A parameter inversion and model training method and system for flow calculation in hydrological monitoring

Abstract

This invention discloses a method and system for parameter inversion and model training in flow calculation during hydrological monitoring, belonging to the field of hydrological monitoring technology. The method includes collecting basic hydrological data and calculating derived hydrological parameters; performing moving average and filtering on water level variability to obtain a standardized hydrological parameter set; then, using multi-dimensional feature vectors as input, employing machine learning combined with hydraulic physical equations to invert fixed feature parameters of the monitoring station; optimizing the physical equations based on the fixed parameters and calibrating the exponential correction parameters; finally, using the fixed parameters, exponential parameters, and optimized physical equations as constraints, constructing and training a machine learning model that integrates physical constraints to obtain a real-time flow calculation model. This invention achieves accurate inversion of station-specific parameters and personalized optimization of physical equations, taking into account both hydraulic laws and the nonlinear temporal characteristics of data, significantly improving the accuracy and applicability of flow calculation in complex hydrological scenarios, and can be used for real-time flow calculation and short-term forecasting.

Description

Technical Field

[0001] This invention relates to the field of hydrological monitoring technology, specifically to a method and system for parameter inversion and model training in flow calculation during hydrological monitoring. Background Technology

[0002] Flow calculation in hydrological monitoring is a core technical link in flood control and disaster reduction, water conservancy project scheduling, water resources management, and hydrological and water resources research. The accuracy of flow calculation directly determines the scientificity and reliability of hydrological analysis and engineering decision-making.

[0003] Existing methods for calculating hydrological flow can be mainly divided into two categories: one is based on traditional hydraulic equations, with Manning's formula as the core, which calculates flow through factors such as riverbed slope, roughness, and cross-sectional parameters. However, this type of method has obvious drawbacks, such as low accuracy in determining the core fixed parameters, and the fixed form of traditional hydraulic equations makes it poorly adaptable to complex hydrological scenarios such as unsteady flow, flood fluctuations, backwater, and reciprocating flow caused by dam and gate scheduling, and it cannot match the nonlinear characteristics of actual hydrological data. The other type is flow prediction methods based on pure machine learning models, which calculate flow by mining the temporal features of hydrological data. However, this type of method lacks the constraints of hydraulic physical laws, and the model output results are prone to deviating from actual hydrological laws. Furthermore, it does not customize parameters for the individual characteristics of different stations, resulting in poor universality across stations and scenarios.

[0004] Meanwhile, existing technologies lack standardized smoothing and filtering methods for processing derived hydrological parameters such as water level variability and dynamic gradient. Data fluctuations and measurement errors are easily introduced into the calculation process, further reducing the accuracy of flow calculation. Moreover, an integrated technical solution has not yet been formed, which cannot take into account the individual characteristics of the station, the laws of hydraulic physics, and the nonlinear and temporal characteristics of hydrological data. As a result, the accuracy and applicability of flow calculation in complex hydrological scenarios cannot meet the actual engineering needs. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for parameter inversion and model training in flow calculation during hydrological monitoring, so as to solve the technical problems of low parameter determination accuracy, poor adaptability to complex scenarios, insufficient integration of physical laws and machine learning, and large data processing errors in existing hydrological monitoring flow calculation technologies.

[0006] To achieve the above objectives, the present invention provides the following technical solution: In a first aspect, the present invention provides a method for parameter inversion and model training in flow calculation during hydrological monitoring, comprising the following steps: Basic hydrological data from hydrological stations are collected, and derived hydrological parameters are calculated based on the basic hydrological data. The water level variability in the derived hydrological parameters is processed by moving average and filtering to obtain a standardized set of hydrological parameters. Using multidimensional feature vectors from a standardized set of hydrological parameters as model input, a fusion method combining machine learning and hydraulic physical equations is employed to invert and obtain the fixed feature parameters of hydrological stations. Based on the fixed characteristic parameters obtained by inversion, the form of the hydraulic physical equation is optimized. The fusion method of machine learning and hydraulic physical equation is adopted to calibrate the exponential correction parameters of the optimized hydraulic physical equation, thus completing the personalized optimization of the hydraulic physical equation for hydrological stations. Using the fixed characteristic parameters obtained from inversion, the exponential correction parameters obtained from calibration, and the optimized hydraulic physical equations as physical constraints, a machine learning model integrating physical constraints is constructed. The standardized set of hydrological parameters is used as the model input, and after iterative training and convergence, a model for real-time calculation of hydrological flow is obtained.

[0007] Furthermore, the basic hydrological data includes water level Z, cross-sectional area A, wetted perimeter P, and average cross-sectional velocity V. a The water surface width B and the riverbed slope S0; the derived hydrological parameters include the water level variability Z. cr Water level acceleration Z cr2 Observational gradient S m Power ratio reduction S d Additional ratio S r .

[0008] Furthermore, the calculation formula for the derived hydrological parameters is as follows: Where ΔZ is the difference between the current water level and the water level 5 minutes ago. τ is a 5-minute time step.

[0009] Furthermore, the multidimensional feature vector is [Z(t), A(t), P(t), Z cr V a The fixed characteristic parameters of the hydrological station obtained by inversion include the riverbed slope S0 and the equivalent roughness characteristic parameter k.

[0010] Furthermore, the machine learning model uses a BP neural network, a sequential quadratic programming model, or an SVM support vector machine. The input layer receives multi-dimensional feature vectors, the LSTM layer / hidden layer adopts a multi-layer stacked structure, and the physical constraint layer maps the output of the machine learning model to the hydraulic physical equations.

[0011] Furthermore, the hydraulic physical equations are as follows: in, Here, k represents the overall resistance parameter, and k represents the equivalent roughness characteristic parameter. The optimized hydraulic physical equations are as follows: Where α, β, and γ are exponential correction parameters.

[0012] Furthermore, the specific method for fusing the physical constraints is as follows: The residual between the flow rate calculated by the optimized hydraulic physical equations and the flow rate predicted by the machine learning model is used as one of the loss terms in the model training. This residual is then fused with the model's prediction loss to form the total loss function, which is: Total loss function = MAE(model prediction Q, actual monitoring Q) + ω × MAE(physical equation calculation Q, model prediction Q) Where MAE is the mean absolute error and ω is the physical constraint weight, with a value range of 0.1-0.5. The combination of physical laws and machine learning models is achieved through iterative optimization of the total loss function.

[0013] Secondly, the present invention provides a parameter inversion and model training system for flow calculation in hydrological monitoring, including a parameter preprocessing module, a fixed parameter inversion module, an exponential parameter calibration module and a physical constraint training module, wherein each module realizes data interaction through a hydrological data bus; The parameter preprocessing module is used to collect basic hydrological data, calculate derived hydrological parameters, and perform moving average and filtering processing on water level variability to output a smoothed standardized set of hydrological parameters. The fixed parameter inversion module is connected to the parameter preprocessing module. It takes a multi-dimensional feature vector as input and outputs the station's fixed feature parameters through a fusion method of machine learning and hydraulic physical equations. The exponential parameter calibration module is connected to the fixed parameter inversion module. It optimizes the hydraulic physical equations based on fixed characteristic parameters, calibrates and outputs exponential correction parameters, and automatically generates the optimized hydraulic physical equations. The physical constraint training module is connected to the fixed parameter inversion module, the exponential parameter calibration module, and the parameter preprocessing module, respectively. It takes the hydrological parameter set as input, and the fixed feature parameters, exponential correction parameters, and optimized physical equations as physical constraints to construct and train a machine learning model that integrates physical constraints, and outputs a flow calculation model.

[0014] Based on the above technical solution, the embodiments of the present invention can produce at least the following technical effects: (1) This invention uses a fusion method of machine learning and hydraulic physical equations to invert the fixed characteristic parameters of the station. The physical equations impose hard constraints on the machine learning model to avoid the parameters driven by pure data from deviating from the actual hydraulic laws. This enables the accurate determination of the station's exclusive parameters, perfectly adapts to the differences in hydrological characteristics of different stations, ensures the accuracy of hydraulic station parameter inversion, and has strong personalized adaptability.

[0015] (2) This invention optimizes the form of the traditional basic hydraulic equations by calibrating the exponent parameters and corrects the power coefficients of the equations so that the optimized equations can accurately fit the actual hydrological laws of the target station. It is applicable to various complex hydrological scenarios such as natural rivers, flood rise and fall, backwater, downstream of dams, and reciprocating flow, and solves the core problem of the fixed form and poor adaptability of traditional equations.

[0016] (3) This invention integrates fixed feature parameters and optimized physical equations as physical constraints into the training of machine learning models. By using loss function fusion, model coupling, and residual establishment, it takes into account the nonlinear and temporal characteristics of hydrodynamic physical laws and hydrological data. This avoids the problems of overfitting and deviation from reality in pure machine learning models, and solves the defects of pure physical equations in being unable to mine complex data features. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0018] Figure 1 This is an overall flowchart of the method described in this invention; Figure 2 A schematic diagram of the module composition of the system of the present invention. Detailed Implementation

[0019] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. In addition, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0020] Example 1 like Figure 1 As shown, this embodiment provides a method for parameter inversion and model training in flow calculation during hydrological monitoring, including the following steps: Step 1: Hydrological parameter preprocessing: Collect basic hydrological data from hydrological stations, including water level Z, cross-sectional area A, wetted perimeter P, and average cross-sectional velocity V. a The water surface width (B) and riverbed slope (S0) are calculated based on the basic hydrological data. Derived hydrological parameters include water level variability (Z). cr Water level acceleration Z cr2 Observational gradient S m Power ratio reduction S d Additional ratio S r ; for water level variability Z cr A standardized set of hydrological parameters was obtained by using moving average and filtering.

[0021] In this embodiment, the filtering process is Gaussian filtering, and the moving average window is set to 5 time steps.

[0022] Specifically, the formulas for calculating the derived hydrological parameters are as follows: Where ΔZ is the difference between the current water level and the water level 5 minutes ago. τ is a 5-minute time step.

[0023] Step 2, Inversion of Fixed Feature Parameters at a Single Station: Using the multidimensional feature vector [Z(t), A(t), P(t), Z] from the standardized hydrological parameter set... cr V a Using [the model input] as input, a fusion method combining machine learning and hydraulic physical equations is employed to invert and obtain the fixed characteristic parameters of the hydrological station, including the riverbed slope S0 and the equivalent roughness characteristic parameter k.

[0024] It should be noted that the machine learning model preferably uses a backpropagation (BP) neural network, but sequential quadratic programming and support vector machines (SVM) can also be used; the model structure for the single-station fixed feature parameter inversion is as follows: the input layer receives a multi-dimensional feature vector [Z(t), A(t), P(t), Z...]. cr, V a The LSTM layer / hidden layer adopts a multi-layer stacked structure, and the physical constraint layer maps the output of the machine learning model to the hydraulic physics equation.

[0025] Specifically, the hydraulic physical equations are: in, Here, k represents the overall resistance parameter, and k represents the equivalent roughness characteristic parameter. Step 3, Single-station index parameter calibration: Based on the fixed characteristic parameters obtained by inversion, the hydraulic physical equation is formally optimized. The fusion method of machine learning and hydraulic physical equation is used to calibrate the index correction parameters of the optimized hydraulic physical equation, thus completing the personalized optimization of the hydrological station hydraulic physical equation. It should be noted that the machine learning model primarily uses LSTM neural networks, but CNN convolutional neural networks can also be used. The fusion of physical constraints includes model coupling, residual generation, and incorporating the physical equation error into the model loss function. Specifically, the fusion of physical constraints involves using the residual between the flow rate calculated from the optimized hydraulic fundamental equations and the flow rate predicted by the machine learning model as one of the loss terms in model training. This residual is fused with the model's prediction loss to form the total loss function: Total Loss Function = MAE(Model Prediction Q, Actual Monitoring Q) + ω × MAE(Physical Equation Calculation Q, Model Prediction Q), where MAE is the mean absolute error, and ω is the weight of the physical constraint, ranging from 0.1 to 0.5. The combination of physical laws and the machine learning model is achieved through iterative optimization of the total loss function.

[0026] The optimized hydraulic physical equations are as follows: Where α, β, and γ are exponential correction parameters.

[0027] Step 4: Model training under physical constraints: Using the fixed feature parameters obtained from inversion, the exponential correction parameters obtained from calibration, and the optimized hydraulic physical equations as physical constraints, a machine learning model integrating physical constraints is constructed. The standardized hydrological parameter set is used as the model input, and after iterative training and convergence, a model for real-time calculation of hydrological flow is obtained.

[0028] Example 2 like Figure 2 As shown, this embodiment provides a parameter inversion and model training system for flow calculation in hydrological monitoring. The system is developed based on the Python language, integrates the TensorFlow / PyTorch deep learning framework, and adopts a modular design, including a parameter preprocessing module, a fixed parameter inversion module, an exponential parameter calibration module, and a physical constraint training module. Each module realizes data interaction through a hydrological data bus. The parameter preprocessing module integrates a hydrological monitoring sensor data acquisition interface (supporting BeiDou / 4G / 5G transmission), and has built-in derived parameter calculation algorithms and moving average and Gaussian filtering data processing algorithms. It can automatically collect, calculate, and process hydrological data, output a standardized hydrological parameter set in CSV format, and support data visualization. This module is used to collect basic hydrological data, calculate derived hydrological parameters, and perform moving average and filtering processing on water level variability to output a smoothed standardized hydrological parameter set. The fixed parameter inversion module is connected to the parameter preprocessing module. It builds a BP neural network model based on the TensorFlow framework and has a built-in hydraulic physical equation constraint module. It can read the hydrological parameter set of the parameter preprocessing module, take multi-dimensional feature vectors as input, and automatically train and output the station's fixed feature parameters S0 and k through the fusion method of machine learning and hydraulic physical equations. It supports the export and visualization storage of parameter results in Excel. The exponential parameter calibration module is connected to the fixed parameter inversion module. It optimizes the hydraulic physical equations based on fixed characteristic parameters, calibrates and outputs exponential correction parameters, and automatically generates the optimized hydraulic physical equations. The physical constraint training module is connected to the fixed parameter inversion module, the exponential parameter calibration module, and the parameter preprocessing module. It constructs an LSTM model based on the PyTorch framework and includes a built-in loss function fusion module. It can read the output parameters of the fixed parameter inversion module and the exponential parameter calibration module, as well as the hydrological parameter set from the parameter preprocessing module. It automatically trains a flow calculation model that integrates physical constraints, supporting model saving (.pth format), retrieval, and real-time inference. It can achieve real-time flow calculation and 1-3 hour short-term prediction, with prediction results supporting visualization and data export. Using the hydrological parameter set as input and fixed feature parameters, exponential correction parameters, and optimized physical equations as physical constraints, it constructs and trains a machine learning model that integrates physical constraints, outputting a flow calculation model.

[0029] The system can be deployed on hydrological monitoring terminals (embedded devices), edge computing servers, or cloud servers. It supports Windows / Linux operating systems, supports parallel computing across multiple stations, and can process up to 1,000 data points per second per station, fully meeting the real-time requirements of hydrological monitoring.

[0030] In the actual application of the system in this embodiment at a hydrological station in the middle reaches of the Yangtze River, it operates stably, processes data accurately, and has high accuracy in flow calculation and short-term forecasting. It can be directly connected to the existing hydrological monitoring system, providing scientific and technical support for the station's flood control scheduling and water conservancy project management.

[0031] In practical implementation, based on hydrological monitoring data from a downstream hydrological station of a dam from 2020 to 2024, with a time step τ = 5 minutes, the parameter inversion and model training method for flow calculation in hydrological monitoring of this invention is implemented. The specific steps are as follows: Step 1: Hydrological parameter preprocessing 1. Basic Data Collection: Collect basic hydrological data for this station from 2020 to 2023, including water level Z, cross-sectional area A, wetted perimeter P, water surface width B, and average cross-sectional velocity V. a The data sampling interval was 5 minutes, and a total of 525,600 valid data entries were collected; 2. Calculation of derived parameters: Based on the basic data, the effective water flow area A is calculated according to the formula defined in this invention. e Effective hydraulic radius R e Average water depth H a Water level variability Z cr Water level acceleration Z cr2 Observational gradient S m Power ratio reduction S d Additional ratio S r ; 3. Data smoothing: Smoothing the water level variability Z cr A 5-step moving average and Gaussian filtering process is used to eliminate data ripple and measurement errors, resulting in a continuously smooth 5-minute Z-axis. cr The mean square error of the filtered data was reduced by 68% compared to the original data.

[0032] Step 2: Inversion of fixed characteristic parameters at a single station 1. Data partitioning: The preprocessed hydrological data were divided into a training set (357,920 records) and a validation set (157,680 records) in a 7:3 ratio. 2. Model Construction: A backpropagation (BP) neural network was selected as the core machine learning model, and a fusion model consisting of an input layer, hidden layer, physical constraint layer, and output layer was constructed. Input layer: Multidimensional feature vector [Z(t), A(t), P(t), Z cr V a ]; Hidden layers: 3 stacked layers, 64 neurons per layer, with ReLU activation function; Physical constraint layer: embedding core physical equations ; Output layer: Fixed feature parameters [S0,k] of the station; 3. Model Training: The mean square error (MSE) between the flow rate calculated by the physical equation and the actual monitored flow rate is used as the loss function. The Adam optimizer is used with a learning rate of 0.001. The model is trained iteratively for 80 rounds until the loss function converges. 4. Results output: The fixed characteristic parameters of the downstream monitoring station of the dam are: S0=0.0012, k=0.018.

[0033] Step 3: Single-station index parameter calibration 1. Equation Optimization: Based on the fixed characteristic parameters S0 and k obtained in step 2, the optimized hydraulic physical equations are constructed: 2. Model reuse: Reuse the BP neural network model from step 2, only replacing the output layer with exponential correction parameters [α,β,γ]; 3. Calibration training: Using the mean absolute error (MAE) between the flow rate calculated by the optimized equation and the actual monitored flow rate as the loss function, iterative training is performed for 60 rounds until convergence; 4. Results output: The exponential correction parameters are obtained as follows: α=0.02, β=0.015, γ=0.03, completing the personalized optimization of the hydraulic equation.

[0034] Step 4: Model training under physical constraints 1. Basic Model Construction: LSTM neural network is selected as the core model, and two LSTM layers are constructed, with 128 neurons in each layer. The dropout coefficient is set to 0.2 to prevent overfitting. 2. Physical constraint fusion: The loss function fusion method is adopted, and the total loss function = MAE(model prediction Q, actual monitoring Q) + 0.3×MAE(physical equation calculation Q, model prediction Q) is used, where the physical constraint weight ω=0.3; 3. Model training: The Adam optimizer was used with a learning rate of 0.001. The training was iterated for 100 rounds. During the training process, a validation set was used for early stopping to avoid overfitting. 4. Model Validation: After training, the model was tested on the validation set. The mean absolute error of the traffic calculation was 0.85m. 3 The model achieves a flow rate of / s, reducing the error by 42% compared to the traditional Manning formula method and by 18% compared to the pure LSTM model. Furthermore, the model exhibits good adaptability to reciprocating flows caused by dam scheduling and unsteady flows during flood rise and fall periods, enabling short-term flow predictions for the next 1-3 hours with an accuracy of over 93%.

[0035] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for parameter inversion and model training in flow calculation during hydrological monitoring, characterized in that, Includes the following steps: Basic hydrological data from hydrological stations are collected, and derived hydrological parameters are calculated based on the basic hydrological data. The water level variability in the derived hydrological parameters is processed by moving average and filtering to obtain a standardized set of hydrological parameters. Using multidimensional feature vectors from a standardized set of hydrological parameters as model input, a fusion method combining machine learning and hydraulic physical equations is employed to invert and obtain the fixed feature parameters of hydrological stations. Based on the fixed characteristic parameters obtained by inversion, the form of the hydraulic physical equation is optimized. The fusion method of machine learning and hydraulic physical equation is adopted to calibrate the exponential correction parameters of the optimized hydraulic physical equation, thus completing the personalized optimization of the hydraulic physical equation for hydrological stations. Using the fixed characteristic parameters obtained from inversion, the exponential correction parameters obtained from calibration, and the optimized hydraulic physical equations as physical constraints, a machine learning model integrating physical constraints is constructed. The standardized set of hydrological parameters is used as the model input, and after iterative training and convergence, a model for real-time calculation of hydrological flow is obtained.

2. The parameter inversion and model training method for flow calculation in hydrological monitoring according to claim 1, characterized in that, The basic hydrological data includes water level Z, cross-sectional area A, wetted perimeter P, and average cross-sectional velocity V. a The water surface width B and the riverbed slope S0; the derived hydrological parameters include the water level variability Z. cr Water level acceleration Z cr2 Observational gradient S m Power ratio reduction S d Additional ratio S r .

3. The parameter inversion and model training method for flow calculation in hydrological monitoring according to claim 2, characterized in that, The formulas for calculating the derived hydrological parameters are as follows: ; ; ; ; ; Where ΔZ is the difference between the current water level and the water level 5 minutes ago. τ is a 5-minute time step.

4. The parameter inversion and model training method for flow calculation in hydrological monitoring according to claim 3, characterized in that, The multidimensional feature vector is [Z(t), A(t), P(t), Z cr V a The fixed characteristic parameters of the hydrological station obtained by inversion include the riverbed slope S0 and the equivalent roughness characteristic parameter k.

5. The parameter inversion and model training method for flow calculation in hydrological monitoring according to claim 1, characterized in that, The machine learning model uses a BP neural network, a sequential quadratic programming model, or an SVM support vector machine. The input layer receives multi-dimensional feature vectors, the LSTM layer / hidden layer adopts a multi-layer stacked structure, and the physical constraint layer maps the output of the machine learning model to the hydraulic physical equations.

6. The method for parameter inversion and model training in flow calculation during hydrological monitoring according to claim 1, characterized in that, The hydraulic physical equations are as follows: ; in, Here, k represents the overall resistance parameter, and k represents the equivalent roughness characteristic parameter. The optimized hydraulic physical equations are as follows: ; Where α, β, and γ are exponential correction parameters.

7. The parameter inversion and model training method for flow calculation in hydrological monitoring according to claim 1, characterized in that, The specific method for fusing the physical constraints is as follows: The residual between the flow rate calculated by the optimized hydraulic physical equations and the flow rate predicted by the machine learning model is used as one of the loss terms in the model training. This residual is then fused with the model's prediction loss to form the total loss function, which is: Total loss function = MAE(model prediction Q, actual monitoring Q) + ω × MAE(physical equation calculation Q, model prediction Q) Where MAE is the mean absolute error and ω is the physical constraint weight, with a value range of 0.1-0.

5. The combination of physical laws and machine learning models is achieved through iterative optimization of the total loss function.

8. A parameter inversion and model training system for flow calculation in hydrological monitoring, characterized in that, The method for implementing any one of claims 1-7 includes a parameter preprocessing module, a fixed parameter inversion module, an exponential parameter calibration module, and a physical constraint training module, wherein each module interacts with data through a hydrological data bus. The parameter preprocessing module is used to collect basic hydrological data, calculate derived hydrological parameters, and perform moving average and filtering processing on water level variability to output a smoothed standardized set of hydrological parameters. The fixed parameter inversion module is connected to the parameter preprocessing module. It takes a multi-dimensional feature vector as input and outputs the station's fixed feature parameters through a fusion method of machine learning and hydraulic physical equations. The exponential parameter calibration module is connected to the fixed parameter inversion module. It optimizes the hydraulic physical equations based on fixed characteristic parameters, calibrates and outputs exponential correction parameters, and automatically generates the optimized hydraulic physical equations. The physical constraint training module is connected to the fixed parameter inversion module, the exponential parameter calibration module, and the parameter preprocessing module, respectively. It takes the hydrological parameter set as input, and the fixed feature parameters, exponential correction parameters, and optimized physical equations as physical constraints to construct and train a machine learning model that integrates physical constraints, and outputs a flow calculation model.

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