Hydrological data prediction method based on convolutional neural network and physical information neural network

Abstract

The invention discloses a hydrological data prediction method based on a convolutional neural network and a physical information neural network. Comprising the following steps: 1) utilizing an HEC-RAS hydrological model to generate cross sectional area, flow and water level elevation data of a key section of a target river channel in a preset time period as training reference truth values; 2) constructing space-time coordinates corresponding to the data as an input data set; 3) space-time input is processed through a physical information neural network model based on a convolutional neural network, the convolutional neural network extracts local heterogeneity features of a spatial position of a river channel through one-dimensional convolution, and a continuity equation and a momentum equation of a Saint-Venant equation set are embedded into a training process in a residual form through physical information constraint; and 4) finally, directly outputting a hydrological parameter prediction result of the target river channel. Therefore, under the environment that the observation data is deficient or missing, the hydrological parameter prediction precision and the model generalization ability are remarkably improved by fusing the simulation data and the physical mechanism constraint.

Description

TECHNICAL FIELD

[0001] The application relates to the cross field of environmental big data and deep learning, and particularly relates to a hydrological data prediction method based on a convolutional neural network and a physical information neural network. BACKGROUND

[0002] Precise prediction of hydrological parameters is the key to coping with flood disasters and water resource management. However, climate change leads to frequent extreme hydrological events, and traditional physical models (such as HEC-RAS) have clear physical mechanisms but are heavily dependent on high-precision input data (such as river geometry and boundary conditions), and their applicability is significantly reduced in basins with insufficient monitoring data or remote areas. On the other hand, data-driven machine learning models can handle complex nonlinear relationships, but they are prone to overfitting or physical distortion when data is scarce, especially when it is difficult to capture spatial heterogeneity in hydrological processes. In recent years, physical information neural networks (PINN) have provided a new approach to data-lacking scenarios by embedding physical equations into network training. However, traditional PINN mostly uses a multi-layer perceptron (MLP) structure, which has difficulty in effectively extracting spatial local features due to its global connection characteristics, resulting in insufficient modeling capability for complex hydrological spatial patterns. In addition, existing PINN research mostly focuses on data-rich environments, and how to balance physical constraints and data fitting in real data-lacking conditions is still a challenge. Therefore, a new prediction framework that combines local spatial feature extraction and physical law constraints is still needed to improve the accuracy and physical rationality of hydrological prediction in data-lacking environments. SUMMARY

[0003] The present application provides a hydrological data prediction method based on a convolutional neural network and a physical information neural network to address the physical distortion of traditional hydrological models and the insufficient spatial feature capture of machine learning methods in data-lacking environments. This method replaces the MLP basic network of PINN with a CNN architecture, couples the Saint-Venant equation physical constraint with HEC-RAS simulation data-driven, and realizes high-precision prediction of parameters such as flow and water level in data-lacking scenarios.

[0004] The technical solution to achieve the purpose of the present application is:

[0005] The hydrological data prediction method based on a convolutional neural network and a physical information neural network includes the following steps:

[0006] 1) HEC-RAS benchmark data construction: Use the hydrological model software HEC-RAS to simulate the hydrodynamic process of the target river, generate hydrological data of key sections within a predetermined time period as training benchmark true values, and perform preprocessing, specifically:

[0007] 1-1) Data generation: input the target river channel topographic parameters (including river channel geometry, cross-section size) and hydrological boundary conditions (upstream flow, downstream water level) into the HEC-RAS hydrodynamic model, simulate the generation of cross-sectional area, flow, water level elevation hydrological true value data set at multiple cross-sections at consecutive time steps by numerically solving the Saint-Venant equation set;

[0008] 1-2) Data processing: standardize the hydrological data generated by HEC-RAS, and repair missing values and remove outliers; missing values include cases where there is no valid record during the monitoring period, and outliers include data points that exceed the physically reasonable range; the standardization calculation is shown in formula (1):

[0009]

[0010] Where X represents the original data value output by HEC-RAS, μ represents the mean of the parameter data set, σ represents the standard deviation of the parameter data set, and Z represents the standardized data value;

[0011] 2) Construction of spatio-temporal input data set: construct a spatio-temporal coordinate data set corresponding to the HEC-RAS output data in step 1), the data set has spatial location sequence and time stamp as characteristic dimensions, and each data point corresponds to an (x, t) coordinate;

[0012] 3) Model core module and principle: the CNN-PINN model realizes high-precision prediction of hydrological parameters through the synergistic effect of the following three core modules, especially suitable for data missing environment:

[0013] 3-1) Spatial local feature extraction module: this module is used to replace the fully connected network (MLP) in traditional PINN, and its core function is to capture the local change pattern and heterogeneity characteristics of river hydrological parameters along the spatial dimension (x). Its working principle is to use one-dimensional convolution kernel to perform local sliding calculation on the spatial coordinate sequence. The mathematical expression of convolution operation is:

[0014]

[0015] Where F represents the input feature vector, K represents the learnable convolution kernel weight vector, w represents the half-width of the convolution kernel (i.e. the receptive field size), and i represents the current calculation spatial position index. A nonlinear activation function is applied after each convolution layer. By stacking multiple convolution layers, this module can abstract and extract the correlation, gradient change and high-frequency detail information of the hydrological parameters within the spatial neighborhood layer by layer. Compared with the global connection of MLP, the local receptive field characteristic of CNN can more effectively represent the spatial non-uniformity and local mutation characteristics of the river cross-section along the river, which is the key to improving the prediction accuracy of the model in the spatial dimension.

[0016] 3-2) Physical law constraint module: This module uses the Saint-Venant equations, which describe the one-dimensional unsteady flow dynamics of river channels, as physical prior knowledge to embed the training process of the neural network in the form of soft constraints, ensuring that the model's prediction results comply with basic fluid mechanics laws. The core is to calculate the residual of the physical control equation at the output value of the model prediction:

[0017] (1) Continuity equation

[0018] The continuity equation describes mass conservation, ensuring that the flow changes continuously in space and time. Its mathematical expression is as follows:

[0019]

[0020] where A is the flow cross-sectional area, Q is the flow (i.e., the volume of fluid per unit time), is a proportionality factor used to distinguish the flow distribution of the main channel and the floodplain, x c represents the distance in the direction of the main channel, x f represents the distance in the direction of the floodplain, and t is time.

[0021] (2) Momentum conservation equation

[0022] The momentum equation describes the conservation of momentum, expressing the balance between the acceleration of water flow and external forces. Its mathematical expression is as follows:

[0023]

[0024] In this equation, A c and A f represent the cross-sectional areas of the main channel and the floodplain, g is the acceleration of gravity, Z is the water level elevation, S c and S f are the friction slopes of the main channel and the floodplain, respectively, used to represent the energy loss due to friction in the flow.

[0025] The core role of this module is to force the model to not only fit the limited observation data during training, but also to meet the physical conservation laws that control the movement of water flow as much as possible. This greatly enhances the physical reasonableness and generalization ability of the model's prediction, especially in areas where real observation data is scarce or missing. Physical constraints can effectively guide the model to generate prediction results that conform to physical common sense, avoiding the physical distortion that may be produced by purely data-driven models.

[0026] 3-3) Multi-objective collaborative optimization module:

[0027] The module is used for coordinating and balancing multiple key objectives in the model training process, defining a total optimization objective function of the CNN-PINN model is a weighted sum of the following four loss terms:

[0028]

[0029]

[0030] is a physical loss, is a data loss, and are boundary condition and initial condition losses, respectively, and α, β, γ and η are weight coefficients; by adjusting the weight coefficients α, β, γ and η, the module can dynamically balance the importance between the data-driven fitting accuracy, the physical law consistency, the boundary condition satisfaction, and the initial condition accuracy. This multi-objective optimization strategy is the core mechanism for the model to generate high-precision and physically reasonable prediction results in a data missing environment.

[0031] 4) Model training and hydrological parameter prediction: using the CNN-PINN model defined in step 3) to process the spatio-temporal input dataset constructed in step 2), completing the hydrological parameter prediction task at any spatio-temporal location of the target river. Specifically:

[0032] 4-1) A multi-layer one-dimensional convolutional network is used to replace the fully connected structure of the traditional PINN, and a sliding convolution kernel is used along the spatial dimension x to extract local heterogeneity features of the river.

[0033] 4-2) Taking the spatio-temporal coordinates (x, t) constructed in step 2) as input, the trained model directly outputs the corresponding cross-sectional area, flow, and water level elevation prediction values. The model training process optimizes the network parameters by minimizing the total loss function (5), which coordinates data fitting, physical law constraints, boundary condition and initial condition satisfaction;

[0034] 4-3) The training process is based on the Adam optimizer for parameter adjustment, and the early stopping method is used to monitor the performance of the validation set to prevent overfitting, ensuring that the model learns an effective representation that integrates the HEC-RAS benchmark true value and the Saint-Venant equation physical constraint, ultimately obtaining a prediction model with high precision and physical reasonableness in a data missing environment.

[0035] 5) Effect verification: the prediction results obtained in step 4) are evaluated by root mean square error RMSE, mean absolute error MAE and mean absolute percentage error SMAPE, the data is normalized during model training, the parameters of the deep learning model are adjusted according to the Adam optimizer, the weight decay method, the packet loss method and the early stopping method are combined to suppress overfitting, and the effect is verified by comparison with other benchmark models, including:

[0036] 5-1) RMSE, MAE and SMAPE represent the deviation between the prediction results and the true results, the smaller the better, wherein,

[0037] The calculation of RMSE is shown in formula (10):

[0038]

[0039] The calculation of MAE is shown in formula (11):

[0040]

[0041] The calculation of SMAPE is shown in formula (12):

[0042]

[0043] Wherein, y′ i The predicted value of the i-th sample, y i is the true value of the i-th sample, and n is the number of samples

[0044] 5-2) Normalization: normalization is to scale the data to the range of 0 to 1, which can accelerate the training process of the model and improve the convergence speed;

[0045] 5-3) Weight decay method: add a regularization term to the objective function to prevent model overfitting;

[0046] 5-4) Packet loss method: randomly discard neural network units to prevent model overfitting;

[0047] 5-5) Early stopping method: monitor the performance of the model on the validation set and stop training when overfitting begins to occur;

[0048] 5-6) Other benchmark models: compared with traditional PINN (based on MLP) and data-driven models (LSTM, CNN) for verification;

[0049] Compared with the prior art, the technical scheme proposes a hybrid model framework combining convolutional neural network and physical information neural network, aiming to solve the problem of hydrological prediction in data-deficient or missing areas. CNN has good local feature extraction capability, can effectively process complex structural information in spatial dimension, and capture spatial variation patterns in hydrological processes, while PINN introduces hydrological physical laws to ensure that the model prediction results are physically reasonable and maintain high accuracy in data-scarce conditions. By combining the advantages of the two, the CNN-PINN model shows strong adaptability and robustness in hydrological parameter prediction tasks. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 A flowchart of the method in the embodiment is shown.

[0051] Figure 2 A watershed monitoring station distribution map of the research area in the embodiment is shown.

[0052] Figure 3 A PINN architecture diagram in the embodiment is shown.

[0053] Figure 4 A CNN-PINN prediction model architecture diagram in the embodiment is shown. DETAILED DESCRIPTION

[0054] The content of the present application will be further described below in conjunction with the drawings and embodiments, but is not limited to the present application.

[0055] Embodiment:

[0056] Reference Figure 1 A hydrological data prediction method based on convolutional neural network and physical information neural network, comprising the following steps:

[0057] 1) HEC-RAS benchmark data construction: using the hydrological model software HEC-RAS to simulate the hydrodynamic process of the target river, generating the hydrological data of the key section in the predetermined time period as the training benchmark true value, and preprocessing, specifically:

[0058] 1-1) Data generation: input the topographic parameters of the target river (including river geometry, section size) and hydrological boundary conditions (upstream flow, downstream water level) to the HEC-RAS hydrodynamic model, solve the Saint-Venant equation set by numerical method, simulate to generate a plurality of key section cross-sectional area, flow, water level elevation hydrological true value data set at continuous time step;

[0059] 1-2) Data processing: Standardize the data generated by HEC-RAS, and repair missing values and remove outliers; missing values include cases where there is no valid record within the monitoring period, and outliers include data points that are beyond the physically reasonable range; the standardization calculation is shown in equation (1):

[0060]

[0061] where X represents the original data value (A, Q or Z) output by HEC-RAS, μ represents the mean of the parameter data set, σ represents the standard deviation of the parameter data set, and Z represents the standardized data value. In this example, the Li River in Guilin, Guangxi, China is selected as the study area. The study area is the Li River in Guilin, China, which has typical subtropical climate characteristics. The river channel is complex in structure, and the water quality process is influenced by multiple factors such as agricultural activities, industrial emissions, tourism and natural environment. The study area selected in this paper includes the river section from section S1 (the uppermost upstream) to section S13 (the most downstream), with a total length of about 20 kilometers, a straight flow direction, a slope of 0.001, and a reservoir intake point at the most downstream. The datum elevation is 100 meters, the water level is up to 111.3913 meters, and the river width at the upstream S1 section is about 12 meters (as shown in Figure 2 The main focus of the study is the hydrological indicators such as flow, flow velocity and water level in the river channel. As shown in Table 1, the table lists the hydrological parameters and corresponding parameter variables in the basic data set. The hydrological data generated by HEC-RAS is used as the baseline training data set for model training and testing. The data set is divided into three parts: training set, validation set and test set. The training set is used to train the model, the validation set is used for hyperparameter search, and the best model obtained by hyperparameter search is applied to the test set to evaluate the performance of the model.

[0062] Table 1. Variable data

[0063]

[0064] 2) Construction of spatio-temporal input dataset: A spatio-temporal coordinate dataset corresponding to the HEC-RAS output data in step 1) is constructed, which has spatial position sequence (x) and time stamp (t) as characteristic dimensions, and each data point corresponds to an (x, t) coordinate; in specific implementation, first, the specific division and purpose of the key sections of the study river reach are determined. This study involves 13 key sections, based on these spatial divisions, a spatio-temporal input dataset is further constructed. This dataset corresponds to the standardized hydrological parameters output by the HEC-RAS model in step 1, and its core characteristic dimensions are the spatial position sequence (x) of the river and the time stamp (t), and each data point corresponds to a unique (x, t) coordinate. The spatial position x is defined as the cumulative distance of the key sections along the river flow direction (from upstream to downstream), with the most upstream section S1 as the spatial origin (x = 0 m), and the downstream sections are assigned incrementally according to their actual river mileage, for example, the most downstream section S13 corresponds to x = 20000 m, covering the entire 20 km long study river reach. The time dimension t uses continuous simulated hourly time stamps, covering the entire simulation period of HEC-RAS, and the time origin (t = 0) is set to the start time of the simulation. By strictly associating each spatio-temporal coordinate point (x, t) with the standardized hydrological true value parameters generated in step 1, a complete sample set is formed. The spatial division of this dataset strictly follows the aforementioned section purpose: the training set contains data from all sections S1 to S10, the validation set is composed of data from sections S12 and S13, and the test set uses data from section S11 that the model has not seen before. In the time dimension, the division process ensures the continuity of the time series, avoiding the time series discontinuity that may be caused by random splitting. The training set is used to optimize the model parameters, the validation set is used to monitor the overfitting phenomenon during training and adjust the model hyperparameters, and the test set is used to evaluate the generalization performance of the model.

[0065] 3) Model core module and principle: The CNN-PINN model (as shown in Figure 4 ) achieves high-precision prediction of hydrological parameters through the synergistic effect of the following three core modules, especially suitable for data missing environments:

[0066] 3-1) Spatial local feature extraction module: This module is used to replace the fully connected network (MLP) in traditional PINN, and its core function is to capture the local variation pattern and heterogeneity characteristics of river hydrological parameters along the spatial dimension (x). Its working principle is to use one-dimensional convolution kernel to perform local sliding calculation on the spatial coordinate sequence. The mathematical expression of convolution operation is:

[0067]

[0068] where F denotes the input feature vector, K denotes the learnable convolution kernel weight vector, w denotes the half-width of the convolution kernel (i.e., the receptive field size), and i denotes the index of the spatial position being computed. A nonlinear activation function is applied after each convolution layer. By stacking multiple convolution layers, this module can abstract and extract the correlation, gradient variation, and high-frequency detail information of hydrological parameters within the spatial neighborhood layer by layer. Compared to the global connection of MLP, the local receptive field characteristic of CNN enables it to more effectively represent the spatial non-uniformity and local mutation characteristics existing along the river cross-section, which is the key to improving the prediction accuracy of the model in the spatial dimension.

[0069] 3-2) Physical Law Constraint Module: This module is used to embed the Saint-Venant Equations, which describe the one-dimensional unsteady flow dynamics of the river, as physical prior knowledge into the training process of the neural network in the form of soft constraints, ensuring that the model prediction results comply with the basic laws of fluid mechanics. The core is to calculate the residual of the physical control equation at the output value of the model prediction:

[0070] (1) Continuity Equation

[0071] The continuity equation describes the conservation of mass, ensuring that the flow changes continuously in space and time. Its mathematical expression is as follows:

[0072]

[0073] where A is the flow cross-sectional area, Q is the flow (i.e., the volume of fluid per unit time), is a proportionality factor used to distinguish the flow distribution of the main river and the floodplain, x c represents the distance in the direction of the main river, f x represents the distance in the direction of the floodplain, and t is the time.

[0074] (2) Momentum Conservation Equation

[0075] The momentum equation describes the conservation of momentum, expressing the balance between the acceleration of the water flow and external forces. Its mathematical expression is as follows:

[0076]

[0077] In this equation, A c and A f represent the cross-sectional areas of the main river and the floodplain, g is the acceleration of gravity, z is the water depth, S c and S f are the friction slopes of the main river and the floodplain, respectively, used to represent the energy loss due to friction in the flow.

[0078] The core role of this module is that during the training process, the hydrodynamic state predicted by the forced model not only fits the limited observation data, but also must meet the physical conservation law controlling the flow motion as much as possible. This greatly enhances the physical rationality and generalization ability of the model prediction, especially in areas where real observation data is scarce or missing. Physical constraints can effectively guide the model to generate predictions that conform to physical common sense, avoiding the physical distortion that purely data-driven models may produce.

[0079] 3-3) Multi-objective collaborative optimization module: This module is used to coordinate and balance multiple key objectives in the model training process, and to define the total optimization objective function of the CNN-PINN model is the weighted sum of the following four loss terms:

[0080]

[0081]

[0082] is the physical loss, is the data loss, and are the boundary condition and initial condition losses, respectively, and α, β, γ, and η are weight coefficients; by adjusting the weight coefficients α, β, γ, and η, this module can dynamically balance the importance of data-driven fitting accuracy, physical law consistency, boundary condition satisfaction, and initial condition accuracy. This multi-objective optimization strategy is the core mechanism for the model to generate high-precision and physically reasonable prediction results in a data-lacking environment. In this example, for the hydrological prediction task of the 20-kilometer-long river section from S1 to S13 of the Lijiang River, the model achieves high-precision prediction through the cooperation of the three core modules. In the spatial local feature extraction module (3-1), a three-layer one-dimensional convolution layer is used instead of the traditional PINN's fully connected network (MLP), with the first layer configured with 64 3x1 convolution kernels, a step size of 1, and a spatial padding of 1; the second layer is expanded to 128 channels, and the third layer is increased to 256 channels, each followed by a ReLU activation function. By sliding the convolution kernel along the river space sequence x, the local gradient changes and heterogeneity features of the cross-section flow area A, flow Q, and water level Z are captured layer by layer. In the physical law constraint module (3-2), the Saint-Venant equation set is embedded in the training process in the form of soft constraints, and based on the prediction output, the continuity equation residual (Formula 3) and momentum equation residual (Formula 4) are calculated in real time. In this example, the main river flow distribution factor friction slope S c and S f are derived from the HEC-RAS terrain parameters, and the gravitational acceleration g = 9.81 m / s 2The partial derivatives are calculated by automatic differentiation technique, ensuring that the residual term enforces the constraint of water mass and momentum conservation. The multi-objective collaborative optimization module (3-3) designs a dynamic weight strategy: in the initial stage, the physical loss weight a = 0.6, the data loss b = 0.3, the boundary loss g = 0.05, and the initial condition loss h = 0.05. During training, the balance ratio of a and b is adjusted according to the performance of the validation set. Finally, the four types of losses are optimized simultaneously by the Adam optimizer, so that the model reaches an optimal balance between spatial heterogeneity description and physical law compliance.

[0083] 4) Model training and hydrological parameter prediction: use the CNN-PINN model defined in step 3) to process the spatio-temporal input dataset constructed in step 2), and complete the task of predicting hydrological parameters at any spatio-temporal location in the target river. Specifically:

[0084] 4-1) Use a multi-layer one-dimensional convolutional network to replace the traditional PINN's fully connected structure (as shown in Figure 3 ), and slide the convolution kernel along the spatial dimension x to extract local heterogeneity features of the river.

[0085] 4-2) Take the spatio-temporal coordinates (x, t) constructed in step 2) as input, and directly output the predicted values of cross-sectional area, flow, and water level elevation through the trained CNN-PINN model. The model training process optimizes the network parameters by minimizing the total loss function (5), which coordinates data fitting, physical law constraints, boundary condition and initial condition satisfaction;

[0086] 4-3) The training process is based on the Adam optimizer for parameter adjustment, and the early stopping method is used to monitor the performance of the validation set to prevent overfitting, ensuring that the model learns an effective representation that integrates the HEC-RAS benchmark true value and the Saint-Venant equation physical constraint. Finally, a prediction model with high precision and physical reasonableness is obtained in the environment of data missing.

[0087] 5) Effect verification: use the root mean square error RMSE, mean absolute error MAE, and mean absolute percentage error SMAPE to evaluate the prediction results obtained in step 4). During the model training process, the data is normalized, the deep learning model is adjusted according to the Adam optimizer, and the weight decay method, packet loss method, and early stopping method are combined to suppress overfitting. The results are compared with other benchmark models, including:

[0088] 5-1) RMSE, MAE, and SMAPE represent the deviation between the predicted results and the true results. The smaller the value, the more accurate the result. Among them,

[0089] The calculation of RMSE is shown in formula (10):

[0090]

[0091] The calculation of MAE is shown in equation (11):

[0092]

[0093] The calculation of SMAPE is shown in equation (12):

[0094]

[0095] where y' is the predicted value of the i-th sample, y is the true value of the i-th sample, and n is the number of samples i The predicted value of the i-th sample, y i is the true value of the i-th sample, and n is the number of samples

[0096] 5-2) Normalization: Normalization is to scale the data to the range of 0 to 1, to accelerate the training process of the model and improve the convergence speed;

[0097] 5-3) Weight decay method: Add a regularization term in the objective function to prevent model overfitting;

[0098] 5-4) Packet loss method: Randomly discard the way of neural network unit to prevent model overfitting;

[0099] 5-5) Early stopping method: Monitor the performance of the model on the validation set, and stop training when overfitting begins to appear;

[0100] 5-6) Other benchmark models: Compared with traditional PINN (based on MLP) and pure data-driven models (LSTM, CNN) for verification; Specifically:

[0101] The proposed CNN-PINN model shows significant advantages in the hydrological parameter prediction of a 20-kilometer river section in the Lijiang River Basin. As shown in Table 2, in the prediction tasks of cross-sectional area, flow, and water level elevation, the RMSE, MAE, and SMAPE of CNN-PINN are significantly lower than those of traditional PINN and purely data-driven models (LSTM, CNN). Taking flow prediction as an example, the RMSE (0.289) and MAE (0.196) of CNN-PINN are reduced by 47.6% and 42.9% compared with traditional PINN (based on MLP), and by 97.0% and 97.4% compared with LSTM model. This result verifies the core advantages of CNN-PINN in terms of high precision and physical rationality in the environment of data deficiency. LSTM and CNN models rely entirely on data statistical rules, and perform poorly in training data scarcity. For example, the prediction RMSE of LSTM for flow is as high as 9.623, and its global connection structure is difficult to capture the spatial heterogeneity of the river channel, and the lack of physical constraints leads to non-physical interpretation (such as negative flow) in the section without monitoring data. Although CNN improves the spatial feature extraction ability through convolution operation, it still ignores the laws of fluid mechanics, and the error accumulates significantly in the prediction of dynamic processes such as flood wave propagation. Traditional PINN improves the physical rationality by embedding the Saint-Venant equation, but the global receptive field of its MLP structure leads to insufficient capture of local spatial features. Especially when facing the sudden change of river section size in the Lijiang River, MLP cannot effectively represent the local hydraulic gradient change, resulting in a higher SMAPE (0.201) of cross-sectional area prediction than CNN-PINN (0.088).

[0102] Table 2. Prediction results of other models

[0103]

[0104] The validation of this method in the Lijiang River Basin shows that the CNN-PINN model can well fit the simulation results of HEC-RAS, while solving the limitation of traditional hydrological models that are highly dependent on terrain data. For remote areas lacking detailed terrain parameters, when precise simulation using HEC-RAS is not possible, this model can still achieve high-precision hydrological parameter prediction by combining physical constraints with local feature extraction. This feature makes it particularly suitable for hydrological simulation and flood warning in areas with scarce terrain data, providing a new technical means for water resources management and disaster prevention. In the future, it can be further applied to complex scenarios such as ecological flow regulation and inter-basin water transfer, promoting the transformation of hydrological prediction from "data-dependent" to "mechanism-driven".

Claims

1. A hydrological data prediction method based on convolutional neural networks and physical information neural networks, characterized in that, Includes the following steps: 1) HEC-RAS Baseline Data Construction: The hydrodynamic processes of the target river channel are simulated using the hydrological modeling software HEC-RAS to generate hydrological data of key sections within a predetermined time period as training baseline values. Preprocessing is then performed, specifically as follows: 1-1) Data generation: Input the target river channel topographic parameters (including river channel geometry and cross-sectional dimensions) and hydrological boundary conditions (upstream flow and downstream water level) into the HEC-RAS hydrodynamic model. Through numerical solution of the Saint-Venant equations, simulate and generate a true hydrological dataset of cross-sectional area, flow, and water level elevation of multiple sections at continuous time steps. 1-2) Data Processing: The hydrological data generated by HEC-RAS is standardized, and missing values ​​are repaired and outliers are removed. Missing values ​​include cases where there are no valid records within the monitoring period, and outliers include data points that exceed the physical reasonable range. The standardization calculation is shown in formula (1): Where X represents the original data value output by HEC-RAS, μ represents the mean of the parameter dataset, σ represents the standard deviation of the parameter dataset, and Z represents the standardized data value. 2) Constructing a spatiotemporal input dataset: Construct a spatiotemporal coordinate dataset corresponding to the HEC-RAS output data in step 1). The dataset uses the spatial location sequence of the river channel and the timestamp as feature dimensions, and each data point corresponds to a (x,t) coordinate. 3) Core Modules and Principles of the Model: The CNN-PINN model achieves high-precision prediction of hydrological parameters through the synergistic effect of the following three core modules, making it particularly suitable for environments with missing data: 3-1) Spatial Local Feature Extraction Module: This module replaces the fully connected network (MLP) in traditional PINN. Its core function is to capture the local variation patterns and heterogeneity features of river hydrological parameters along the spatial dimension (x). Its working principle is to perform local sliding calculations on the spatial coordinate sequence using a one-dimensional convolution kernel. The mathematical expression for the convolution operation is: Where F represents the input feature vector, K represents the learnable convolutional kernel weight vector, w represents the half-width of the convolutional kernel (i.e., the receptive field size), and i represents the spatial location index of the current computation. A non-linear activation function is applied after each convolutional layer. By stacking multiple convolutional layers, this module can abstract and extract the correlation, gradient changes, and high-frequency detail information of hydrological parameters within the spatial neighborhood layer by layer. Compared to the global connectivity of MLPs, the local receptive field characteristics of CNNs enable them to more effectively characterize the spatial non-uniformity and local abrupt changes along the river cross-section, which is key to improving the model's prediction accuracy in the spatial dimension. 3-2) Physical Law Constraint Module: This module uses the Saint-Venant Equations, which describe the one-dimensional unsteady flow dynamics of a river, as prior physical knowledge. It embeds these equations into the neural network training process as soft constraints, ensuring that the model's predictions conform to fundamental fluid dynamics laws. Its core function is to calculate the residuals of the physical governing equations at the model's predicted output values. (1) Continuity equation The continuity equation describes the conservation of mass, ensuring that the variation of flow rate in space and time is continuous. Its mathematical expression is as follows: Where A is the cross-sectional area of ​​the flow, and Q is the flow rate (i.e., the volume of fluid per unit time). It is a scaling factor used to distinguish the flow distribution between the main channel and the floodplain, x c x represents the distance along the main channel. f t represents the distance in the direction of the floodplain, and t represents time. (2) Momentum conservation equation The momentum equation describes the conservation of momentum and expresses the balance between the acceleration of water flow and external forces. Its mathematical expression is: In this equation, A c and A f Let S represent the cross-sectional areas of the main channel and the floodplain, respectively; g be the acceleration due to gravity; Z be the water level elevation; and S be the cross-sectional area of ​​the floodplain. c and S f These are the friction slopes of the main channel and the floodplain, respectively, used to represent the energy loss caused by friction during flow. The core function of this module is to force the model, during training, to not only fit the limited observation data in its predicted hydrodynamic states, but also to satisfy the physical conservation laws governing water flow as much as possible. This greatly enhances the physical rationality and generalization ability of the model's predictions. Especially in areas where real observation data is scarce or missing, physical constraints can effectively guide the model to generate predictions that conform to physical common sense, avoiding the physical distortion that may occur with a purely data-driven model. 3-3) Multi-objective collaborative optimization module: This module is used to coordinate and balance multiple key objectives during model training, defining the overall optimization objective function of the model. The weighted sum of the following four loss terms: For physical loss, For data loss, and The boundary conditions and initial conditions are represented by their respective losses, with α, β, γ, and η serving as weighting coefficients. By adjusting these weighting coefficients, this module dynamically balances the importance of four objectives: data-driven fitting accuracy, consistency with physical laws, satisfaction of boundary conditions, and accuracy of initial conditions. This multi-objective optimization strategy is the core mechanism by which the model can generate high-precision and physically reasonable predictions even in environments with missing data. 4) Model Training and Hydrological Parameter Prediction: The CNN-PINN model defined in step 3) is used to process the spatiotemporal input dataset constructed in step 2) to complete the task of predicting hydrological parameters at any spatiotemporal location of the target river channel. Specifically: 4-1) A multi-layer one-dimensional convolutional network is used to replace the fully connected structure of the traditional PINN, and the convolutional kernel is slidable along the spatial dimension x to extract the local heterogeneity features of the river channel. 4-2) Using the spatiotemporal coordinates (x,t) constructed in step 2) as input, the corresponding cross-sectional area, flow rate, and water level elevation prediction values ​​are directly output through the trained CNN-PINN model. The model training process optimizes the network parameters by minimizing the total loss function (5), which coordinates the data fitting, physical law constraints, boundary conditions, and initial condition satisfaction. 4-3) The training process is based on the Adam optimizer to adjust parameters and uses early stopping to monitor the performance of the validation set to prevent overfitting. This ensures that the model learns an effective representation that integrates the HEC-RAS benchmark true value and the physical constraints of the Saint-Venant equation, and finally obtains a prediction model with high accuracy and physical rationality in the absence of data. 5) Performance Validation: The prediction results obtained in step 4) were evaluated using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (SMAPE). During model training, data normalization was performed, and the parameters of the deep learning model were adjusted using the Adam optimizer. Overfitting was suppressed using a combination of weight decay, packet loss, and early stopping methods. Performance was validated by comparing the model with other benchmark models, including: 5-1) RMSE, MAE, and SMAPE represent the deviation between the predicted and actual results. The smaller the deviation, the more accurate the result. The RMSE is calculated as shown in formula (10): The calculation of MAE is shown in formula (11): The calculation of SMAPE is shown in formula (12): Where, y′ i The predicted value of the i-th sample, y i It is the true value of the i-th sample, and n is the number of samples. 5-2) Normalization: Normalization scales the data to a range between 0 and 1, accelerating the training process of the model and improving the convergence speed. 5-3) Weight decay method: Add a regularization term to the objective function to prevent the model from overfitting; 5-4) Packet dropping method: Randomly discarding neural network units to prevent model overfitting; 5-5) Early stopping method: Monitor the model's performance on the validation set and stop training when overfitting begins to occur; 5-6) Other benchmark models: Compare and validate with traditional PINN (based on MLP) and pure data-driven models (LSTM, CNN).

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