Underground water level early warning method based on pore water pressure

Abstract

The invention relates to an underground water level early warning method based on pore water pressure, and belongs to the technical field of computer deep learning. According to the method, long-time-sequence layered underground water level data and layered pore water pressure data are obtained based on a ground subsidence monitoring station; according to the principles of fluid statics and groundwater dynamics, the measured value of the pore water pressure is converted into the water level elevation. On the basis, information such as various hydrogeological parameters and source and sink items of a region is integrated, an underground water kinetic equation is used as a physical constraint, an LSTM-PINNs model with a physical mechanism is constructed, and underground water level early warning based on pore water pressure is achieved. Compared with a traditional numerical model, the method can more accurately capture single-point layered groundwater dynamic characteristics, has strong data missing complementation capability and long-time scale prediction capability, and meanwhile has the advantages of being easy to implement, efficient in operation and high in expansibility; the method can be widely applied to the fields of underground water resource management, environmental protection, geological disaster prevention and control and the like.

Description

Technical Field

[0001] The present invention belongs to the technical field of hydrogeology and groundwater resource management, and particularly relates to an early warning method for groundwater level based on pore water pressure, which is used for monitoring the dynamic change of groundwater level, predicting leakage recharge and carrying out early warning of groundwater level. Background Technique

[0002] Early Warning systems (EWSs) are mainly applied in aspects such as medicine, food security, and geological disasters (Quansah et al., 2010; Esposito et al., 2022). In the medical aspect, the main issue of the TREWS Early Warning System pointed out in the main issue of "Nature" can help doctors identify sepsis patients as early as possible (Adams et al., 2022). In terms of food security, international academic institutions such as the United Nations Environment Program (UNEP) have carried out a series of early warning programs. Currently, early warning of geological disasters is only applied in the fields of floods, earthquakes, tsunamis, and landslides. The main issue of "Science" pointed out that Google's Android earthquake early warning system, using a public-private partnership model, uses the accelerometer built into mobile phones to detect earthquakes and issue early warnings (M. Allen et al., 2022).

[0003] Early warning and regulation of groundwater level are important components of water resource management. The change of groundwater level plays an important role in the occurrence, development, and evolution of a series of geological disasters (such as land subsidence, ground fissures, karst collapse, etc.). It is necessary to carry out research on early warning of groundwater level, but this research has not been proposed yet. To sum up, it is necessary to carry out research on early warning of groundwater level from the perspective of pore water pressure.

[0004] Currently, the overexploitation of groundwater has led to many geological environment problems, including land subsidence and irrecoverable loss of groundwater resources. However, traditional groundwater level prediction models (such as numerical simulation and empirical regression models) usually require a large amount of regional data support and lack of description of the hydraulic relationship of single-point multi-layers. Especially under complex hydrogeological conditions, it is difficult to achieve high-precision early warning of groundwater.

[0005] Pore water pressure is a key indicator of groundwater dynamics. Especially in the semi-pervious layer, its change can reflect the hydraulic gradient and recharge-discharge relationship between the upper and lower aquifers. From the perspective of early warning of groundwater level based on pore water pressure, the present invention proposes an LSTM-PINNs model combining physical constraints and data-driven methods, which can effectively solve the problem of scarce groundwater level data and carry out early warning of groundwater level from the perspective of mechanism-data drive. Summary of the Invention

[0006] The object of the present invention is to propose an early warning method for groundwater level based on pore water pressure to solve the problems raised in the background art. For the first time, from the perspective of early warning of groundwater level based on pore water pressure, an LSTM-PINNs model combining physical constraints and data-driven is constructed, effectively solving the problem of scarce water level data and realizing early warning of groundwater level based on pore water pressure.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] An early warning method for groundwater level based on pore water pressure, comprising the following steps:

[0009] S1. Data collection: Through the ground settlement monitoring station, obtain the groundwater level data, stratified pore water pressure values and pumping volume under the preset time period of the site.

[0010] S2. Data conversion: Based on the principles of hydrostatics and groundwater dynamics, convert the measured pore water pressure value into the water level elevation.

[0011] S3. Equation construction: Obtain the basic geological and hydrogeological data of the study area, and construct a one-dimensional vertical saturated seepage equation of groundwater in the study area according to Darcy's law and the law of mass conservation.

[0012] S4. Model construction: Based on the groundwater level data and pumping volume data obtained in S1 and the pore water level data of different layers obtained in S2, construct an LSTM-PINNs groundwater level model based on pore water pressure and train the constructed model to obtain the data error term between the neural network output and the training data.

[0013] S5. Physical constraint: Substitute the network output result obtained from the model in S4 into the groundwater dynamics equation in S3, and solve the physical equation by the difference method to obtain the physical residual term of the LSTM-PINNs network.

[0014] S6. Model optimization: Superimpose the data error term obtained in S4 and the physical residual term obtained in S5 to obtain the overall loss of the LSTM-PINNs model of the groundwater level based on pore water pressure, and backpropagate it to the neural network model, continuously iterate and optimize, the model gradually learns, and an optimized groundwater level early warning model is obtained to realize the simulation and early warning of the groundwater level based on pore water pressure.

[0015] Preferably, in S1, when obtaining the stratified groundwater level data and stratified pore water pressure values of the site, geoscience and statistics related processing software, including software such as ArcGIS, QGIS or EXCEL, etc., is adopted.

[0016] Preferably, the conversion of the pore water pressure measurement value into the equivalent water column elevation in S2 specifically includes:

[0017] According to the principles of hydrostatics and groundwater dynamics, convert the pore water pressure measurement value into the water level elevation, and calculate the equivalent water column elevations at the depths of m meters and n meters where the piezometers are buried respectively:

[0018]

[0019] In the formula, h represents the equivalent water column elevation (m) converted from the pore water pressure; H represents the elevation of the measurement point (m); D represents the buried depth of the pore water pressure measurement point (m); u represents the pore water pressure measurement value (Pa); r w represents the unit weight of water (N / m 3 ).

[0020] Preferably, S3 specifically includes the following steps:

[0021] S3.1. Based on Darcy's law, in a saturated porous medium, the one-dimensional vertical flow equation of groundwater can be expressed as:

[0022]

[0023] In the formula, q z represents the vertical unit area flow rate; K z represents the vertical permeability coefficient; h represents the groundwater level; z represents the vertical coordinate.

[0024] S3.2. Based on the law of conservation of mass, in a saturated porous medium, the continuity equation of one-dimensional vertical flow of groundwater can be expressed as:

[0025]

[0026] In the formula, S s represents the storage coefficient of the aquifer; h represents the groundwater level; t represents time; q z represents the vertical unit area flow rate; z represents the vertical coordinate; Q represents the source-sink term, indicating the influence of external recharge or pumping.

[0027] S3.3. Substitute Darcy's law into the continuity equation to obtain the one-dimensional vertical saturated seepage equation of groundwater in the study area, which describes the variation of the water head with time and depth:

[0028]

[0029] In the formula, S s represents the storage coefficient of the aquifer; h represents the groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source-sink term.

[0030] S3.4. The equation satisfies under the initial conditions

[0031] h| t＝0 = h0 (5)

[0032] Preferably, the said S4 specifically includes the following steps:

[0033] S4.1. The input layer of the LSTM-PINNs groundwater level model receives time series data, including: aquifer water level, equivalent water column elevation of stratified pore water, and pumping volume. The data is organized according to the number of samples for batch training (batch_size), time step dimension (time_steps), and input feature dimension (input_features) to support batch training and time series modeling.

[0034] S4.2. In the LSTM-PINNs network, the input layer is connected to the hidden layer through a weight matrix. In the hidden layer, each LSTM cell follows the following calculation formula:

[0035] First, calculate which information needs to be forgotten through the forget gate f t :

[0036] f t = σ(W f ·[h t-1 , x t + b f ) (6)

[0037] Secondly, calculate which new information needs to be updated through the input gate i t :

[0038] i t = σ(W i ·[h t-1 , x t + b i ) (7)

[0039] Thirdly, calculate the candidate memory cell C~ t , generate a new potential information as an update candidate, and update the cell state accordingly:

[0040]

[0041] Finally, through the output gate o t , and calculate the new hidden state of the cell.

[0042] o t = σ(W o ·[h t-1 , x t + b o ) (10)

[0043] h t = o t *tanh(C t ) (11)

[0044] In the formula, x t represents the input data at the current time step; h t-1 represents the hidden state at the previous moment; C t-1 represents the cell state at the previous moment; h t represents the hidden state at the current moment; C t the cell state at the current moment; W f , W i , W C , W o respectively represent the weight matrices of the forget gate, input gate, candidate memory unit, and output gate; b f , b i , b C , b o respectively represent the bias vectors of the forget gate, input gate, candidate memory unit, and output gate; σ represents the sigmoid activation function and the tanh activation function respectively.

[0045] S4.3. The LSTM is a time-recurrent network. Multiple LSTM units are connected in series on the time axis to form an LSTM layer. At each time step t, the LSTM unit receives the input x t at the current time step and the hidden state h t-1 and cell state C t-1 passed from the previous time step, and calculates the hidden state h t at the current time step and the cell state C t at the current time step using the steps described in S4.2, and passes them to the next time step t + 1 for recursive calculation, that is:

[0046] (h t , C t ) = LSTM(x t , h t-1 , C t-1 ) (12)

[0047] In the formula, x t represents the input data at the current time step; h t-1 represents the hidden state at the previous moment; C t-1 represents the cell state at the previous moment; h t represents the hidden state at the current moment; C t the cell state at the current moment.

[0048] S4.4. When there are multiple LSTM layers, each LSTM layer is connected in series in the depth direction, and the hidden state of the previous LSTM layer will be passed as input to the next LSTM layer, and the output of the last LSTM layer is then mapped to the output layer through a fully connected layer (Dense):

[0049] y = Dense(h T ) = W out ·h T + b out (13)

[0050] where hT represents the hidden state of the last time step of the last layer of LSTM; W out represents the weight of the fully connected layer; b out represents the bias of the fully connected layer; y represents the predicted value of the network.

[0051] S4.5. Finally, calculate the data error term between the network predicted data and the true training data, and use the mean square error (MSE) for calculation. The formula is as follows:

[0052]

[0053] where N represents the number of samples; Y i represents the true value of the groundwater level; y i represents the predicted value of the groundwater level model; MSE NN represents the data error term of the network.

[0054] Preferably, S5 specifically includes the following steps:

[0055] S5.1. Use the groundwater level value predicted by the LSTM network, combine it with the one-dimensional vertical saturated seepage equation of groundwater in the study area, and calculate the partial derivative by the finite difference method:

[0056]

[0057] where h represents the predicted value of the groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; h m represents the equivalent water column elevation at the depth of m meters of the piezometer; h n represents the equivalent water column elevation at the depth of n meters of the piezometer; h t+1 represents the predicted value of the water level at the previous moment; h t represents the predicted value of the water level at the current moment.

[0058] S5.2. Substitute the partial derivative calculated in S5.1 back into the physical equation to calculate the physical residual term:

[0059]

[0060] In the formula, MSE PDE represents the cyber-physical residual term; N represents the number of samples; S s represents the aquifer storage coefficient; h represents the groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source-sink term.

[0061] Preferably, the S6 includes the following steps:

[0062] S6.1. The total loss function of the LSTM-PINNs network is composed of the weighted physical residual term of the model and the data error term of the network:

[0063] MSE = λ1MSE NN + λ2MSE PDE (18)

[0064] In the formula, MSE represents the total loss term of the LSTM-PINNs model, and λ1 and λ2 are weight coefficients used to balance the data error term and the physical residual, MSE PDE and MSE NN are the physical residual term and the data error term of the model, respectively.

[0065] S6.2. The LSTM-PINNs network starts from the last time step and gradually calculates the gradient forward through the Backpropagation Through Time (BPTT) algorithm, and backpropagates the gradient to all time steps. First, calculate the output layer gradient and backpropagate the error from the network prediction value back to the LSTM network:

[0066]

[0067] Second, calculate the hidden layer gradient and propagate the error from the next time step to the current time step to accumulate the error:

[0068]

[0069] Third, calculate the gating mechanism gradient, that is, calculate the gradients of the forget gate, input gate, candidate memory unit, and output gate respectively:

[0070]

[0071] Then, calculate the cell state gradient and the input gradient:

[0072]

[0073]

[0074] Finally, calculate the parameter gradient:

[0075]

[0076] In the formula, MSE represents the loss function of the LSTM - PINNs network; y represents the network prediction value; W out represents the weight of the fully - connected layer; N represents the number of samples; f t , i t , o t represent the forget gate, input gate, candidate memory cell, and output gate respectively; h t represents the hidden state at the current moment; h t+1 represents the hidden state at the next moment; y t represents the network prediction result at time step t; x t represents the input data at time step t; W f , W i , W C , W o represent the weight matrices of the forget gate, input gate, candidate memory cell, and output gate respectively; b f , b i , b C , b o represent the bias vectors of the forget gate, input gate, candidate memory cell, and output gate respectively; tanh represents the tanh activation function.

[0077] S6.3. After calculating all gradients, update the network weights through an optimization algorithm (such as SGD or Adam), continuously iterate and train, and minimize the model loss function until the model converges:

[0078]

[0079]

[0080] In the formula, W represents the weight parameters of the LSTM - PINNs network (such as: W f , W i , W o , W C ); b represents the bias parameters of the LSTM - PINNs network (such as: b f , b i , b o , b C ); η represents the learning rate of the model; MSE represents the network loss function.

[0081] Preferably, the calculations of the above formulas (1) to (36) are obtained through Python or Matlab calculations.

[0082] Compared with the prior art, the present invention provides an early warning method for groundwater level based on pore water pressure, which has the following beneficial effects:

[0083] The change of groundwater level or the total stress of soil mass causes the dissipation or increase of pore water pressure. The change of effective stress controls the deformation caused by the reorganization of soil particles. The deformation cannot be identified by layer marks or InSAR monitoring in the short term. The present invention proposes an early warning method for groundwater level based on pore water pressure. For the first time, from the perspective of groundwater regulation and early warning, a LSTM-PINNs groundwater level simulation model based on pore water pressure is constructed, which can be used to monitor the dynamic change of groundwater level, predict leakage recharge and carry out early warning of groundwater level, providing a scientific basis for the risk assessment and prevention decision of geological disasters (such as land subsidence, ground fissures, etc.), and having a wide application prospect. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings involved in the embodiments are briefly introduced below. Obviously, the accompanying drawings in the following description are only schematic illustrations of some embodiments of the present invention. For those skilled in the art, other forms of drawings can be constructed based on these drawings without creative efforts.

[0085] Figure 1 It is a schematic flow chart of an early warning method for groundwater level based on pore water pressure proposed in Embodiment 1 of the present invention;

[0086] Figure 2 It is a structural diagram of an early warning model for groundwater level based on pore water pressure and LSTM-PINNs proposed in Embodiment 1 of the present invention;

[0087] Figure 3 It is a schematic diagram of the equivalent water level height by layer proposed in Embodiment 2 of the present invention;

[0088] Figure 4 It is a schematic diagram of the water level simulation and prediction result proposed in Embodiment 2 of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0089] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments.

[0090] Embodiment 1:

[0091] Please refer to Figure 1 , an early warning method for groundwater level based on pore water pressure, including the following steps:

[0092] S1. Obtain the groundwater level data, stratified pore water pressure values, and pumping volume at each site on a 5-day scale through a land subsidence monitoring station. Geoscience and statistical processing software, including ArcGIS, QGIS, or EXCEL software, etc., are required.

[0093] S2. According to the principles of hydrostatics and groundwater dynamics, convert the measured pore water pressure value into the water level elevation, and calculate the equivalent water column elevations at the depths of 78.5 m and 96.5 m of the piezometer respectively:

[0094]

[0095] In the formula, h represents the equivalent water column elevation (m) converted from the pore water pressure; H represents the elevation of the measurement point (m); D represents the buried depth of the pore water pressure measurement point (m); u represents the measured pore water pressure value (Pa); r w represents the unit weight of water (N / m 3 ).

[0096] S3. Obtain the basic geological and hydrogeological data of the study area, and construct a one-dimensional vertical saturated seepage equation for groundwater in the study area according to Darcy's law and the law of conservation of mass. The specific steps are as follows:

[0097] S3.1. Based on Darcy's law, in a saturated porous medium, the one-dimensional vertical flow equation of groundwater can be expressed as:

[0098]

[0099] In the formula, q z represents the vertical unit area flow velocity; K z represents the vertical permeability coefficient; h represents the groundwater level; z represents the vertical coordinate.

[0100] S3.2. Based on the law of conservation of mass, in a saturated porous medium, the continuity equation of one-dimensional vertical flow of groundwater can be expressed as:

[0101]

[0102] In the formula, S s represents the storage coefficient of the aquifer; h represents the groundwater level; t represents time; q z represents the vertical unit area flow velocity; z represents the vertical coordinate; Q represents the source-sink term, indicating the influence of external recharge or pumping.

[0103] S3.3. Substitute Darcy's law into the continuity equation to obtain the one-dimensional vertical saturated seepage equation of groundwater in the study area, which describes the variation of the water head with time and depth:

[0104]

[0105] In the formula, S s represents the aquifer storage coefficient; h represents the groundwater level; t represents time; K z represents the vertical hydraulic conductivity; z represents the vertical coordinate; Q represents the source-sink term.

[0106] S3.4. The equation satisfies the following under the initial conditions

[0107] h| t＝0 = h0 (5)

[0108] S4. Use the groundwater level data and pumping volume data obtained in S1, and the pore water level data of different layers obtained in S2 as inputs to train the LSTM-PINNs model with physical constraints, and obtain the data error term between the neural network output and the training data, which specifically includes the following steps:

[0109] S4.1. The input layer of the LSTM-PINNs groundwater level model receives time series data, including: aquifer water level, equivalent water column elevation of layered pore water, and pumping volume. The data is organized according to the number of samples in batch training (batch_size), the time step dimension (time_steps), and the input feature dimension (input_features) to support batch training and time series modeling.

[0110] S4.2. In the LSTM-PINNs network, the input layer is connected to the hidden layer through a weight matrix. Each LSTM unit in the hidden layer follows the following calculation formula:

[0111] First, calculate which information needs to be forgotten through the forget gate f t :

[0112] f t = σ(W f · [h t-1 , x t + b f ) (6)

[0113] Secondly, calculate which new information needs to be updated through the input gate i t :

[0114] i t = σ(W i · [h t-1 , x t + b i ) (7)

[0115] Thirdly, calculate the candidate memory unit Generate a new potential information as an update candidate to update the cell state:

[0116]

[0117] Finally, through the output gate o t , the new hidden state of the cell is calculated.

[0118] o t = σ(W o · [h t-1 , x t + b o ) (10)

[0119] h t = o t * tanh(C t ) (11)

[0120] In the formula, x t represents the input data at the current time step; h t-1 represents the hidden state at the previous moment; C t-1 represents the cell state at the previous moment; h t represents the hidden state at the current moment; C t represents the cell state at the current moment; W f , W i , W C , W o represent the weight matrices of the forget gate, input gate, candidate memory unit, and output gate respectively; b f , b i , b C , b o represent the bias vectors of the forget gate, input gate, candidate memory unit, and output gate respectively; σ and tanh represent the sigmoid activation function and tanh activation function respectively.

[0121] S4.3. The LSTM is a time-recursive network, and multiple LSTM units are connected in series on the time axis to form an LSTM layer. At each time step t, the LSTM unit receives the input x t at the current time step, as well as the hidden state h t-1 and cell state C t-1 passed from the previous time step, and calculates the hidden state h t at the current time step and the cell state C t at the current time step using the steps described in S4.2, and passes them to the next time step t + 1 for recursive calculation, that is:

[0122] (h t , C t ) = LSTM(x t , h t-1 , C t-1 ) (12)

[0123] where x t represents the input data at the current time step; h t-1 represents the hidden state at the previous moment; C t-1 represents the cell state at the previous moment; h t represents the hidden state at the current moment; C t is the cell state at the current moment.

[0124] S4.4. When there are multiple LSTM layers, each LSTM layer is connected in series in the depth direction, and the hidden state of the previous LSTM layer will be passed as input to the next LSTM layer, and the output of the last LSTM layer is then mapped to the output layer through a fully connected layer (Dense):

[0125] y = Dense(h T ) = W out ·h T + b out (13)

[0126] where hT represents the hidden state of the last time step of the last layer of LSTM; W out represents the weight of the fully connected layer; b out represents the bias of the fully connected layer; y represents the predicted value of the network.

[0127] S4.5. Finally, calculate the data error term between the network prediction data and the true training data, and use the mean squared error (MSE) for calculation. The formula is as follows:

[0128]

[0129] where N represents the number of samples; Y i represents the true value of the groundwater level; y i represents the predicted value of the groundwater level model; MSE NN represents the data error term of the network.

[0130] S5. Substitute the network output result obtained by the model in S4 into the groundwater dynamics equation in S3, and solve the physical equation by the finite difference method to obtain the physical residual term of the LSTM-PINNs network, which specifically includes the following steps:

[0131] S5.1. Using the groundwater level value predicted by the LSTM network, combined with the one-dimensional vertical saturated seepage equation of groundwater in the study area, calculate the partial derivative by the finite difference method:

[0132]

[0133] where h represents the predicted value of the groundwater level; t represents time; K zrepresents the vertical hydraulic conductivity; z represents the vertical coordinate; h m represents the equivalent water column elevation at the depth of m meters where the piezometer is buried; h n represents the equivalent water column elevation at the depth of n meters where the piezometer is buried; h t+1 represents the predicted water level value at the previous moment; h t represents the predicted water level value at the current moment.

[0134] S5.2. Substitute the partial derivatives calculated in S5.1 back into the physical equation to calculate the physical residual term:

[0135]

[0136] In the formula, MSE PDE represents the network physical residual term; N represents the number of samples; S s represents the aquifer storage coefficient; h represents the groundwater level; t represents time; K z represents the vertical hydraulic conductivity; z represents the vertical coordinate; Q represents the source-sink term.

[0137] S6. Superimpose the data error term obtained in S4 and the physical residual term obtained in S5 to obtain the overall loss of the LSTM-PINNs model for groundwater level based on pore water pressure, and backpropagate it to the neural network model for continuous iterative optimization. The model gradually learns to obtain the optimized groundwater level warning model (whose structure is as Figure 2 shown). To realize the simulation of groundwater level based on pore water pressure, the specific steps are as follows:

[0138] S6.1. The total loss function of the LSTM-PINNs network is composed of the weighted physical residual term and the data error term of the model:

[0139] MSE = λ1MSE NN + λ2MSE PDE (18)

[0140] In the formula, MSE represents the total loss term of the LSTM-PINNs model, λ1 and λ2 are weight coefficients used to balance the data error term and the physical residual, MSE PDE and MSE NN are the physical residual term and the data error term of the model respectively.

[0141] S6.2. The LSTM-PINNs network starts from the last time step and calculates the gradient step by step forward through the Backpropagation Through Time (BPTT) algorithm, and backpropagates the gradient to all time steps. First, calculate the output layer gradient and backpropagate the error from the network prediction value back to the LSTM network:

[0142]

[0143] Secondly, calculate the hidden layer gradients, and propagate the error from the next time step to the current time step to accumulate the error:

[0144]

[0145] Thirdly, calculate the gating mechanism gradients, that is, calculate the gradients of the forget gate, input gate, candidate memory unit, and output gate respectively:

[0146]

[0147] Then, calculate the cell state gradient and the gradient of the input:

[0148]

[0149] Finally, calculate the gradients of the parameters:

[0150]

[0151]

[0152] In the formula, MSE represents the loss function of the LSTM-PINNs network; y represents the network prediction value; W out represents the weight of the fully connected layer; N represents the number of samples; f t , i t , o t represent the forget gate, input gate, candidate memory unit, and output gate respectively; h t represents the hidden state at the current moment; h t+1 represents the hidden state at the next moment; y t represents the network prediction result at time step t; x t represents the input data at time step t; W f , W i , W C , W o represent the weight matrices of the forget gate, input gate, candidate memory unit, and output gate respectively; b f , b i , b C , b o represent the bias vectors of the forget gate, input gate, candidate memory unit, and output gate respectively; tanh represents the tanh activation function.

[0153] S6.3. After calculating all gradients, update the network weights through an optimization algorithm (such as SGD or Adam), continuously iterate and train, and minimize the model loss function until the model converges:

[0154]

[0155] In the formula, \(W\) represents the weight parameters of the LSTM - PINNs network (e.g., \(W\) f , \(W\) i , \(W\) o , \(W\) C ); \(b\) represents the bias parameters of the LSTM - PINNs network (e.g., \(b\) f , \(b\) i , \(b\) o , \(b\) C ); \(\eta\) represents the learning rate of the model; MSE represents the network loss function.

[0156] Example 2:

[0157] Based on Example 1, the difference is that in this example, the layered pore water pressure values and groundwater level information in the plain area of a certain city are obtained through the national land subsidence monitoring project, and the acquisition period is from April 5, 2004 to October 1, 2014. After identifying the hydrogeological and engineering geological conditions in this area, integrating various hydrogeological parameters, source - sink terms and other information in the region, and taking the groundwater dynamics equation as the physical constraint, an LSTM - PINNs model with physical mechanisms is constructed to realize the early warning of groundwater level based on pore water pressure. The main steps are as follows:

[0158] Step 1: Through the land subsidence monitoring station, obtain the groundwater level, layered pore water pressure values and pumping volume data at the scale of every 5 days for each site;

[0159] Step 2: According to the principles of hydrostatics and groundwater dynamics, convert the measured values of layered pore water pressure into water level elevations, and the equivalent water level heights of each layer are as Figure 3 shown;

[0160] Step 3: Obtain the vertical permeability coefficient and aquifer storage coefficient of the study area, and construct a one - dimensional vertical saturated seepage equation of groundwater in the study area based on Darcy's law and the law of mass conservation.

[0161] Step 4: Use the groundwater level data and pumping volume data obtained in Step 1, and the equivalent water level data of pore water at different layers obtained in Step 2 as inputs to train the LSTM - PINNs model with physical constraints, and obtain the data error term between the neural network output and the training data;

[0162] Step 5: Substitute the network output result obtained by the model in Step 4 into the groundwater dynamics equation constructed in Step 3 as its physical constraint equation, and solve the physical equation by the difference method to obtain the physical residual term of the LSTM - PINNs network;

[0163] Step 6: Superimpose the data error term obtained in Step 4 and the physical residual term obtained in Step 5 to obtain the overall loss of the LSTM-PINNs groundwater level model based on pore water pressure, and backpropagate it back to the neural network model. Through continuous iterative optimization, the model gradually learns to achieve the simulation of groundwater level based on pore water pressure. The prediction results are as Figure 4 shown.

[0164] The above is only a preferred specific implementation manner 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, according to the technical solution and inventive concept of the present invention, makes equivalent substitutions or changes, and should be covered by the protection scope of the present invention.

Claims

1. A groundwater level early warning method based on pore water pressure, characterized in that: The steps include: S1. Data collection: Through the ground subsidence monitoring station, the groundwater level data, stratified pore water pressure values ​​and pumping volume at the preset time period of the site are obtained; S2. Data conversion: Based on the principles of fluid statics and groundwater dynamics, the pore water pressure measurement value is converted into water level elevation; S3, equation construction: obtain basic geological and hydrogeological data of the study area, and construct the one-dimensional vertical saturated seepage equation of groundwater in the study area according to Darcy's law and the law of conservation of mass; S4, model construction: Based on the groundwater level data and pumping volume data obtained in S1 and the pore water level data of different layers obtained in S2, a LSTM-PINNs groundwater level model based on pore water pressure is constructed and the constructed model is trained to obtain the data error term between the neural network output and the training data; S5, physical constraints: Substitute the network output results obtained from the model in S4 into the groundwater dynamics equation in S3, solve the physical equation by the difference method, and obtain the physical residual term of the LSTM-PINNs network; S6, model optimization: superimpose the data error term obtained by S4 and the physical residual term obtained by S5 to obtain the overall loss of the groundwater level LSTM-PINNs model based on pore water pressure, and backpropagate it back to the neural network model. Continuously iterate and optimize, the model gradually learns, and the optimized groundwater level early warning model is obtained to realize groundwater level simulation and early warning based on pore water pressure.

2. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: In S1, when obtaining the site stratified groundwater level data and stratified pore water pressure values, geoscientific and statistical related processing software is used, including ArcGIS, QGIS or EXCEL software.

3. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: The conversion of the measured pore water pressure value into the equivalent water column elevation described in S2 specifically includes: According to the principle of water head and fluid statics, calculate the equivalent water column elevation of the piezometer at a depth of m meters and n meters: Where, h represents the equivalent water column elevation converted from pore water pressure, m; H represents the elevation of the measuring point, m; D represents the buried depth of the pore water pressure measuring point, m; u represents the measured value of pore water pressure, Pa; r w Indicates the unit weight of water, N / m 3 .

4. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: The construction of the one-dimensional vertical saturated seepage equation of groundwater in the study area described in S3 specifically includes the following steps: S3.

1. Based on Darcy's law, the one-dimensional vertical flow equation of groundwater in saturated porous media is expressed as: In the formula, q z Indicates the vertical flow velocity per unit area; K z represents the vertical permeability coefficient; h represents the groundwater level; z represents the vertical coordinate; S3.

2. Based on the law of conservation of mass, the continuity equation for one-dimensional vertical flow of groundwater in saturated porous media is expressed as: In the formula, S s represents the aquifer storage coefficient; h represents the groundwater level; t represents time; q z represents the vertical flow velocity per unit area; z represents the vertical coordinate; Q is the source-sink term, which represents the influence of external recharge or pumping. S3.

3. Substituting Darcy's law into the continuity equation, we obtain the one-dimensional vertical saturated seepage equation of groundwater in the study area, which is used to describe the change of water head with time and depth: In the formula, S s represents the water storage coefficient of the aquifer; h represents the groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source and sink term; S3.4, the equation satisfies the initial conditions h| t＝0 =h0 (5)。 5. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: The construction of the LSTM-PINNs groundwater level model based on pore water pressure described in S4 specifically includes the following steps: S4.

1. The input layer of the LSTM-PINNs groundwater level model receives time series data, including aquifer water level, stratified pore water equivalent water column elevation, and pumping volume; the data is organized according to the number of samples for batch training, the time step dimension, and the input feature dimension to support batch training and time series modeling; S4.

2. In the LSTM-PINNs network, the input layer is connected to the hidden layer through a weight matrix. Each LSTM unit in the hidden layer follows the following calculation formula: First, through the forget gate f t Calculate the forgotten information: f t =σ(W f ·[h t-1 ,x t ]+b f ) (6) Secondly, through the input gate i t Calculate the updated new information: i t =σ(W i ·[h t-1 ,x t ]+b i ) (7) Again, calculate the candidate memory cells Generate a new potential information as an update candidate to update the cell state: Finally, through the output gate o t , calculate the new hidden state of the cell: the t =σ(W o ·[h t-1 ,x t ]+b o ) (10) h t =o t *tanh(C t ) (11) In the formula, x t Represents the input data of the current time step; h t-1 Indicates the hidden state of the previous moment; C t-1 Indicates the cell state at the previous moment; h t Indicates the hidden state at the current moment; C t The cell state at the current moment; W f , W i , W C , W o Represents the weight matrices of the forget gate, input gate, candidate memory unit, and output gate respectively; b f , b i , b C , b o Respectively represent the bias vectors of the forget gate, input gate, candidate memory unit, and output gate; σ and tanh represent the sigmoid activation function and tanh activation function respectively; S4.3, LSTM is a time recursive network. Multiple LSTM units are connected in series on the time axis to form an LSTM layer. At each time step t, the LSTM unit receives the input x of the current time step. t and the hidden state h passed from the previous time step t-1 and cell state C t-1 , use the steps described in S4.2 to calculate the hidden state h of the current time step t and the cell state C at the current time step t , and pass it to the next time step t+1 for recursive calculation, namely: (h t ,C t )=LSTM(x t ,h t-1 ,C t-1 ) (12) In the formula, x t Represents the input data of the current time step; h t-1 Indicates the hidden state of the previous moment; C t-1 Indicates the cell state at the previous moment; h t Indicates the hidden state at the current moment; C t The cell state at the current moment; S4.

4. When there are multiple LSTM layers, each LSTM layer is connected in series in the depth direction, and the hidden state of the previous LSTM layer The output of the last LSTM layer is passed as input to the next LSTM layer, and the output is mapped to the output layer through a fully connected layer: y=Dense(h T )=W out ·h T +b out (13) Where hT represents the hidden state of the last time step of the last layer of LSTM; W out represents the weight of the fully connected layer; b out represents the bias of the fully connected layer; y represents the predicted value of the network; S4.

5. Calculate the data error term between the network prediction data and the actual training data using the mean square error. The formula is as follows: In the formula, N represents the number of samples; Y i Represents the true value of the groundwater level; y i Represents the predicted value of groundwater level model; MSE NN represents the data error term of the network.

6. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: The process of solving the physical equation described in S5 to obtain the calculation of the physical residual term specifically includes the following steps: S5.

1. Using the groundwater level predicted by the LSTM network and the one-dimensional vertical saturated seepage equation of groundwater in the study area, the partial derivatives are calculated by the finite difference method: In the formula, h represents the predicted value of groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; h m Indicates the equivalent water column elevation of the piezometer at a buried depth of m meters; h n Indicates the equivalent water column elevation of the piezometer at a depth of n meters; h t+1 Indicates the predicted water level value at the previous moment; h t Indicates the predicted water level value at the current moment; S5.

2. Substitute the partial derivatives calculated in S5.1 back into the physical equation and calculate the physical residual term: In the formula, MSE PDE represents the network physical residual term; N represents the number of samples; S s represents the water storage coefficient of the aquifer; h represents the groundwater level; t represents time; K z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source and sink terms.

7. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that: As described in S6, the overall loss of the obtained LSTM-PINNs model is back-propagated back to the neural network model and parameter iteration for learning, which specifically includes the following steps: S6.

1. The total loss function of the LSTM-PINNs network is composed of the weighted physical residual term of the model and the data error term of the network: MSE=λ1MSE NN +λ2MSE PDE (18) Where, MSE represents the total loss term of the LSTM-PINNs model, λ1 and λ2 are weight coefficients used to balance the data error term and the physical residual, and MSE PDE and MSE NN are the physical residual term and data error term of the model respectively; S6.2, LSTM-PINNs network uses the time back propagation algorithm to calculate the gradient step by step from the last time step, and back propagates the gradient to all time steps, including the following: First, the output layer gradients are calculated to backpropagate the error from the network predictions back to the LSTM network: Second, the hidden layer gradients are calculated to propagate the error from the next time step to the current time step to accumulate the error: Again, calculate the gradient of the gating mechanism, that is, calculate the gradient of the forget gate, input gate, candidate memory unit and output gate respectively: Then, calculate the gradient of the cell state and the gradient of the input: Finally, the gradient of the parameters is calculated: Where, MSE represents the LSTM-PINNs network loss function; y represents the network prediction value; W out represents the weight of the fully connected layer; N represents the number of samples; f t 、i t , o t They represent the forget gate, input gate, candidate memory unit and output gate respectively; h t Indicates the hidden state at the current moment; h t+1 Indicates the hidden state at the next moment; y t represents the network prediction result at time step t; x t represents the input data at time step t; W f , W i , W C , W o Represent the weight matrices of the forget gate, input gate, candidate memory unit and output gate respectively; b f ,b i ,b C ,b o represents the bias vectors of the forget gate, input gate, candidate memory unit and output gate respectively; tanh represents the tanh activation function; S6.

3. After calculating all gradients, update the network weights through the optimization algorithm, continuously iterate the training, minimize the model loss function, until the model converges: In the formula, W represents the weight parameter of the LSTM-PINNs network; b represents the bias parameter of the LSTM-PINNs network; η represents the learning rate of the model; MSE represents the network loss function.

8. The method for early warning of groundwater level based on pore water pressure according to any one of claims 2 to 6 is characterized in that the calculation of equations (1) to (36) is completed by Python or Matlab.

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