permeability coefficient; source and sink information. The input layer is connected to the first hidden layer through a fully connected layer. If the input data is denoted as X, the output data is:

[0023]

[0024] Where, represents the output of the first hidden layer, represents the weight matrix of the first layer, represents the hidden layer bias vector of the first layer, Represents the activation function of the network, common ones include ReLU, Sigmoid and Tanh;

[0025] S3.2 When there are multiple hidden layers, each hidden layer is connected to the previous layer through a fully connected layer:

[0026]

[0027] Where, represents the output of the nth hidden layer, represents the weight matrix of the nth hidden layer, represents the bias vector of the nth hidden layer, represents the activation function of the network, represents the input of the nth hidden layer, which is also the output of the n-1th hidden layer;

[0028] S3.3. Finally, the hidden layer and the output layer are also connected through a fully connected layer. The output layer can be expressed as:

[0029]

[0030] Where, is the prediction result of the network, is the weight matrix of the output layer, is the bias vector of the output layer, is the output of the last hidden layer;

[0031] S3.4. Through automatic differentiation technology, the output layer prediction results are connected with the physical equations, and the physical equations are used as global constraints to guide the neural network training process.

[0032] Preferably, the data error term between the PINNs model network output and the training data in S4 is calculated by the model MSE:

[0033]

[0034] Where, represents the true value of the groundwater level, represents the groundwater value predicted by the model, and N represents the sample size.

[0035] Preferably, the process of solving the physical equation in S5 to obtain the calculation of the physical residual term specifically includes the following steps:

[0036] S5.1. Water level values ​​predicted using the network , and calculate the partial derivatives by automatic differentiation:

[0037] Compute the first-order derivative: 、 and ;

[0038] Compute the second-order derivative:

[0039] S5.2. Calculate the loss function:

[0040] Substitute the derivative of the network output into the physical equation to calculate:

[0041]

[0042] In the formula, N represents the sample size.

[0043] Preferably, the model back propagation and parameter iterative update in S6 specifically include the following steps:

[0044] S6.1. The total loss function of the PINNs network is composed of two weighted parts: the physical residual term of the physical equation and the data error term of the neural network model:

[0045]

[0046] Where, represents the total loss term of the PINNs model, and is the weight coefficient used to balance the data error term and the physical residual, and are the physical residual term and the data error term respectively;

[0047] S6.2, in the process of back propagation, The gradient of the loss with respect to the network output is calculated:

[0048]

[0049] Where, represents the total loss term of the PINNs model, and is the weight coefficient, and are the physical residual term and the data error term respectively, N is the sample size, Indicates the true groundwater level value, represents the groundwater level value predicted by the model;

[0050] S6.3. Using the calculated gradient, update the weights in the network through an optimization algorithm, and continuously iterate the training until the model loss function converges; the optimization algorithm is SGD or Adam.

[0051] Compared with the existing technology, the present invention provides a new simulation method for characterizing the spatiotemporal evolution of groundwater levels, which has the following beneficial effects:

[0052] The present invention obtains regional precipitation information based on remote sensing image data; further obtains long-term groundwater level data based on long-term groundwater observation wells; based on the groundwater level information, multiple hydrogeological parameter information is superimposed, and a two-dimensional groundwater seepage equation is used as the physical mechanism to construct a regional neural network model with a physical mechanism, thereby efficiently simulating the temporal and spatial evolution of the regional groundwater level, providing a scientific basis for protecting the safety of regional groundwater and promoting the sustainable development of groundwater resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is a flow chart of a novel simulation method for characterizing the spatiotemporal evolution of groundwater levels proposed in the present invention;

[0054] Figure 2 This is the network structure of a new neural network model for simulating the spatiotemporal evolution of groundwater levels proposed by the present invention;

[0055] Figure 3 Schematic diagram of the contribution of each characteristic factor to the network. DETAILED DESCRIPTION

[0056] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0057] As one of the most important freshwater resources on Earth, groundwater not only affects the ecological environment but also controls social and economic development. Traditional groundwater simulation methods mainly rely on numerical solutions such as the finite element method (FEM), the finite difference method (FDM), and the finite volume method (FVM). Such methods require explicit discretization of equations and construction of networks based on the geometric structure and boundary conditions of the problem. For complex geometries or multi-physics field coupling problems, modeling and solving can be very complex and time-consuming. In addition, such methods show obvious limitations in dealing with nonlinear problems. To address the above problems, the present invention proposes a new simulation method for characterizing the spatiotemporal evolution of groundwater levels.

[0058] PINNs is a neural network framework that integrates physical knowledge into deep learning. The core idea of ​​PINNs is to embed physical laws into the loss function of the neural network, so that the model can not only fit the data, but also guide the training process through the constraints of physical processes, thereby enhancing the interpretability of the model. The prediction results of the model can be explained by physical laws, thereby reducing the "black box" characteristics of the neural network. Because the embedded physical mechanism is universal, PINNs can show better generalization ability when dealing with unseen physical scenarios or conditions. Although PINNs have achieved success in many fields, their application research in groundwater simulation is still relatively limited. This paper generalizes the groundwater flow in the study area as a two-dimensional unsteady flow problem, integrates data such as groundwater time series water level, water level geological parameters and precipitation, and explores the potential of PINNs in groundwater level simulation and prediction by using the basic differential equations of groundwater unsteady motion as neural network constraints, providing a new method for accurate and efficient groundwater simulation. Specifically including the following content.

[0059] Example 1:

[0060] See also Figure 1 , a new simulation method for characterizing the spatiotemporal evolution of groundwater levels, including the following steps;

[0061] S1. Obtain monthly precipitation data covering the study area through long-term satellite remote sensing precipitation products; obtain monthly groundwater level information through long-term groundwater observation wells within the study area.

[0062] When S1 acquires remote sensing image data, it needs to use geoscience related software for processing. The geoscience related processing software is ArcGIS, QGIS or GeoScence .

[0063] S2. Obtain basic hydrogeological data within the study area and, combined with the groundwater control equation, construct a two-dimensional groundwater seepage equation for the study area.

[0064] S2.1. Based on the groundwater governing equation, determine the physical laws used for the neural network constraints. Under heterogeneous isotropic conditions, the basic differential equation for the unstable motion of groundwater can be expressed as:

[0065]

[0066] Where h represents the groundwater level, Represents the water storage rate of the water-containing medium; and are the permeability coefficients of the aquifer in the x and y directions, respectively. Under isotropic conditions, for any point in space, the value of K is the same in all directions ( ), but in space, the value of K will vary - heterogeneity; W is the source and sink term, which represents the impact of external recharge or pumping;

[0067] S2.2, the equation satisfies the following conditions under the initial conditions:

[0068]

[0069] Where, represents the groundwater level at the initial moment of the model; t represents time;

[0070] S3. Based on the two-dimensional groundwater seepage equation in the study area and combined with the neural network model, a deep learning neural network model incorporating physical knowledge - the PINNs model - is constructed.

[0071] S3.1. Construct the model input layer to receive groundwater level information including: water level height, water point location; physical parameter information including permeability coefficient; source and sink information. The input layer is connected to the first hidden layer through a fully connected layer. If the input data is denoted as X, the output data is:

[0072]

[0073] Where, represents the output of the first hidden layer, represents the weight matrix of the first layer, represents the hidden layer bias vector of the first layer, Represents the activation function of the network, common ones include ReLU, Sigmoid and Tanh;

[0074] S3.2 When there are multiple hidden layers, each hidden layer is connected to the previous layer through a fully connected layer:

[0075]

[0076] Where, represents the output of the nth hidden layer, represents the weight matrix of the nth hidden layer, represents the bias vector of the nth hidden layer, represents the activation function of the network, represents the input of the nth hidden layer, which is also the output of the n-1th hidden layer;

[0077] S3.3. Finally, the hidden layer and the output layer are also connected through a fully connected layer. The output layer can be expressed as:

[0078]

[0079] Where, is the prediction result of the network, is the weight matrix of the output layer, is the bias vector of the output layer, is the output of the last hidden layer;

[0080] S3.4. Through automatic differentiation technology, the output layer prediction results are connected with the physical equations, and the physical equations are used as global constraints to guide the neural network training process.

[0081] Preferably, the data error term between the PINNs model network output and the training data in S4 is calculated by the model MSE:

[0082]

[0083] Where, represents the true value of the groundwater level, represents the groundwater value predicted by the model, and N represents the sample size.

[0084] S4. The precipitation data and groundwater level data obtained in S1 are used as input to train the deep learning neural network model integrating physical knowledge - PINNs model, and the data error term between the network output and the training data is obtained.

[0085] The data error term between the PINNs model network output and the training data in S4 is calculated by the model MSE:

[0086]

[0087] Where, represents the true value of the groundwater level, represents the groundwater value predicted by the model, and N represents the sample size.

[0088] S5. Substitute the network output result obtained from the model in S4 into the physical equation, solve the physical equation through automatic differentiation, and obtain the physical residual term.

[0089] S5.1. Water level values ​​predicted using the network , and calculate the partial derivatives by automatic differentiation:

[0090] Compute the first-order derivative: 、 and ;

[0091] Compute the second-order derivative:

[0092] S5.2. Calculate the loss function:

[0093] Substitute the derivative of the network output into the physical equation to calculate:

[0094]

[0095] In the formula, N represents the sample size.

[0096] S6. Superimpose the data error term obtained in S4 and the physical residual term obtained in S5 to obtain the overall loss of the PINNs model, which is back-propagated back to the neural network model. Learning is performed through model iteration to achieve accurate simulation of the groundwater level.

[0097] S6.1. The total loss function of the PINNs network is composed of two weighted parts: the physical residual term of the physical equation and the data error term of the neural network model:

[0098]

[0099] Where, represents the total loss term of the PINNs model, and is the weight coefficient used to balance the data error term and the physical residual, and are the physical residual term and the data error term respectively;

[0100] S6.2, in the process of back propagation, The gradient of the loss with respect to the network output is calculated:

[0101]

[0102] Where, represents the total loss term of the PINNs model, and is the weight coefficient, and are the physical residual term and the data error term respectively, N is the sample size, Indicates the true groundwater level value, represents the groundwater level value predicted by the model;

[0103] S6.3. Using the calculated gradient, update the weights in the network through an optimization algorithm, and continuously iterate the training until the model loss function converges; the optimization algorithm is SGD or Adam.

[0104] Example 2:

[0105] This method is based on Example 1, but differs in that it uses ERA5-Land remote sensing imagery on the GEE platform to obtain monthly precipitation data from 1984 to 2023. Linyi City, Shandong Province, is used as the study area. Water levels at existing long-term groundwater observation wells are obtained, their temporal evolution patterns are analyzed, and simulations are performed. The main steps are as follows:

[0106] Step 1:

[0107] Through long-term satellite remote sensing precipitation products, monthly precipitation data with a resolution of 0.1° covering the study area are obtained; combined with long-term groundwater observation wells within the study area, monthly groundwater level information is obtained.

[0108] Step 2:

[0109] The basic hydrogeological data within the study area were obtained and combined with the groundwater control equation to construct a two-dimensional groundwater seepage equation for the study area.

[0110] The permeability coefficient describes the permeability of underground media and directly affects the velocity and flow distribution of groundwater. The water storage capacity describes the ability of a column of unit horizontal area and height equal to the aquifer thickness to release or store water volume per unit height drop or rise in the aquifer head.

[0111] Step 3:

[0112] Based on the two-dimensional groundwater seepage equation in the study area and combined with the neural network model, a deep learning neural network model integrating physical knowledge - the PINNs model was created.

[0113] Step 4:

[0114] Input data, train the deep learning neural network model that integrates physical knowledge, and obtain the data error term between the network output and the training data.

[0115] Step 5:

[0116] The network output results obtained by the model are substituted into the physical equation, and the physical equation is solved through automatic differentiation to obtain the physical residual term.

[0117] Step 6:

[0118] The data error term of the neural network is superimposed with the physical residual term in the process of solving the physical equation to obtain the overall loss term of the PINNs model, which is backpropagated back to the neural network model. Learning is performed through model iteration to achieve accurate simulation of the groundwater level.

[0119] Step 7:

[0120] The constructed groundwater level simulation model was interpreted using SHAP, and the importance of each characteristic factor in the PINNs network was obtained.

[0121] The model simulation results show that the PINNs model score is 0.87 and the RMSE is 1.98 m. The results show that the model has a good fit and high accuracy in groundwater level simulation.

[0122] Depend on Figure 3 It can be seen that in the PINNs network, the contribution of the permeability coefficients Kx and Ky to the network is significantly higher than that of the traditional MLP network, followed by the spatial position. This shows that for the PINNs network, compared with the spatial position, the model prediction pays more attention to the physical process, which enhances the interpretability of the model prediction results.

[0123] The above are only preferred specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with this technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solutions and inventive concepts of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels, characterized by: The following steps are included: S1. Obtain monthly precipitation data covering the study area through long-term satellite remote sensing precipitation products; obtain monthly groundwater level information through long-term groundwater observation wells within the study area; S2. Obtain basic hydrogeological data within the study area and, combined with the groundwater control equation, construct a two-dimensional groundwater seepage equation for the study area; S3. Based on the two-dimensional groundwater seepage equation in the study area and combined with the neural network model, a deep learning neural network model incorporating physical knowledge, namely the PINNs model, was created. S4, using the precipitation data and groundwater level data obtained in S1 as input to train a deep learning neural network model that integrates physical knowledge, and obtaining a data error term between the network output and the training data; S5, bringing the network output result obtained from the model in S4 into the physical equation, solving the physical equation through automatic differentiation to obtain the physical residual term; S6. Superimpose the data error term obtained in S4 and the physical residual term obtained in S5 to obtain the overall loss of the PINNs model, which is back-propagated back to the neural network model. Learning is performed through model iteration to achieve accurate simulation of the groundwater level.

2. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 1, characterized in that: When S1 obtains long-term satellite remote sensing precipitation products, it uses geoscience-related software for processing. The geoscience-related software is ArcGIS, QGIS or GeoScence .

3. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 1, characterized in that: The S2 construction of the two-dimensional groundwater seepage equation in the study area specifically includes the following steps: S2.

1. Based on the groundwater control equation, determine the physical laws used for the neural network constraints. Under heterogeneous isotropic conditions, the basic differential equation for the unstable motion of groundwater is expressed as: Where h represents the groundwater level, Represents the water storage rate of the water-containing medium; and are the permeability coefficients of the aquifer in the x and y directions, respectively. Under isotropic conditions, for any point in space, the value of K is the same in all directions, that is, , but in space, the value of K will be heterogeneous; W is the source and sink term, which represents the influence of external recharge or pumping; S2.2, the equation satisfies the following conditions under the initial conditions: Where, represents the groundwater level at the initial moment of the model; t represents time.

4. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 1, characterized in that: S3 constructs a deep learning neural network model that incorporates physical knowledge, specifically including the following steps: S3.

1. Construct the model input layer to receive groundwater level information including: water level height, water point location; physical parameter information including permeability coefficient; source and sink information. The input layer is connected to the first hidden layer through a fully connected layer. If the input data is denoted as X, the output data is: Where, represents the output of the first hidden layer, represents the weight matrix of the first layer, represents the hidden layer bias vector of the first layer, Represents the activation function of the network, common ones include ReLU, Sigmoid and Tanh; S3.2 When there are multiple hidden layers, each hidden layer is connected to the previous layer through a fully connected layer: Where, represents the output of the nth hidden layer, represents the weight matrix of the nth hidden layer, represents the bias vector of the nth hidden layer, represents the activation function of the network, represents the input of the nth hidden layer, which is also the output of the n-1th hidden layer; S3.

3. Finally, the hidden layer and the output layer are connected through a fully connected layer; the output layer is represented as: Where, is the prediction result of the network, is the weight matrix of the output layer, is the bias vector of the output layer, is the output of the last hidden layer; S3.

4. Through automatic differentiation technology, the output layer prediction results are connected with the physical equations, and the physical equations are used as global constraints to guide the neural network training process.

5. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 1, characterized in that: The data error term between the PINNs model network output and the training data in S4 is calculated by the model MSE: Where, represents the true value of the groundwater level, represents the groundwater value predicted by the model, and N represents the sample size; represents the data error term.

6. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 5, characterized in that: The process of solving the physical equation in S5 to obtain the calculation of the physical residual term specifically includes the following steps: S5.

1. Water level values ​​predicted using the network , and calculate the partial derivatives by automatic differentiation: Compute the first-order derivative: 、 and ; Compute the second-order derivative: and ; S5.

2. Calculate the loss function: Substitute the derivative of the network output into the physical equation to calculate: Where N represents the sample size; represents the physical residual term.

7. A new simulation method for characterizing the spatiotemporal evolution of groundwater levels according to claim 6, characterized in that: The overall loss of the PINNs model is obtained in S6, which is back-propagated back to the neural network model and parameter iteration for learning, specifically including the following steps: S6.

1. The total loss function of the PINNs network is composed of two weighted parts: the physical residual term of the physical equation and the data error term of the neural network model: Where, represents the total loss term of the PINNs model, and is the weight coefficient used to balance the data error term and the physical residual, and are the physical residual term and the data error term respectively; S6.2, in the process of back propagation, The gradient of the loss with respect to the network output is calculated: Where, represents the total loss term of the PINNs model, and is the weight coefficient, and are the physical residual term and the data error term respectively, N is the sample size, Indicates the true groundwater level value, represents the groundwater level value predicted by the model; S6.

3. Using the calculated gradient, update the weights in the network through an optimization algorithm, and continuously iterate the training until the model loss function converges; the optimization algorithm is SGD or Adam.

8. A novel simulation method for characterizing the spatiotemporal evolution of groundwater levels according to any one of claims 2 to 6, characterized in that: The calculation of the formula is obtained through Python or Matlab.

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