An early warning method for groundwater level based on pore water pressure

Abstract

This invention relates to an early warning method for groundwater levels based on pore water pressure, belonging to the field of deep learning technology. Based on ground subsidence monitoring stations, this invention acquires long-term, layered groundwater level data and layered pore water pressure data. According to the principles of fluid statics and groundwater dynamics, the measured pore water pressure values ​​are converted into water level elevations. On this basis, by integrating various regional hydrogeological parameters, source and sink terms, and other information, and using groundwater dynamics equations as physical constraints, an LSTM-PINNs model with a physical mechanism is constructed to achieve early warning of groundwater levels based on pore water pressure. Compared with traditional numerical models, this invention can more accurately capture the dynamic characteristics of single-point layered groundwater, possesses strong data completion capabilities and long-term prediction capabilities, and also has the advantages of simple implementation, high efficiency, and strong scalability, making it widely applicable to groundwater resource management, environmental protection, and geological disaster prevention and control.

Description

Technical Field

[0001] This invention belongs to the field of hydrogeology and groundwater resource management technology, specifically relating to an early warning method for groundwater level based on pore water pressure, used to monitor dynamic changes in groundwater level, predict overflow recharge, and conduct early warning of groundwater level. Background Technology

[0002] Early Warning Systems (EWSs) are primarily used in medicine, food security, and geological disasters (Quansah et al., 2010; Esposito et al., 2022). In medicine, *Nature* reports that the TREWS early warning system can help doctors identify sepsis patients early (Adams et al., 2022). Regarding food security, international academic organizations such as the United Nations Environment Programme (UNEP) have implemented a series of early warning programs. Currently, early warning systems for geological disasters are only applied to floods, earthquakes, tsunamis, and landslides. *Science* reports that Google's Android earthquake early warning system, using a public-private partnership, utilizes the phone's built-in accelerometer to detect earthquakes and issue warnings (M. Allen et al., 2022).

[0003] Early warning and regulation of groundwater levels are crucial components of water resource management. Changes in groundwater levels play a significant role in the occurrence, development, and evolution of a series of geological disasters (such as land subsidence, ground fissures, and karst collapses). Early warning research on groundwater levels is necessary, but such research has not yet been proposed. Therefore, it is essential to conduct early warning research on groundwater levels from the perspective of pore water pressure.

[0004] Currently, excessive groundwater extraction has led to numerous geological and environmental problems, including land subsidence and irreversible depletion of groundwater resources. However, traditional groundwater level prediction models (such as numerical simulation and empirical regression models) typically require extensive regional data and are insufficient in characterizing the hydraulic relationships at single points and multiple layers. Especially under complex hydrogeological conditions, they struggle to achieve high-precision early warning of groundwater levels.

[0005] Pore ​​water pressure is a key indicator of groundwater dynamics, especially in weakly permeable aquifers, where its changes reflect the hydraulic gradient and recharge-discharge relationship between upper and lower aquifers. From the perspective of early groundwater level warning based on pore water pressure, this invention proposes an LSTM-PINNs model combining physical constraints and data-driven approaches. This model effectively addresses the scarcity of groundwater level data and provides early groundwater level warning from a mechanism-data-driven perspective. Summary of the Invention

[0006] The purpose of this invention is to propose an early warning method for groundwater levels based on pore water pressure to address the problems mentioned in the background art. This invention, for the first time, constructs an LSTM-PINNs model combining physical constraints and data-driven approaches from the perspective of early warning of groundwater levels based on pore water pressure. This effectively solves the problem of scarce water level data and realizes early warning of groundwater levels based on pore water pressure.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

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

[0009] S1. Data Acquisition: Obtain groundwater level data, pore water pressure values ​​of different layers, and pumping volume through the ground settlement monitoring station at a preset time period.

[0010] S2. Data Conversion: Based on the principles of fluid statics and groundwater dynamics, the measured pore water pressure values ​​are converted into water level elevations.

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

[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, an LSTM-PINNs groundwater level model based on pore water pressure is constructed and trained to obtain the data error term between the neural network output and the training data.

[0013] 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 using the difference method, and obtain the physical residual terms of the LSTM-PINNs network.

[0014] S6. Model Optimization: The data error term obtained in S4 and the physical residual term obtained in S5 are superimposed to obtain the overall loss of the LSTM-PINNs model of groundwater level based on pore water pressure. This loss is then backpropagated to the neural network model for continuous iteration and optimization. The model learns step by step to obtain the optimized groundwater level early warning model, realizing groundwater level simulation and early warning based on pore water pressure.

[0015] Preferably, in S1, when acquiring the stratified groundwater level data and stratified pore water pressure values ​​at the site, geoscience and statistical processing software, including ArcGIS, QGIS, or EXCEL, is used.

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

[0017] Based on the principles of fluid statics and groundwater dynamics, the pore water pressure measurements are converted into water level elevations, and the equivalent water column elevations at piezometer burial depths of m meters and n meters are calculated respectively:

[0018]

[0019] In the formula, h represents the equivalent water column elevation (m) for pore water pressure conversion; H represents the elevation of the measuring point (m); D represents the burial depth of the pore water pressure measuring point (m); u represents the measured value of pore water pressure (Pa); r w The unit weight of water (N / m³) 3 ).

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

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

[0022]

[0023] In the formula, q z K represents the vertical velocity per unit area. z denoted by , h represents the vertical permeability coefficient; represents the groundwater level; and z represents the vertical coordinate.

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

[0025]

[0026] In the formula, S s The aquifer storage coefficient is represented by h; the groundwater level is represented by t; and time is represented by q. z The vertical velocity per unit area is represented by z; the vertical coordinate is represented by Q; and the source / sink term represents the effect of external recharge or pumping.

[0027] S3.3 Substituting Darcy's law into the continuity equation, we obtain the one-dimensional vertical saturated seepage equation for groundwater in the study area, which describes the change of hydraulic head with time and depth:

[0028]

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

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

[0031] h| t＝0 =h0 (5)

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

[0033] The input layer of the S4.1 LSTM-PINNs groundwater level model receives time-series data, including aquifer level, equivalent water column elevation of stratified pore water, and pumping volume. The data is organized according to the batch size, time steps, and input features to support batch training and time-series modeling.

[0034] S4.2 In LSTM-PINNs networks, the input layer is connected to the hidden layer through a weight matrix. Each LSTM unit in the hidden layer follows the calculation formula:

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

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

[0037] Secondly, by inputting gate i t Calculate which new information needs to be updated:

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

[0039] Next, calculate the candidate memory cell C~ t This generates a new potential information as an update candidate, thereby updating the cell state:

[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 This represents the input data at the current time step; h t-1 Indicates the hidden state at 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 Current cell state; W f W i W C W o b represents the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively; f ,b i ,b C ,b o σ represents the bias vectors of the forget gate, input gate, candidate memory unit, and output gate, respectively; σ represents the sigmoid activation function and the tanh activation function, respectively.

[0045] S4.3 LSTM is a time-recursive network, where multiple LSTM units are cascaded along 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 The hidden state h at the current time step is calculated using the steps described in S4.2. t and the current time step cell state C t And pass it 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 This represents the input data at the current time step; h t-1 Indicates the hidden state at 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 current state of the cell.

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

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

[0050] In the formula, h T W represents the hidden state at the last time step of the last LSTM layer. out Indicates the weights of the fully connected layer; b out y represents the bias of the fully connected layer; y represents the network's prediction.

[0051] S4.5 Finally, calculate the data error term between the network's predicted data and the actual training data, using the mean squared error (MSE) as follows:

[0052]

[0053] In the formula, N represents the sample size; Y i This represents the actual value of the groundwater level; y i This represents the predicted value of the groundwater level model; MSE NN This represents the data error term in the network.

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

[0055] S5.1 Using the groundwater level predicted by the LSTM network and the one-dimensional vertical saturated seepage equation for groundwater in the study area, the partial derivatives are calculated using the finite difference method:

[0056]

[0057] In the formula, h represents the predicted groundwater level; t represents time; K z Represents the vertical permeability coefficient; z represents the vertical coordinate; h m h represents the equivalent water column elevation at a piezometer burial depth of m meters; n h represents the equivalent water column elevation at a piezometer depth of n meters; t+1 h represents the predicted water level at the previous moment. t This represents the predicted water level at the current moment.

[0058] S5.2 Substitute the partial derivatives obtained in S5.1 back into the physical equations to calculate the physical residual terms:

[0059]

[0060] In the formula, MSE PDE Represents the network physical residual term; N represents the sample size; S s The aquifer storage coefficient is represented by h; the groundwater level is represented by t; and K represents time. z denoted by , z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source-sink term.

[0061] Preferably, S6 includes the following steps:

[0062] S6.1 The total loss function of the LSTM-PINNs network is a weighted sum of the 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 weighting coefficients used to balance the data error term and the physical residual. PDE and MSE NN These are the physical residual term and the data error term of the model, respectively.

[0065] S6.2 The LSTM-PINNs network uses the Backpropagation-Time (BPTT) algorithm to progressively calculate gradients from the last time step forward and backpropagate these gradients to all time steps. First, the output layer gradient is calculated, and the error is backpropagated from the network's predictions back to the LSTM network.

[0066]

[0067] Secondly, the gradient of the hidden layer is calculated, and the error is propagated from the next time step to the current time step, thereby accumulating the error:

[0068]

[0069] Next, calculate the gradients of the gating mechanism, 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 gradient of the parameters:

[0075]

[0076] In the formula, MSE represents the loss function of the LSTM-PINNs network; y represents the network prediction value; W out The weights of the fully connected layer are represented by N; the number of samples is represented by f. t i t , o t These 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 x represents the network prediction result at time step t; t Indicates the input data for time step t; W f W i W C W o b represents the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively; f ,b i ,b C ,b o represents the bias vectors for the forget gate, input gate, candidate memory unit, and output gate, respectively; tanh represents the tanh activation function.

[0077] S6.3 After calculating all gradients, update the network weights using an optimization algorithm (such as SGD or Adam), iterate through the training process, 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 (e.g., Wi). f W i W o W C ); b represents the bias parameter of the LSTM-PINNs network (e.g., b f b i b o b C ); η represents the model's learning rate; MSE represents the network loss function.

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

[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] Changes in groundwater level or total soil stress can cause pore water pressure to dissipate or increase. Effective stress changes control soil particle reorganization and deformation. This deformation cannot be identified by stratification markers or InSAR monitoring in the short term. This 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, an LSTM-PINNs groundwater level simulation model based on pore water pressure is constructed. This model can be used to monitor dynamic changes in groundwater level, predict overflow recharge, and conduct early warning of groundwater level. It provides a scientific basis for risk assessment and prevention decisions of geological disasters (land subsidence, ground fissures, etc.) and has broad application prospects. Attached Figure Description

[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 now briefly described. Obviously, the drawings in the following description are merely illustrative 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 effort.

[0085] Figure 1 This is a schematic flowchart of an early warning method for groundwater level based on pore water pressure proposed in Embodiment 1 of the present invention.

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

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

[0088] Figure 4 This is a schematic diagram of the water level simulation prediction results proposed in Embodiment 2 of the present invention. Detailed Implementation

[0089] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0090] Example 1:

[0091] Please see Figure 1 An early warning method for groundwater levels based on pore water pressure includes the following steps:

[0092] S1. By using ground subsidence monitoring stations, obtain groundwater level data, stratified pore water pressure values, and pumping volume data at the station every 5 days. This requires the use of geoscience and statistical software, including ArcGIS, QGIS, or EXCEL.

[0093] S2. Based on the principles of fluid statics and groundwater dynamics, the pore water pressure measurements are converted into water level elevations. The equivalent water column elevations at piezometer depths of 78.5m and 96.5m are calculated respectively.

[0094]

[0095] In the formula, h represents the equivalent water column elevation (m) for pore water pressure conversion; H represents the elevation of the measuring point (m); D represents the burial depth of the pore water pressure measuring point (m); u represents the measured value of pore water pressure (Pa); r w The unit weight of water (N / m³) 3 ).

[0096] S3. Obtain basic geological and hydrogeological data for the study area, and construct a one-dimensional vertical saturated seepage equation for groundwater in the study area based on Darcy's law and the law of conservation of mass. This includes the following steps:

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

[0098]

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

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

[0101]

[0102] In the formula, S s The aquifer storage coefficient is represented by h; the groundwater level is represented by t; and time is represented by q. z The vertical velocity per unit area is represented by z; the vertical coordinate is represented by Q; and the source / sink term represents the effect of external recharge or pumping.

[0103] S3.3 Substituting Darcy's law into the continuity equation, we obtain the one-dimensional vertical saturated seepage equation for groundwater in the study area, which describes the change of hydraulic head with time and depth:

[0104]

[0105] In the formula, S s The aquifer storage coefficient is represented by h; the groundwater level is represented by t; and K represents time. z denoted by , z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source-sink term.

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

[0107] h| t＝0 =h0 (5)

[0108] S4. Using the groundwater level data and pumping volume data obtained in S1, and the pore water level data of different strata obtained in S2, as inputs, train the physically constrained LSTM-PINNs model, and obtain the data error term between the neural network output and the training data. This specifically includes the following steps:

[0109] The input layer of the S4.1 LSTM-PINNs groundwater level model receives time-series data, including aquifer level, equivalent water column elevation of stratified pore water, and pumping volume. The data is organized according to the batch size, time steps, and input features to support batch training and time-series modeling.

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

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

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

[0113] Secondly, by inputting gate i t Calculate which new information needs to be updated:

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

[0115] Next, calculate candidate memory units. Generate new potential information as an update candidate to update the cell state:

[0116]

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

[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 This represents the input data at the current time step; h t-1 Indicates the hidden state at 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 Current cell state; W f W i W C W o b represents the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively; f ,b i ,b C ,b o σ and tanh 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 the tanh activation function, respectively.

[0121] S4.3 LSTM is a time-recursive network, where multiple LSTM units are cascaded along 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 The hidden state h at the current time step is calculated using the steps described in S4.2. t and the current time step cell state C t And pass it 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] In the formula, x t This represents the input data at the current time step; h t-1 Indicates the hidden state at 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 current state of the cell.

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

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

[0126] In the formula, h T W represents the hidden state at the last time step of the last LSTM layer. out Indicates the weights of the fully connected layer; b out y represents the bias of the fully connected layer; y represents the network's prediction.

[0127] S4.5 Finally, calculate the data error term between the network's predicted data and the actual training data, using the mean squared error (MSE) as follows:

[0128]

[0129] In the formula, N represents the sample size; Y i This represents the actual value of the groundwater level; y i This represents the predicted value of the groundwater level model; MSE NN This represents the data error term in the network.

[0130] S5. Substitute the network output results obtained from the model in S4 into the groundwater dynamics equation in S3, and solve the physical equation using the finite difference method to obtain the physical residual terms of the LSTM-PINNs network. This includes the following steps:

[0131] S5.1 Using the groundwater level predicted by the LSTM network and the one-dimensional vertical saturated seepage equation for groundwater in the study area, the partial derivatives are calculated using the finite difference method:

[0132]

[0133] In the formula, h represents the predicted groundwater level; t represents time; K z Represents the vertical permeability coefficient; z represents the vertical coordinate; h m h represents the equivalent water column elevation at a piezometer burial depth of m meters; n h represents the equivalent water column elevation at a piezometer depth of n meters; t+1 h represents the predicted water level at the previous moment. t This represents the predicted water level at the current moment.

[0134] S5.2 Substitute the partial derivatives obtained in S5.1 back into the physical equations to calculate the physical residual terms:

[0135]

[0136] In the formula, MSE PDE Represents the network physical residual term; N represents the sample size; S s The aquifer storage coefficient is represented by h; the groundwater level is represented by t; and K represents time. z denoted by , z represents the vertical permeability coefficient; z represents the vertical coordinate; Q represents the source-sink term.

[0137] S6. By superimposing the data error term obtained in S4 and the physical residual term obtained in S5, the overall loss of the LSTM-PINNs model for groundwater level based on pore water pressure is obtained. This loss is then backpropagated to the neural network model for iterative optimization. The model learns gradually and obtains the optimized groundwater level early warning model (its structure is as follows). Figure 2 (As shown). The simulation of groundwater level based on pore water pressure includes the following steps:

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

[0139] MSE=λ1MSE NN +λ2MSE PDE (18)

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

[0141] S6.2 The LSTM-PINNs network uses the Backpropagation-Time (BPTT) algorithm to progressively calculate gradients from the last time step forward and backpropagate these gradients to all time steps. First, the output layer gradient is calculated, and the error is backpropagated from the network's predictions back to the LSTM network.

[0142]

[0143] Secondly, the gradient of the hidden layer is calculated, and the error is propagated from the next time step to the current time step, thereby accumulating the error:

[0144]

[0145] Next, calculate the gradients of the gating mechanism, 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 input gradient:

[0148]

[0149] Finally, calculate the gradient 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 The weights of the fully connected layer are represented by N; the number of samples is represented by f. t i t , o t These 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 x represents the network prediction result at time step t; t Indicates the input data for time step t; W f W i W C W o b represents the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively; f b i b C b o represents the bias vectors for 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 using an optimization algorithm (such as SGD or Adam), iterate through the training process, 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., Wi). f W i W o W C ); b represents the bias parameter of the LSTM-PINNs network (e.g., b f b i b o b C ); η represents the model's learning rate; MSE represents the network loss function.

[0156] Example 2:

[0157] Based on Example 1, the difference lies in that this example obtains stratified pore water pressure values ​​and groundwater level information in a plain area of ​​a certain city through the National Ground Subsidence Monitoring Project, covering the period from April 5, 2004 to October 1, 2014. After identifying the hydrogeological and engineering geological conditions of the area, and integrating various regional hydrogeological parameters, source and sink information, an LSTM-PINNs model with a physical mechanism is constructed using the groundwater dynamics equation as a physical constraint to achieve early warning of groundwater level based on pore water pressure. The main steps are as follows:

[0158] Step 1: Obtain groundwater level, pore water pressure values ​​and pumping volume data at the ground settlement monitoring station every 5 days.

[0159] Step Two: Based on the principles of fluid statics and groundwater dynamics, convert the measured values ​​of stratified pore water pressure into water level elevations. The equivalent water level elevations for each stratum are as follows: Figure 3 As shown;

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

[0161] Step 4: Use the groundwater level data and pumping volume data obtained in Step 1, and the equivalent pore water level data of different strata obtained in Step 2 as input to train the physically constrained LSTM-PINNs model, and obtain the data error term between the neural network output and the training data.

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

[0163] Step Six: Superimpose the data error term obtained in Step Four and the physical residual term obtained in Step Five to obtain the overall loss of the LSTM-PINNs groundwater level model based on pore water pressure. Backpropagate this loss back to the neural network model for iterative optimization. The model learns gradually, achieving groundwater level simulation based on pore water pressure. The prediction results are as follows: Figure 4 As shown.

[0164] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A groundwater level early warning method based on pore water pressure, characterized in that, Includes the following steps; S1. Data Acquisition: Obtain groundwater level data, pore water pressure values ​​of different layers, and pumping volume through the ground settlement monitoring station at a preset time period. S2. Data Conversion: Based on the principles of fluid statics and groundwater dynamics, the measured pore water pressure values ​​are converted into equivalent water column elevations. S3. Equation Construction: Obtain basic geological and hydrogeological data of the study area, and construct a one-dimensional vertical saturated seepage equation for groundwater in the study area based on 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 strata obtained in S2, an LSTM-PINNs groundwater level model based on pore water pressure is constructed and trained to obtain the data error term between the neural network output and the training data. S5. Physical Constraints: Substitute the network output obtained from the model in S4 into the groundwater dynamics equation in S3, and solve the physical equation using the finite difference method to obtain the physical residuals of the LSTM-PINNs network; specifically, this includes the following steps: S5.1 Using the groundwater level predicted by the LSTM network and the one-dimensional vertical saturated seepage equation for groundwater in the study area, the partial derivatives are calculated using the finite difference method: （15） （16） In the formula, h Indicates the groundwater level; t Indicates time; K z Indicates the vertical permeability coefficient; z Represents the vertical coordinate; h m This indicates the equivalent water column elevation at a piezometer burial depth of m meters; h n This represents the equivalent water column elevation at a piezometer burial depth of n meters; h t+1 This indicates the predicted water level at the next moment. h t This represents the predicted water level at the current moment. S5.2 Substitute the partial derivatives obtained in S5.1 back into the physical equations to calculate the physical residual terms: （17） In the formula, MSE PDE Represents the network physical residual term; N Indicates the number of samples; S s Indicates the aquifer water storage coefficient; Q Indicates source and sink terms; S6. Model Optimization: The data error term obtained in S4 and the physical residual term obtained in S5 are superimposed to obtain the overall loss of the LSTM-PINNs model of groundwater level based on pore water pressure. This loss is then backpropagated to the neural network model for continuous iteration and optimization. The model learns step by step to obtain the optimized groundwater level early warning model, realizing 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 acquiring the stratified groundwater level data and stratified pore water pressure values ​​at the site, geographical and statistical processing software, including ArcGIS, QGIS, or EXCEL software, is used.

3. The method for early warning of groundwater level based on pore water pressure according to claim 1, characterized in that, The conversion of pore water pressure measurements to equivalent water column elevation as described in S2 specifically includes: Based on the principles of head and hydrostatics, calculate the equivalent water column elevation: （1） In the formula, h The equivalent water column elevation, in meters, represents the pressure conversion of pore water. H Indicates the elevation of the measuring point, in meters (m). D The depth of the pore water pressure measuring point is indicated in meters (m). u This represents the measured pore water pressure, in Pa. r w 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 for groundwater in the study area, as described in S3, specifically includes the following steps: S3.1 Based on Darcy's law, the one-dimensional vertical flow equation of groundwater in a saturated porous medium can be expressed as: （2） In the formula, q z This indicates the vertical velocity per unit area. K z Indicates the vertical permeability coefficient; h Indicates 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 a saturated porous medium is expressed as: （3） In the formula, S s Indicates the aquifer water storage coefficient; Q For source and sink terms, it indicates the impact of external supply or pumping; S3.3 Substituting Darcy's law into the continuity equation, we obtain the one-dimensional vertical saturated seepage equation for groundwater in the study area, which describes the change of hydraulic head with time and depth: （4） S3.4, The equation satisfies the following conditions under the initial conditions. （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, as described in S4, specifically includes the following steps: The input layer of the S4.1 LSTM-PINNs groundwater level model receives time series data, including: aquifer 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, time step dimension, and input feature dimension to support batch training and time series modeling. S4.2 In LSTM-PINNs networks, the input layer is connected to the hidden layer through a weight matrix. Each LSTM unit in the hidden layer follows the calculation formula: First, through the Gate of Oblivion f t Calculate forgotten information: （6） Secondly, through the input gate i t Calculate the updated information: （7） Next, calculate candidate memory units. This generates a new potential information as an update candidate, thereby updating the cell state: （8） （9） Finally, through the output gate o t Calculate the new hidden state of the cell: （10） （11） In the formula, x t This represents the input data at the current time step. h t-1 Indicates the hidden state at the previous moment; C t-1 It indicates the cell state at the previous moment; h t Indicates the current hidden state; C t The current state of the cell; W f , W i , W C , W o These represent the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively. b f , b i , b C , b o These represent the bias vectors for the forget gate, input gate, candidate memory unit, and output gate, respectively. and tanh represent the sigmoid activation function and the tanh activation function, respectively; S4.3 LSTM is a time-recursive network, where multiple LSTM units are cascaded on the time axis to form an LSTM layer; at each time step t The LSTM unit receives the input of the current time step. x t And the hidden state transmitted from the previous time step h t-1 and cell state C t-1 The hidden state at the current time step is calculated using the steps described in S4.

2. h t and the current time step cell state C t and pass it to the next time step. t +1, perform recursive calculation, that is: ( h t , C t )= LSTM ( x t , h t-1 , C t-1 )（12） S4.4 When there are multiple LSTM layers, each LSTM layer is concatenated in the depth direction, and the hidden state of the previous LSTM layer is... The output of the last LSTM layer is passed as input to the next LSTM layer, and the output of the last LSTM layer is mapped to the output layer through a fully connected layer. y = Dense ( h T )= W out · h T + b out （13） In the formula, h T This represents the hidden state at the last time step of the last LSTM layer; W out Indicates the weights of the fully connected layer; b out Indicates the bias of the fully connected layer; y This represents the network's predicted value; S4.5 Calculate the data error term between the network's predicted data and the actual training data, using the mean squared error (MSE) formula as follows: （14） In the formula, N Indicates the number of samples; Y i This represents the actual value of the groundwater level; y i This represents the predicted value of the groundwater level model; MSE NN This represents the data error term in 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 described in S6 involves backpropagating the overall loss of the obtained LSTM-PINNs model back to the neural network model and iterating the parameters for learning. This specifically includes the following steps: S6.1 The total loss function of the LSTM-PINNs network is a weighted sum of the physical residual term of the model and the data error term of the network: （18） In the formula, MSE This represents the total loss term of the LSTM-PINNs model. and These are weighting coefficients used to balance the data error term and the physical residual. MSE PDE and MSE NN These are the physical residuals and data error terms of the model, respectively. S6.2 The LSTM-PINNs network uses the time backpropagation algorithm to calculate the gradient step by step from the last time step and backpropagate the gradient to all time steps. Specifically, it includes the following: First, calculate the gradient of the output layer and backpropagate the error from the network's predictions back to the LSTM network: （19） Secondly, the gradient of the hidden layer is calculated, and the error is propagated from the next time step to the current time step to accumulate the error: （20） Next, calculate the gradients of the gating mechanism, that is, calculate the gradients of the forget gate, input gate, candidate memory unit, and output gate respectively: （21） （22） （23） （24） Then, calculate the cell state gradient and the input gradient: （25） （26） Finally, calculate the gradient of the parameters: （27） （28） （29） （30） （31） （32） （33） （34） In the formula, y Indicates the network prediction value; W out Indicates the weights of the fully connected layer; N Indicates the number of samples; f t , i t , , o t These represent the forget gate, input gate, candidate memory unit, and output gate, respectively. h t Indicates the current hidden state; h t+1 Indicates the hidden state in the next moment; y t Indicates time step t Network prediction results; x t Indicates time step t Input data; W f , W i , W C , W o These represent the weight matrices for the forget gate, input gate, candidate memory units, and output gate, respectively. b f , b i , b C , b o denoted as the bias vectors for 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 using an optimization algorithm, iterate through training, and minimize the model loss function until the model converges: （35） （36） In the formula, W Represents the weight parameters of the LSTM-PINNs network; b Indicates the bias parameters of the LSTM-PINNs network; This represents the model's learning rate.

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