River network hydrodynamic force multi-scene intelligent prediction method
By constructing a multi-input operator network with an encoder-decoder architecture and combining it with the MIONet and PINN models, the problems of low computational efficiency and insufficient generalization ability in real-time prediction of hydrodynamics in river networks under multiple scenarios are solved, and efficient and accurate hydrodynamic prediction is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HOHAI UNIV
- Filing Date
- 2026-01-15
- Publication Date
- 2026-04-28
AI Technical Summary
Existing technologies suffer from low computational efficiency, insufficient generalization ability across multiple scenarios, and poor physical consistency in real-time prediction of hydrodynamics in river networks, making it difficult to cope with changes in heterogeneous boundary conditions in complex river network systems.
A multi-input operator network with an encoder-decoder architecture, combined with MIONet and PINN models, generates high-fidelity datasets for multiple scenarios through an automated workflow. The model is trained using a composite loss function to achieve the mapping from boundary parameters to hydrodynamic solution fields, and integrates heterogeneous boundary conditions while ensuring physical constraints.
It achieves second-level multi-scenario simulation, has multi-scenario generalization capability, maintains physical consistency, improves prediction accuracy and training efficiency, and adapts to the real-time changes of complex river network systems.
Smart Images
Figure CN121936724A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a prediction method, and more particularly to an intelligent prediction method for multi-scenario hydrodynamics in river networks. Background Technology
[0002] The hydrodynamic processes in plain river networks are highly nonlinear and dynamic in both time and space. The rivers are gently sloping and susceptible to the combined effects of upstream floods, localized torrential rains, and downstream tides, exhibiting a complex, bidirectional, and non-constant flow pattern. Furthermore, frequent regulation of water conservancy projects such as sluice gates and pumping stations in this region further exacerbates the dynamic variability of boundary conditions. Therefore, developing high-precision river network hydrodynamic simulation technology capable of rapidly responding to changes in multiple operating conditions is of significant practical importance for flood control early warning, optimized water resource allocation, and smart water management.
[0003] Currently, river network hydrodynamic simulation mainly relies on three technical approaches: 1. Numerical models based on physical processes. These models typically use the finite difference method to discretize and solve the one-dimensional Saint-Venant equations describing river network flow motion, providing high-fidelity simulation results. 2. Data-driven deep learning models. These methods aim to learn the complex mapping relationship between input and output from a large amount of historical observation data, thereby avoiding the computational cost of solving physical equations. 3. Physical Information Neural Networks (PINN). PINN was proposed to incorporate physical constraints into data-driven models. It embeds the residual terms of partial differential equations (PDEs) into the loss function through automatic differentiation, enabling the model to be trained even under sparse data conditions while maintaining physical interpretability. 4. Operator learning. Neural operators mainly include DeepONet and Fourier Neural Operators (FNO), whose goal is to learn the mapping relationship from any input function to the corresponding solution function.
[0004] Despite their individual characteristics, the aforementioned technologies all exhibit significant limitations when addressing the specific need for real-time prediction of multi-scenario hydrodynamics in river networks: 1. Traditional numerical models suffer from low computational efficiency. These models require re-running for each scheduling condition in engineering applications, leading to high computational costs. In real-time applications such as optimized scheduling in water conservancy projects, which require rapid analysis of numerous conditions, the computation time of several hours or even days makes it difficult to meet the timeliness requirements of decision-making. 2. Data-driven models suffer from poor physical consistency and rely heavily on large amounts of data. The prediction results of purely data-driven models lack physical interpretability, and their generalization ability is questionable in conditions not covered by training data. However, in real river network systems, hydrological stations are sparsely distributed, making it difficult to obtain data sufficient for training deep learning models, greatly limiting their application in practical engineering. 3. Standard PINN models lack multi-scenario generalization ability. Standard PINN models are designed for fixed boundary and initial conditions. Once the boundary conditions change, the model must be retrained. This deficiency makes them unsuitable for multi-scenario prediction tasks requiring real-time response to changing boundaries. 4. Existing operator networks lack the ability to handle heterogeneous boundary conditions. The single-branch structure commonly used in existing DeepONet systems makes it difficult to effectively decouple and integrate boundary conditions with different physical meanings and dynamic characteristics in river network systems. This structural limitation affects the model's learning efficiency and prediction accuracy for complex physical mapping relationships. Summary of the Invention
[0005] Purpose of the invention: The purpose of this invention is to provide a multi-scenario intelligent prediction method for river network hydrodynamics that has high computational efficiency, strong multi-scenario generalization ability, good physical consistency, and intelligent heterogeneous boundary processing.
[0006] Technical solution: The intelligent prediction method for multi-scenario hydrodynamics of river networks described in this invention includes the following steps:
[0007] (1) The key driving factors of river network hydrodynamics are parameterized into a set of boundary parameters, and a mapping operator learning task is defined from the boundary parameters to the hydrodynamic solution field; the boundary conditions corresponding to the boundary parameters include the non-constant flow rate controlled by the upstream gate and the downstream tide;
[0008] (2) By calling numerical simulation software through an automated workflow, different combinations of the boundary parameters are run in batches to generate a high-fidelity dataset covering multiple scenarios in the parameter space;
[0009] (3) Construct a network model of encoder-decoder architecture, wherein the encoder is a multi-input operator network for decoupling and fusing heterogeneous boundary conditions, including a backbone network for processing spatiotemporal coordinates and multiple branch networks for independently encoding a set of boundary parameters; the decoder adopts a physical information neural network to receive the potential feature vector after the fusion of the backbone network and branch networks of the encoder, and decode and output the corresponding hydrodynamic state.
[0010] (4) The model is trained by a composite loss function including a supervised loss term and a physical loss term to obtain a generalized solution operator model; the physical loss term is used to constrain the prediction results of the model at the physical location point to conform to the hydrodynamic control equation;
[0011] (5) Use the trained solution operator model to predict the new combination of boundary parameters in real time and output the hydrodynamic solution field in the whole time and space domain.
[0012] Furthermore, the non-constant flow rate in step (1) is parameterized as the key point flow rate value of a piecewise linear function, and the downstream tide is parameterized as the average tidal level, amplitude, and period of a sine function.
[0013] Furthermore, the automated workflow described in step (2) adopts a Python-RAS automated workflow, and the generation of the multi-scenario high-fidelity dataset includes the following steps:
[0014] (21) Different combinations of the boundary parameters are generated in the multidimensional parameter space using Latin hypercube sampling;
[0015] (22) Perform steady-state flow initialization for each combination, and then automatically call the numerical simulation software to run the unsteady flow simulation;
[0016] (23) Automatically extract water level and flow data from the numerical simulation output file and format it as a structured dataset.
[0017] Furthermore, the encoder in step (3) is MIONet, and the decoder is PINN.
[0018] Furthermore, the formula for calculating the latent feature vector in step (3) is as follows:
[0019]
[0020]
[0021] in, The spatiotemporal coordinates of the main network include the spatial location along the river segment, the river channel number, and the time. For upstream flow parameters; For downstream tidal level parameters; and These are the high-dimensional feature vectors mapping the upstream and downstream branch networks, respectively; and These are the trainable parameters for the two branch networks, respectively. Output feature vectors for feature fusion for all branch networks; The high-dimensional feature vectors mapped to the backbone network; These are the trainable parameters for the backbone network; Z represents the Hadamard product; Z is the latent feature vector of the encoder's final output.
[0022] Furthermore, the decoder in step (3) employs a multilayer feedforward neural network (FNN), and the calculation formula for decoding the latent feature vector and outputting the corresponding spatiotemporal coordinates of the hydrodynamic state variables is as follows:
[0023]
[0024]
[0025] in, Water level; For traffic; This is the set of all trainable parameters for the decoder; For the first The output of each hidden layer; This represents the total number of hidden layers. and The first Layer weight matrix and bias vector; For activation functions; For linear transformation of the output layer; The final predicted water level and flow rate by the model.
[0026] Furthermore, the monitoring loss term in step (4) includes the errors of initial conditions, boundary conditions, and sparse internal hydrological station observation points; the physical loss term includes the residual loss of the one-dimensional Saint-Venant equation calculated at the configuration point in the river channel, and the residual loss of the nodal equation calculated at the physical configuration point at the river network node; the one-dimensional Saint-Venant equation includes the continuity equation and the momentum equation; the nodal equation includes the flow conservation equation and the water level continuity equation.
[0027] Furthermore, the formula for calculating the composite loss function in step (4) is as follows:
[0028]
[0029] in, It is a composite loss function; and Let represent the mean square errors of the initial conditions, boundary conditions, internal observations, and physical loss term, respectively. and It is a weighting coefficient that balances the relative importance of each loss term.
[0030] Furthermore, in step (4), when training the model, the input of the supervised data points is normalized by min-max normalization; the original spatiotemporal coordinates are retained when calculating the physical loss term to ensure the correctness of the physical dimensions; a mini-batch random sampling strategy is adopted during training, and Tanh is used as the activation function of the hidden layer.
[0031] Furthermore, when training the model in step (4), the hyperparameters of the model are determined by Bayesian optimization and the validation set loss is used as the objective function. The AdamW optimizer is used in conjunction with the cosine annealing algorithm to dynamically schedule the learning rate. A dual adaptive weight update mechanism is also used, in which the first weight is used to balance the gradients of the physical loss term and each supervised loss term, and the second weight is used to balance the gradients of different physical quantities in the supervised loss term.
[0032] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: (1) Real-time performance and high efficiency are achieved, reducing the simulation time of multiple scenarios from several hours to seconds, and the computation speedup ratio is thousands of times; (2) It has the ability to generalize to multiple scenarios. After the model is trained once, it can respond to any new boundary conditions in the parameter space without retraining; (3) Physical consistency is guaranteed. By incorporating constraints such as the Saint-Venant equation and the node equation, the prediction results still conform to physical laws under sparse data; (4) It has the advantage of heterogeneous boundary processing. The MIONet encoder is used to decouple multi-source dynamic boundaries, thereby improving prediction accuracy and training efficiency. Attached Figure Description
[0033] Figure 1 A conceptual diagram illustrating the operator learning task defined in the method of this invention;
[0034] Figure 2 This is a schematic diagram of the Python-RAS automated generation process of the present invention;
[0035] Figure 3 This is a schematic diagram of the overall architecture of the present invention;
[0036] Figure 4 This is an example diagram of the convergence curve of the composite loss function of the present invention;
[0037] Figure 5 This is a statistical histogram of the predictive performance of the present invention under 1000 operating conditions;
[0038] Figure 6 This is a comparison chart of the training loss and validation loss of the present invention and a comparative example;
[0039] Figure 7 This is a comparison chart of the prediction accuracy of the present invention and the comparative example. Detailed Implementation
[0040] The technical solution of the present invention will be further described below with reference to the accompanying drawings.
[0041] This embodiment is used to achieve the following: Figure 1 Real-time prediction of hydrodynamics in the illustrated ring-shaped river network under multiple scenarios driven by upstream gate-regulated non-constant discharge and downstream tides includes the following steps:
[0042] Step 1: Definition of Multi-Scene Boundary Parameterization and Operator Learning Tasks
[0043] First, define the geometric and physical parameters of the object of study, such as Figure 1 The ring-shaped river network shown in (a) includes four river channels (A-4 to A-1), two connecting nodes, and six hydrological stations. Secondly, as... Figure 1 As shown in (b) and (c), the boundary conditions driving the hydrodynamic field are parameterized. Specifically, the boundary parameterization includes parameterizing the non-constant flow rate of the upstream boundary controlled by the gate into key point flow rate values describing a piecewise linear function. The tidal level at the downstream boundary is parameterized as the mean tidal level describing a sine function. ,amplitude and cycle As shown in the following formula:
[0044]
[0045] in, For upstream flow processes, This refers to the downstream tidal process.
[0046] The range of boundary condition parameters is defined based on real tidal scenarios, covering microtides, mesentery, and strong tides, as well as semi-diurnal and diurnal tides. The specific values are shown in Table 1 below:
[0047]
[0048] As attached Figure 1 As shown in (d), after parameterizing the upstream and downstream boundaries, this invention defines the problem as a task of learning the mapping operator from the boundary condition parameters to the hydrodynamic solution field.
[0049] Step 2: Python-RAS Automated Dataset Generation
[0050] As attached Figure 2 As shown, the Python-RAS automated workflow generates a high-fidelity dataset for subsequent model training, validation, and testing, and includes the following steps:
[0051] (1) Latin hypercube sampling (LHS) was used to generate 10,000 independent parameter combinations in a 6-dimensional parameter space. Parameter combinations that do not conform to physical common sense (such as tidal amplitude exceeding the average level) were excluded. () were excluded. The valid combinations were divided into training, validation, and test sets in an 8:1:1 ratio.
[0052] (2) For each operating condition, the Python script calls HEC-RAS to perform steady flow calculations using the initial values of the subsequent non-steady boundary sequence. Therefore, each set of boundary parameters defines a unique steady-state initial state, corresponding to a single flow field solution. The resulting steady-state flow field provides the initial distribution for non-steady flow simulations, mitigating numerical oscillations and shortening the time to reach stability.
[0053] (3) Using the steady-state initial field generated in (2), the script writes the non-steady boundary parameters of each working condition into HEC-RAS and automatically calls it to perform a 24-hour non-steady flow simulation.
[0054] (4) After each run, the script reads the HEC-RAS binary output HDF5 file, extracts the water level and flow rate at 64 sections for 97 time points, and provides data for each operating condition. = 6208 spatiotemporal samples. After processing all operating conditions, two unified matrices are obtained: water level and traffic Matrix, each matrix is of size .
[0055] Step 3: Model Building
[0056] As attached Figure 3 As shown, this invention constructs an encoder-decoder PI-RONet model. The encoder employs the MIONet architecture.
[0057] (1) Backbone Network (FNN): Receives spatiotemporal query coordinates Specifically, this includes the spatial location along the river section. River channel number and time And map it into a high-dimensional feature vector. .
[0058] (2) Branch Network 1 (FNN): Receives upstream traffic parameters And map it into a high-dimensional feature vector. .
[0059] (3) Branch Network 2 (FNN): Receives downstream tide level parameters And map it into a high-dimensional feature vector. .
[0060] (4) Branch and trunk fusion. The output features of branch networks 1 and 2 are summed element by element, and then the summation is performed with the output features of the trunk network to obtain the latent feature vector. The process is shown in the following formula:
[0061]
[0062]
[0063] in, and These are the trainable parameters for the two branch networks, respectively. Output feature vectors for feature fusion for all branch networks; These are the trainable parameters for the backbone network; This represents the Hadamard product.
[0064] The decoder uses the PINN architecture:
[0065] (1) The decoder network uses an FNN, and its input layer receives the fused latent feature vector. Its output layer outputs the hydrodynamic state variables corresponding to the spatiotemporal coordinates, namely the water level. and traffic The calculation process is as follows:
[0066]
[0067]
[0068] in, This is the set of all trainable parameters for the decoder; For the first Nonlinear transformation of each hidden layer; This represents the total number of hidden layers. and The first Layer weight matrix and bias vector; For activation functions; For linear transformation of the output layer; The final predicted water level and flow rate by the model.
[0069] (2) Composite loss function Includes monitoring loss items ( ) and physical loss item ( ), as shown in the following formula:
[0070]
[0071] in, and Let represent the loss functions (mean squared error, MSE) for the initial condition IC, boundary condition BC, internal observation OBS, and physical condition F, respectively. and It is a weighting coefficient that balances the relative importance of each loss term.
[0072] Physical loss item Specifically, this includes the residual loss of the one-dimensional Saint-Venant equation calculated at physical placement points within the river channel, and the residual loss of the nodal equations (including the flow conservation equation and the water level continuity equation) calculated at physical placement points at river network nodes. The Saint-Venant equation is shown below:
[0073]
[0074] in, Cross-sectional area of the flow (m²) 2 ), The roughness coefficient is Manning's roughness coefficient. The hydraulic radius is (m).
[0075] The nodal equations are shown below:
[0076]
[0077] in, and It is the total inflow and outflow of all river sections connected at the confluence; and This indicates the water levels at both the upstream and downstream ends of the river section connected to the confluence.
[0078] Step 4: Model Training
[0079] The composite loss function curve is attached. Figure 4 As shown, the model constructed in step 3 is trained using a composite loss function. The training configuration and strategy are as follows:
[0080] (1) Data preprocessing: The input data is processed using min-max normalization, but in the calculation When doing so, the original spacetime coordinates must be used.
[0081] (2) Training configuration: small batch random sampling (200 working conditions per batch); the hidden layer uses the Tanh activation function; the structure of the encoder's backbone network and two branch networks is 3 layers, each layer is a fully connected layer with 32 neurons, and the decoder's PINN network is 8 layers, each layer is a fully connected layer with 128 neurons.
[0082] (3) Optimization strategy: The hyperparameters of the model are determined by Bayesian optimization with the validation set loss as the objective function; the AdamW optimizer is used to update the network parameters; the initial learning rate is 10. -3 And a cosine annealing algorithm is used for dynamic scheduling.
[0083] (4) Weighting strategy: A dual adaptive weighting update mechanism is adopted to balance the supervision and physical loss terms, as well as the water level and flow loss terms. For example... Figure 4 As shown, during the training process, all loss terms steadily decreased and converged.
[0084] Step 5: Model Testing and Evaluation
[0085] The model trained in step 4 was then used in 1000 test scenarios that had not been seen during training. (See attached image.) Figure 5 As shown, statistical results indicate that the method of the present invention has high prediction accuracy, with 90.9% of the water level predictions under the test conditions being accurate. Greater than 0.8 Figure 5 (a) 99.5% of test scenario traffic predictions Greater than 0.8 Figure 5 (c) The average RMSE for water level prediction was 0.062 m, and the predicted flow rate was 1.511 m. 3 / s, all remained at a low level.
[0086] In terms of computational efficiency, it takes nearly 3 hours to simulate these 1,000 test scenarios using HEC-RAS, while the model pre-trained using this invention can complete all predictions in just about 2 seconds, achieving a computational speedup of nearly 5,000 times.
[0087] Comparative Example
[0088] To verify the advantages of the MIONet (multi-branch) encoder used in Embodiment 1 of this invention compared to the traditional single-branch encoder (DeepONet), this comparative model uses the same dataset and training conditions as the embodiment, but replaces the encoder with a DeepONet architecture (i.e., uses a single branch network to receive all combinations of the six boundary parameters). The backbone network and decoder PINN structure, training strategy, and hyperparameter settings of both models are consistent with the embodiment.
[0089] It is important to emphasize that this design ensures that the branch networks of the two models have similar widths (see Table 2 below for specific network configurations), thus enabling a fair examination of the differences inherent in the multi-branch architecture itself.
[0090]
[0091] The structure of the branch network is represented as the number of branches × [number of layers × number of neurons], while the backbone network and decoder network are represented as [number of layers × number of neurons].
[0092] The training process is shown in the attached figure. Figure 6 As shown, the training and validation losses of the MIONet encoder used in this invention are both lower than those of the comparison example using DeepONet, indicating that the MIONet encoder converges faster and more stably. Under the same hardware conditions, the training time of MIONet is reduced by 10.7%.
[0093] For example, a comparison of prediction accuracy is attached. Figure 7 As shown in Table 3 and Appendix 4, the MIONet encoder outperforms the comparison example DeepONet in prediction accuracy across 1000 test scenarios. Figure 7 As shown, the average RMSE for water level prediction in this invention is 0.062m, which is 19.5% lower than that of the comparative example (0.077m), and the average RMSE for flow prediction is 10.0% lower.
[0094]
[0095] As can be seen from the comparative examples, the MIONet encoder architecture adopted in this invention can learn complex physical mappings more effectively by decoupling multi-source heterogeneous boundary inputs. This is a key innovation for achieving high precision and high efficiency, and its technical performance is superior to that of the traditional DeepONet coding operator network.
Claims
1. A multi-scenario intelligent prediction method for river network hydrodynamics, characterized in that, Includes the following steps: (1) The key driving factors of river network hydrodynamics are parameterized into a set of boundary parameters, and a mapping operator learning task is defined from the boundary parameters to the hydrodynamic solution field; the boundary conditions corresponding to the boundary parameters include the non-constant flow rate controlled by the upstream gate and the downstream tide; (2) By calling numerical simulation software through an automated workflow, different combinations of the boundary parameters are run in batches to generate a high-fidelity dataset covering multiple scenarios in the parameter space; (3) Construct a network model of encoder-decoder architecture, wherein the encoder is a multi-input operator network for decoupling and fusing heterogeneous boundary conditions, including a backbone network for processing spatiotemporal coordinates and multiple branch networks for independently encoding a set of boundary parameters; the decoder adopts a physical information neural network to receive the potential feature vector after the fusion of the backbone network and branch networks of the encoder, and decode and output the corresponding hydrodynamic state. (4) The model is trained by a composite loss function including a supervised loss term and a physical loss term to obtain a generalized solution operator model; the physical loss term is used to constrain the prediction results of the model at the physical location point to conform to the hydrodynamic control equation; (5) Use the trained solution operator model to predict the new combination of boundary parameters in real time and output the hydrodynamic solution field in the whole time and space domain.
2. The intelligent prediction method according to claim 1, characterized in that, The non-constant flow rate parameterized in step (1) is the key point flow rate value of a piecewise linear function, and the downstream tide parameterized is the average tidal level, amplitude and period of a sine function.
3. The intelligent prediction method according to claim 1, characterized in that, The automated workflow described in step (2) adopts the Python-RAS automated workflow, and the generation of the multi-scenario high-fidelity dataset includes the following steps: (21) Different combinations of the boundary parameters are generated in the multidimensional parameter space using Latin hypercube sampling; (22) Perform steady-state flow initialization for each combination, and then automatically call the numerical simulation software to run the unsteady flow simulation; (23) Automatically extract water level and flow data from the numerical simulation output file and format it as a structured dataset.
4. The intelligent prediction method according to claim 1, characterized in that, The encoder in step (3) is MIONet, and the decoder is PINN.
5. The intelligent prediction method according to claim 1, characterized in that, The formula for calculating the latent feature vector in step (3) is as follows: in, The spatiotemporal coordinates of the main network include the spatial location along the river segment, the river channel number, and the time. For upstream flow parameters; For downstream tidal level parameters; and These are the high-dimensional feature vectors mapping the upstream and downstream branch networks, respectively; and These are the trainable parameters for the two branch networks, respectively. Output feature vectors for feature fusion for all branch networks; The high-dimensional feature vectors mapped to the backbone network; These are the trainable parameters for the backbone network; Z represents the Hadamard product; Z is the latent feature vector of the encoder's final output.
6. The intelligent prediction method according to claim 1, characterized in that, The decoder described in step (3) uses a multilayer feedforward neural network (FNN). The calculation formula for decoding the latent feature vector and outputting the hydrodynamic state variables corresponding to the spatiotemporal coordinates is as follows: in, Water level; For traffic; This is the set of all trainable parameters for the decoder; For the first The output of each hidden layer; This represents the total number of hidden layers. and The first Layer weight matrix and bias vector; For activation functions; For linear transformation of the output layer; The final predicted water level and flow rate by the model.
7. The intelligent prediction method according to claim 1, characterized in that, The monitoring loss term in step (4) includes the errors of initial conditions, boundary conditions, and sparse internal hydrological station observation points; the physical loss term includes the residual loss of the one-dimensional Saint-Venant equation calculated at the placement points in the river channel, and the residual loss of the nodal equation calculated at the physical placement points at the river network nodes; the one-dimensional Saint-Venant equation includes the continuity equation and the momentum equation; the nodal equation includes the flow conservation equation and the water level continuity equation.
8. The intelligent prediction method according to claim 7, characterized in that, The formula for calculating the composite loss function mentioned in step (4) is as follows: in, It is a composite loss function; and Let represent the mean square errors of the initial conditions, boundary conditions, internal observations, and physical loss term, respectively. and It is a weighting coefficient that balances the relative importance of each loss term.
9. The intelligent prediction method according to claim 1, characterized in that, In step (4), when training the model, the input of the supervised data points is normalized by min-max normalization; the original spatiotemporal coordinates are retained when calculating the physical loss term to ensure the correctness of the physical dimensions; a mini-batch random sampling strategy is adopted during training, and Tanh is used as the activation function of the hidden layer.
10. The intelligent prediction method according to claim 9, characterized in that, In step (4), when training the model, the hyperparameters of the model are determined by Bayesian optimization and the validation set loss is used as the objective function. The AdamW optimizer is used in conjunction with the cosine annealing algorithm to dynamically schedule the learning rate. A dual adaptive weight update mechanism is also used, in which the first weight is used to balance the gradients of the physical loss term and each supervised loss term, and the second weight is used to balance the gradients of different physical quantities in the supervised loss term.