Tropospheric wet delay prediction method and system based on physical information neural network

Abstract

This invention discloses a method and system for predicting tropospheric wet delay based on a physical information neural network. The method first collects real-time meteorological observation data from GNSS stations and preprocesses it. Then, it constructs a tropospheric wet delay prediction model, including a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. Next, it trains the tropospheric wet delay prediction model using a training dataset. Finally, it uses the trained tropospheric wet delay prediction model to predict tropospheric wet delay. By explicitly embedding atmospheric refraction physical constraints into the loss function through the physical information neural network framework, it achieves collaborative training and stable extrapolation of data-driven and physical constraints. This invention provides a new paradigm for the unified quantification of physical constraints and uncertainties in high-precision GNSS geodetic data processing and application technology, and offers a more comprehensive and reliable technical solution for the practical application of tropospheric delay correction.

Description

Technical Field

[0001] This invention relates to the field of navigation technology, and more specifically, to a method and system for predicting tropospheric wet delay based on a physical information neural network. Background Technology

[0002] Tropospheric wet delay (ZWD) is one of the key error sources in high-precision GNSS positioning, precise time synchronization, VLBI geodesy, and InSAR deformation monitoring, directly affecting parameter estimation and product quality at the millimeter to centimeter level. Existing technologies mainly fall into two categories: The first category is classical physical modeling methods, including analytical models and mapping functions (such as the NMF, GMF, and VMF families), empirical meteorological field models (such as the GPT series), and numerical weather reanalysis (such as ERA5). These methods are based on atmospheric physical equations and long-term meteorological observation data, calculating tropospheric delay parameters through analytical expressions or lookup tables. The second category is data-driven machine learning methods, employing algorithms such as random forests, CNNs, LSTMs, and Transformers for short-term forecasting, spatiotemporal interpolation, and assimilation preprocessing of ZTD / ZWD. These methods train models using a large amount of historical observation data to capture the nonlinear mapping relationship between input meteorological parameters and tropospheric delay.

[0003] Existing technologies suffer from three significant drawbacks: First, while classical physical modeling methods are robust under standard conditions, they face significant uncertainties and systematic biases in scenarios such as extreme weather with strong convection and rapid changes in water vapor, complex terrain and high-altitude areas, and cross-seasonal migration. They are particularly prone to systematic biases and timeliness limitations under conditions dominated by high humidity gradients and local meteorological boundary layer processes. Second, purely data-driven methods face challenges such as insufficient physical consistency, limited interpretability, and fragile extrapolation generalization. Their performance is sensitive to the distribution of training data, station characteristics, and climatological stability, resulting in insufficient generalization ability across stations, seasons, and even extreme conditions. Third, incomplete feature representations and insufficient understanding of hierarchical coupling of physical quantities also limit the improvement of model accuracy and robustness. The root cause of these problems lies in the fact that physical methods mainly rely on climatological statistical laws and simplified physical assumptions, making it difficult to accurately describe complex atmospheric processes at local and short-term scales; while purely data-driven methods rely solely on statistical correlation learning, failing to explicitly integrate atmospheric physical conservation laws and refraction mechanisms, and lacking the guidance of physical constraints. Summary of the Invention

[0004] This invention aims to address the core technical challenge of efficiently integrating data-driven approaches with physical constraints. Specifically, it addresses two key challenges: first, how to achieve centimeter-level ZWD prediction accuracy by integrating physical constraints within the Physical Information Neural Network (PINN) framework and enhance the model's spatiotemporal generalization performance; and second, how to design hierarchical feature extraction and uncertainty quantification mechanisms to improve the model's robustness under extreme conditions and across different scenarios. To address the shortcomings of existing methods, such as insufficient accuracy under extreme weather conditions, lack of physical consistency in purely data-driven models, and lack of uncertainty quantification capabilities, this invention strives to overcome the extrapolation instability and uninterpretability of black-box models, while simultaneously improving timeliness and cross-scenario robustness through the synergistic optimization of data and physics.

[0005] Based on the above analysis, the purpose of this invention is to develop a deep learning framework that integrates physical constraints to achieve centimeter-level ZWD prediction accuracy and maintain high robustness in time-transfer and cross-site scenarios, while quantifying prediction uncertainty to improve application credibility. Specifically, a Physical Information Neural Network (PINN) framework is constructed, explicitly embedding atmospheric refraction physical constraints (wet refractive index, including empirical coefficients and non-ideal corrections) into the loss function to achieve collaborative training and stable extrapolation of data-driven and physical constraints. A hierarchical feature strategy is established based on feature importance analysis, with water vapor pressure, wet delay coefficient, and temperature as the core layer and hydrostatic parameters as the auxiliary layer, balancing accuracy and efficiency. A Bayesian Neural Network (BNN) is introduced, and Monte Carlo Dropout technology is used to quantify prediction uncertainty, achieving a leap from "point prediction" to "credible probability prediction." This invention provides a new paradigm for the unified quantification of physical constraints and uncertainties in high-precision GNSS geodetic data processing and application technology, offering a more comprehensive and reliable technical solution for the practical application of tropospheric delay correction.

[0006] Specifically, the first aspect of the present invention provides a method for predicting tropospheric wet delay based on a physical information neural network, comprising: Real-time meteorological observation data from GNSS stations are collected, the collected real-time meteorological observation data is preprocessed, and a training dataset is constructed. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay. A tropospheric wet delay prediction model is constructed, comprising a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture to fuse data-driven and physical constraints based on the high-dimensional feature vector. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module. The tropospheric wet delay prediction model was trained using the training dataset; Tropospheric wet delay prediction is performed using a trained tropospheric wet delay prediction model.

[0007] In one implementation, the collected real-time meteorological observation data is preprocessed, including: Outlier removal and missing value completion are performed on the collected real-time meteorological observation data; Six input features in real-time meteorological observation data are standardized. The input features after standardization are analyzed for feature importance and divided into core features, auxiliary features and background features. The core features include water vapor pressure, wet delay coefficient and temperature, the auxiliary features include static delay coefficient, and the background features include zenith static delay and atmospheric pressure.

[0008] In one implementation, the three-level cascaded structure includes a core feature layer, an auxiliary feature layer, and a background feature layer. The hierarchical feature extraction module uses the three-level cascaded structure to process the features in the input meteorological observation data and fuse them to obtain a high-dimensional feature vector, including: The core feature layer uses a fully connected neural network to perform nonlinear transformations on water vapor pressure, wet delay coefficient and temperature, and extracts high-dimensional representations that are directly related to wet delay. The auxiliary feature layer uses the static delay coefficient as a supplementary input and extracts the feature representations that affect atmospheric density stratification and water vapor transport patterns by setting a selective activation mechanism. The first fused feature is obtained by fusing the high-dimensional characterization directly related to wet delay with the feature characterization that affects atmospheric density stratification and water vapor transport patterns. The background feature layer uses zenith static delay and atmospheric pressure as input to extract features indirectly related to the wet delay. After weight modulation, these features are added to the first fusion feature to form the final complete feature vector.

[0009] In one implementation, the dual-branch parallel architecture includes a data-driven branch and a physical constraint branch. The physical information neural network computation module, based on high-dimensional feature vectors, employs a dual-branch parallel architecture to fuse data-driven and physical constraint operations, including: The data-driven branch receives the complete feature vector from the hierarchical feature extraction module and learns the complex nonlinear mapping relationship between the input meteorological parameters and the tropospheric wet delay through the gradient descent algorithm. The physical constraint branch runs in parallel with the data-driven branch, transforming the wet refractive index formula into a differentiable function module that is directly embedded in the calculation, determining whether the model predictions satisfy the atmospheric refractive physical equations, and quantifying the physical equation residuals. The output of the data-driven branch and the physical constraint branch merge at the loss function fusion node.

[0010] In one implementation, training a tropospheric wet delay prediction model using a training dataset includes: Input the training dataset into the tropospheric wet delay prediction model; The data input module receives the training dataset, the hierarchical feature extraction module processes the features in the input meteorological observation data and fuses them to obtain a high-dimensional feature vector, and the physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The data-driven and physical constraints are fused using a dual-loss training process, which employs a dual-loss function mechanism, including data-driven loss and physical constraint loss. The data-driven loss is used to measure the fitting error between the network prediction value and the observation data, and the physical constraint loss is used to quantify the residual of the physical equation. Through iterative optimization, the weight coefficients are adjusted to make the overall loss less than a preset threshold until the model converges, and the trained model is saved.

[0011] In one implementation, the loss function is:

[0012] in, To account for the overall losses, For data-driven loss, For physical constraint loss, These are the weighting coefficients. , Where n is the number of training samples, and i is the i-th training sample. For the ground truth of the i-th training sample, Let i be the neural network predictions for training samples. These are the predicted values ​​from the physical model.

[0013] In one embodiment, the method further includes: The probability distribution of predicted values ​​is obtained through Monte Carlo sampling; Standardized output reports and visualizations are generated based on the model's predicted values ​​and their probability distribution.

[0014] Based on the same inventive concept, a second aspect of the present invention provides a tropospheric wet delay prediction system based on a physical information neural network, comprising: The data acquisition and preprocessing module is used to acquire real-time meteorological observation data from GNSS stations, preprocess the acquired real-time meteorological observation data, and construct a training dataset. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay. The model building module is used to construct a tropospheric wet delay prediction model. It includes a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module. The model training module is used to train the tropospheric wet delay prediction model using the training dataset; The tropospheric wet delay prediction module is used to predict tropospheric wet delay using a trained tropospheric wet delay prediction model.

[0015] Based on the same inventive concept, a third aspect of the present invention provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, provides the tropospheric wet delay prediction method based on a physical information neural network as described in the first aspect.

[0016] Based on the same inventive concept, a fourth aspect of the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the tropospheric wet delay prediction method based on physical information neural network described in the first aspect.

[0017] Compared with the prior art, the advantages and beneficial technical effects of the present invention are as follows: This invention discloses a tropospheric wet delay prediction method based on a physical information neural network. It constructs a tropospheric wet delay prediction model, which is a deep learning framework integrating physical constraints. The model includes a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. Atmospheric refraction physical constraints (wet refractive index, including empirical coefficients and non-ideal corrections) are explicitly embedded into the loss function, achieving collaborative training and stable extrapolation of data-driven and physical constraints. This invention provides a new paradigm for the unified quantification of physical constraints and uncertainties in high-precision GNSS geodetic data processing and application technologies, and offers a more comprehensive and reliable technical solution for the practical application of tropospheric delay correction.

[0018] Furthermore, this invention proposes a hierarchical feature strategy based on feature importance analysis, with water vapor pressure, wet delay coefficient, and temperature as the core layer and static parameters as the auxiliary layer, balancing accuracy and efficiency to achieve centimeter-level ZWD prediction accuracy and maintain high robustness in time migration and cross-site scenarios. At the same time, it quantifies prediction uncertainty to improve application credibility.

[0019] Furthermore, this invention introduces a Bayesian neural network (BNN) and uses Monte Carlo Dropout technology to quantify prediction uncertainty, achieving a leap from "point prediction" to "credible probability prediction". Attached Figure Description

[0020] 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 some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a flowchart of the tropospheric wet delay prediction method based on a physical information neural network in an embodiment of the present invention; Figure 2 This is a detailed technical roadmap of the tropospheric wet delay prediction method based on physical information neural network in this embodiment of the invention; Figure 3 This is a schematic diagram of the training process of the tropospheric wet delay prediction model in an embodiment of the present invention. Detailed Implementation

[0022] First, the relevant concepts and terms involved in the present invention will be explained: (1) Tropospheric Wet Delay (ZWD): Tropospheric wet delay refers to the path delay of electromagnetic waves propagating when a GNSS signal passes through a region of uneven water vapor distribution in the troposphere. Its magnitude is usually between several centimeters and tens of centimeters, and it is one of the main error sources affecting the accuracy of GNSS high-precision positioning, precise time synchronization, VLBI geodesy, and InSAR deformation monitoring. Unlike static delay, wet delay has significant spatiotemporal variability and unpredictability, and its magnitude directly depends on the water vapor pressure, temperature, and vertical distribution characteristics in the atmosphere. Under extreme weather conditions, complex terrain, and high humidity environments, rapid changes in wet delay can lead to positioning errors at the centimeter or even decimeter level.

[0023] (2) Physical Information Neural Network (PINN): PINN is a novel modeling paradigm that combines physical laws with data-driven deep learning. By explicitly embedding physical equations, conservation laws, or prior knowledge into the loss function of the neural network, it achieves synergistic optimization of data fitting and physical constraints. This method retains the powerful ability of deep learning to handle complex nonlinear relationships while overcoming the shortcomings of pure data-driven models in terms of physical consistency, interpretability, and extrapolation generalization.

[0024] (3) Physical constraints: Physical constraints refer to using the physical laws, empirical formulas, and conservation relationships describing the refraction process of the troposphere as explicit prior knowledge, and embedding them into the training process of the deep learning model through mathematical expressions to ensure that the model output conforms to basic physical laws. By adjusting the physical loss weight coefficient, a dynamic balance can be achieved between data fitting accuracy and physical consistency, enabling the model to capture complex local atmospheric processes while maintaining global physical rationality, fundamentally solving the extrapolation instability problem of traditional black-box models.

[0025] (4) Spatiotemporal generalization: Spatiotemporal generalization ability refers to the ability of a model to maintain stable prediction performance under different time periods, geographical locations and climatic conditions. It is a core indicator for evaluating the practical value of tropospheric delay prediction models. Traditional empirical models and pure data-driven methods often perform well within the training data distribution range, but significant accuracy degradation occurs when the model migrates across seasons, extrapolates spatially across sites and under extreme weather conditions.

[0026] (5) Uncertainty Quantification: Uncertainty quantification refers to the quantitative evaluation of the reliability and error distribution of model prediction results through probabilistic statistical methods, enabling the model to transform from outputting deterministic point estimates to providing complete probability distribution information. This invention introduces a Bayesian Neural Network (BNN) framework, using Monte Carlo Dropout technology to perform multiple random samplings during the inference phase, constructing the probability distribution of predicted values ​​and calculating the prediction mean and confidence interval. This mechanism not only provides the predicted values ​​themselves but also gives quantitative indicators of prediction uncertainty, enabling users to identify high-risk scenarios and make corresponding decision adjustments. In extreme weather, data-sparse regions, or model extrapolation scenarios, the uncertainty quantification function is particularly crucial, as it can provide early warnings of model performance degradation and avoid directly applying low-confidence prediction results to high-precision positioning tasks. This leap from "point prediction" to "reliable probability prediction" significantly improves the practicality and safety of the model, providing an important quality control means for the engineering application of tropospheric delay correction.

[0027] (6) Layered Feature Strategy: The layered feature strategy is a feature engineering method based on feature importance analysis and physical mechanism understanding, which divides input meteorological parameters into core, auxiliary, and background layers according to their contribution to wet delay prediction. The core layer includes water vapor pressure, wet delay coefficient, and temperature, which directly determine the main changes in wet refractive index; the auxiliary layer includes secondary influencing factors such as the hydrostatic delay coefficient; and the background layer includes parameters such as hydrostatic delay and air pressure that can be stably estimated by empirical models. This layered structure ensures the model's sensitivity to key physical quantities and avoids the risk of overfitting introduced by redundant features, achieving an optimal balance between accuracy, computational efficiency, and robustness. Through progressive feature ablation experiments, this invention verifies the necessity and sufficiency of features in each layer and provides flexible feature combination schemes for different application scenarios. This strategy not only improves the stability of the model under extreme conditions but also provides an operable technical path for the deep integration of physics-driven and data-driven approaches.

[0028] Example 1 This embodiment provides a tropospheric wet delay prediction method based on a physical information neural network. Please refer to [link to relevant documentation]. Figure 1 ,include: S1: Collect real-time meteorological observation data from GNSS stations, preprocess the collected real-time meteorological observation data, and construct a training dataset. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay.

[0029] Specifically, S1 is data acquisition and data preprocessing, which provides the data foundation for subsequent model training.

[0030] This includes preprocessing the collected real-time meteorological observation data, including: Outlier removal and missing value completion are performed on the collected real-time meteorological observation data; Six input features in real-time meteorological observation data are standardized. The input features after standardization are analyzed for feature importance and divided into core features, auxiliary features and background features. The core features include water vapor pressure, wet delay coefficient and temperature, the auxiliary features include static delay coefficient, and the background features include zenith static delay and atmospheric pressure.

[0031] In practice, reference ZWD ground truth values ​​are obtained through GNSS double-difference solution or PPP technology, which serve as supervision signals for model training.

[0032] Data preprocessing includes quality control: removing outliers that are outside the physically reasonable range (such as e<0 or T<200K), and completing missing data using linear interpolation or cubic spline interpolation.

[0033] Here, 'e' represents water vapor pressure, 'T' represents temperature, and 'K' represents the thermodynamic temperature unit.

[0034] Standardization: Z-score standardization is performed on the six input features respectively, the mean (μ) and standard deviation (σ) of each feature are calculated, and normalization is performed according to the formula to ensure that the numerical units are consistent.

[0035]

[0036] In the formula, The normalized value. For each input value, μ and σ are the mean and standard deviation, respectively.

[0037] S2: Construct a tropospheric wet delay prediction model, including a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives the input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module.

[0038] Specifically, S2 is the construction of the model.

[0039] The three-level cascaded structure includes a core feature layer, an auxiliary feature layer, and a background feature layer. The hierarchical feature extraction module uses this three-level cascaded structure to process the features in the input meteorological observation data and fuse them to obtain a high-dimensional feature vector, including: The core feature layer uses a fully connected neural network to perform nonlinear transformations on water vapor pressure, wet delay coefficient and temperature, and extracts high-dimensional representations that are directly related to wet delay. The auxiliary feature layer uses the static delay coefficient as a supplementary input and extracts the feature representations that affect atmospheric density stratification and water vapor transport patterns by setting a selective activation mechanism. The first fused feature is obtained by fusing the high-dimensional characterization directly related to wet delay with the feature characterization that affects atmospheric density stratification and water vapor transport patterns. The background feature layer uses zenith static delay and atmospheric pressure as input to extract features indirectly related to the wet delay. After weight modulation, these features are added to the first fusion feature to form the final complete feature vector.

[0040] Specifically, the hierarchical feature extraction module is the core feature engineering unit of the system, and its design is based on the system feature importance analysis results. The core feature layer, located at the bottom of the module, receives and processes three primary physical quantities: water vapor pressure, which consistently ranks as the most important predictor across all stations, directly quantifies the total amount of water vapor in the atmospheric column from an atmospheric physics perspective, making it a decisive factor in tropospheric wet delay; the wet delay coefficient, along with temperature, determines the sensitivity of atmospheric refractive index to water vapor and temperature, based on the physical mechanism that wet refractive index is approximately proportional to e / T (e is surface water vapor pressure, and T is surface absolute temperature). This layer uses a fully connected neural network to perform nonlinear transformations on the core features, extracting high-dimensional representations directly related to wet delay. The auxiliary feature layer, located above the core layer, receives the static delay coefficient as supplementary input. This layer employs a selective activation mechanism, dynamically adjusting its participation level according to application scenario requirements. Its output is concatenated with the core layer feature vector at the feature fusion node. The background feature layer, located at the top of the module, processes background parameters such as ZHD and atmospheric pressure. Although these statically related parameters are not directly physically related to the wet delay, they indirectly affect atmospheric density stratification and water vapor transport patterns by constraining the mass distribution of dry air and the atmospheric background field. This primarily enhances the model's stability and robustness under extreme conditions. The background layer employs a residual connection approach; its output, after weight modulation, is added to the fused features of the lower layers to form the final complete feature vector. This hierarchical structure balances prediction accuracy and computational efficiency, ensuring the model's sensitivity to key physical quantities while avoiding the overfitting risk introduced by redundant features.

[0041] The dual-branch parallel architecture includes a data-driven branch and a physical constraint branch. The physical information neural network computation module, based on high-dimensional feature vectors, employs a dual-branch parallel architecture to fuse data-driven and physical constraint operations, including: The data-driven branch receives the complete feature vector from the hierarchical feature extraction module and learns the complex nonlinear mapping relationship between the input meteorological parameters and the tropospheric wet delay through the gradient descent algorithm. The physical constraint branch runs in parallel with the data-driven branch, transforming the wet refractive index formula into a differentiable function module that is directly embedded in the calculation, determining whether the model predictions satisfy the atmospheric refractive physical equations, and quantifying the physical equation residuals. The output of the data-driven branch and the physical constraint branch merge at the loss function fusion node.

[0042] Specifically, the physical information neural network computation module is the core computing engine of the system, employing an innovative dual-branch parallel architecture to achieve a deep integration of data-driven and physical constraints. The data-driven branch consists of a multi-layer fully connected neural network, including an input layer, 3-5 hidden layers, and an output layer. Nonlinear mappings between layers are achieved through the ReLU activation function, and a Batch Normalization layer is used to accelerate convergence and improve training stability. This branch receives complete feature vectors from the hierarchical feature extraction module and learns the complex nonlinear mapping relationship between input meteorological parameters and ZWD (Zero-Wide Dynamics) through gradient descent. The physical constraint branch runs in parallel with the data-driven branch, directly embedding the module that transforms the wet refractive index formula into a differentiable function for computation. This branch calculates whether the model's predicted values ​​satisfy the atmospheric refractive physical equation and quantifies the residuals of the physical equation. The outputs of the two branches converge at the loss function fusion node, where a comprehensive loss function is calculated. This physics-data co-training strategy enables the model to learn statistical regularities from the data while adhering to atmospheric physical mechanisms, maintaining model stability in dynamic atmospheric environments, particularly at high humidity gradients and sharp transitions. In time series segmentation tests, it achieves centimeter-level robust accuracy and lower system bias.

[0043] Please see Figure 2 This is a detailed technical roadmap of the tropospheric wet delay prediction method based on physical information neural network in this embodiment of the invention.

[0044] (1) Explanation of neural network structure and symbols In the figure, the input feature layer contains six feature variables: ZHD represents the zenith static delay; P represents atmospheric pressure; T represents atmospheric temperature; e represents water vapor pressure; C1 and C2 represent the coefficients of the wet-dry mapping function, used to characterize the vertical stratification and water vapor distribution characteristics of the atmosphere. The neural network contains multiple hidden layers, where Net 1 and Net 2 represent the first and second hidden layers, respectively, and σ represents the nonlinear activation function. and These represent the first hidden layer and the... The first neuron and the first Hidden layer Bias terms for each neuron are used to adjust the neuron activation threshold. The network, through multiple layers of nonlinear mapping, ultimately outputs the predicted tropospheric wet delay value. .

[0045] (2) Explanation of physical model formulas and loss function The ZWD physical model is used as the constraint condition in the figure, and the formula is as follows: (1) (2) (3) (4) (5) (6) In the formula, 0.002277 is the empirical constant coefficient. This is the gravity correction function. Let h be the latitude of the station and h be the elevation of the station; Formula (2) is the original ZWD prediction model. By simplifying the ZWD prediction model using Formulas (1) and (2), Formula (3) is derived, which uses the known ZHD value to simplify the calculation process. The training process adopts a dual-objective loss function: physical constraint loss. Quantizing neural network predictions Compared with the physical model predictions Bias; data-driven loss Measure the predicted value Compared with the actual observed value The mean squared error. The comprehensive loss function is defined as... λ is a weighting coefficient with a value range of (0-1), used to balance the data fitting accuracy and physical consistency constraints. The neural network parameters are optimized by minimizing the comprehensive loss function, thereby obtaining high-precision tropospheric wet delay prediction results.

[0046] Regarding neural network predictions ( ) and physical model predictions ( The explanation for ) is as follows: (1) Neural network predicted value ( ) Definition: This refers to the output result obtained by forward propagation calculation of the data-driven branch.

[0047] Generation mechanism: It is based on the input meteorological observation data (water vapor pressure, temperature, etc.), which is processed by the hierarchical feature extraction module and then obtained by the neural network through complex nonlinear mapping of learned weights and biases.

[0048] Features: It has a strong nonlinear fitting ability and can capture statistical laws in data that are difficult to describe with formulas. However, without constraints, it is prone to overfitting or predictions that do not conform to physical common sense.

[0049] (2) Physical model predicted values ​​( ): Definition: refers to the theoretical values ​​calculated by the physical constraint branch based on the classical atmospheric physics equations (specifically the Saastamoinen model and its improved formula in this invention).

[0050] Generation mechanism: It is the result obtained by directly substituting the input meteorological parameters into a preset, fixed physical analytical formula (such as the wet refractive index formula).

[0051] Characteristics: It strictly follows the laws of physical conservation, has strong physical interpretability and boundary stability, but often exhibits systematic bias under extreme weather or complex terrain.

[0052] Neural network prediction ( ), physical model predictions ( The relationship between the model's "final output" and the model's "final output" is explained below: The final predicted value output by the model: The final output value of the model is the "neural network predicted value".

[0053] During the model inference phase, i.e., when the system is actually used for wet delay prediction, the system only outputs the results generated by the neural network branches. The physical model predictions are not output to the user as the final result at this stage.

[0054] The logical relationship between the three: During the training phase (co-optimization relationship): the physical model predictions act as a "mentor" or "regularization constraint." The system constructs a comprehensive loss function. Connecting the two: Neural network predictions are required not only to approximate the true observed values ​​(data-driven loss) ); At the same time, it should approximate the predicted values ​​of the physical model as closely as possible, or satisfy the derivative constraints of the physical equations (physical constraint loss). This mechanism forces the direction of weight updates in neural networks to not deviate too far from physical laws, thereby correcting non-physical predictions that pure neural networks may produce and achieving "synergistic integration of data and physics".

[0055] In the application phase (internalizing relationships): After training, the laws of the physical model have been "internalized" into the weight parameters of the neural network. Therefore, when making predictions, simply running the neural network branches is enough to obtain a final prediction that is both highly accurate (due to data-driven approaches) and highly robust (due to physical constraints).

[0056] The tropospheric wet delay prediction model includes a neural network structure and a physical model. Specifically, the modules belonging to the "neural network structure" constitute the backbone of the model and are responsible for the complete forward propagation path from data input to final output. These modules include the data input module, the hierarchical feature extraction module, the data-driven branch in the physical information neural network calculation module, and the output module.

[0057] Data input module: As the input layer interface of the neural network, it receives preprocessed standardized tensor data.

[0058] Hierarchical Feature Extraction Module: This is a typical neural network feature engineering structure, employing a "three-level cascaded structure" (combination of fully connected layers). It includes a core feature layer (processing water vapor pressure and temperature), an auxiliary feature layer (processing hydrostatic coefficients), and a background feature layer (processing tropospheric dry delay and atmospheric pressure). High-dimensional feature vectors are extracted through neuron connections and activation functions.

[0059] The "data-driven branch" in the physical information neural network computation module is the core of the neural network, typically composed of multiple fully connected hidden layers. It receives high-dimensional feature vectors, performs weight matrix multiplication and non-linear activation, and outputs... .

[0060] Output module: This is the output layer of the neural network, which is usually a single-node linear layer used to output the final wet delay prediction value.

[0061] The modules belonging to the "physical model" do not directly participate in the generation of the final result, but exist as constraints in the training process and are embedded in specific branches of the computation module; the modules belonging to the "physical model" are the physical constraint branches in the physical information neural network computation module.

[0062] The physical constraint branch in the physical information neural network calculation module: Definition: The "physical model" as described in this invention.

[0063] Its specific structure: It is not composed of trainable neurons, but rather a computational graph consisting of fixed mathematical operators. Specifically, it rewrites the Saastamoinen analytical formula (and its relation to the wet refractive index) as differentiable function modules.

[0064] Function: It receives the same input data in parallel, or receives the output of a neural network and its partial derivatives with respect to the input, calculates the residuals of physical equations or theoretical physical values, and is used to construct the loss function. This is used to guide the training of neural network branches.

[0065] S3: Train the tropospheric wet delay prediction model using the training dataset.

[0066] Specifically, S3 is the training of the model. The training process includes: Input the training dataset into the tropospheric wet delay prediction model; The data input module receives the training dataset, the hierarchical feature extraction module processes the features in the input meteorological observation data and fuses them to obtain a high-dimensional feature vector, and the physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The data-driven and physical constraints are fused using a dual-loss training process, which employs a dual-loss function mechanism, including data-driven loss and physical constraint loss. The data-driven loss is used to measure the fitting error between the network prediction value and the observation data, and the physical constraint loss is used to quantify the residual of the physical equation. Through iterative optimization, the weight coefficients are adjusted to make the overall loss less than a preset threshold until the model converges, and the trained model is saved.

[0067] In the specific implementation process, such as Figure 3 As shown, the input layer contains two types of feature variables: x represents spatial coordinates and t represents time, used to describe the spatiotemporal distribution characteristics of the physical field. The hidden layers of the neural network adopt a fully connected feedforward structure. Each neuron node in the layer achieves nonlinear mapping through the activation function σ. The network output u represents the physical quantity to be solved. The core innovation of the network structure lies in the introduction of a partial differential equation constraint module (physical information neural network calculation module), which calculates the partial derivatives of the output u with respect to the input variables, including the time partial derivatives, through automatic differentiation technology. Spatial partial derivatives and higher-order partial derivatives These partial derivative terms are substituted into the physical governing equations. Among them Indicates the first The residuals of the physical equations, where f is a function. These are the relevant physical variables. The training process employs a dual loss function mechanism: data-driven loss. Physical constraint loss measures the fitting error between network predictions and observed data. Quantify the residuals of the physical equations, where n is the number of collocation points. For the theoretical solution, the comprehensive loss function is defined as follows: The training process involves iterative optimization, where λ is an adjustable weight coefficient. By adjusting the weight coefficient, a dynamic balance can be achieved between data fitting accuracy and physical consistency, ensuring that the overall loss MSE is less than a preset threshold ε. This leads to the convergence of neural network parameters, ensuring that the network output meets both the data fitting accuracy requirements and strictly follows the constraints of the physical equations, significantly improving the physical consistency and generalization ability of the model.

[0068] S4: Use the trained tropospheric wet delay prediction model to predict tropospheric wet delay.

[0069] Specifically, S4 is the concrete application of the model.

[0070] In one implementation, tropospheric wet delay prediction based on a physical information neural network further includes: The probability distribution of predicted values ​​is obtained through Monte Carlo sampling; Standardized output reports and visualizations are generated based on the model's predicted values ​​and their probability distribution.

[0071] Specifically, the uncertainty quantification module is built on a Bayesian neural network framework and uses Monte Carlo Dropout technology to achieve the leap from "point prediction" to "credible probability prediction". This module inserts a Dropout layer (Dropout rate 0.1-0.3) after each hidden layer of the PINN computation module. During the training phase, some neurons are randomly deactivated to enhance generalization ability, while during the inference phase, the Dropout layer remains active. For the same input, 50-100 forward propagations are performed, each producing different prediction results due to the different randomly deactivated neurons. The statistical computation unit receives the outputs from multiple propagations and calculates the prediction mean, variance, and confidence interval. It not only provides the prediction value itself but also gives a quantitative indicator of prediction uncertainty, enabling it to identify high-risk scenarios and provide early warnings of model performance degradation in extreme weather, sparse data regions, or model extrapolation scenarios.

[0072] The complete workflow of the method proposed in this invention is as follows: raw meteorological data undergoes quality control and standardization processing by the data input module → the hierarchical feature extraction module processes the data step by step in the order of core layer - auxiliary layer - background layer to output high-dimensional feature vectors → the PINN calculation module performs parallel calculations in two branches and performs collaborative optimization at the loss function fusion node → the uncertainty quantification module obtains the probability distribution of predicted values ​​through Monte Carlo sampling → the prediction output module integrates the predicted values ​​and uncertainty information to generate a complete report. Specifically, after standardized preprocessing, the data is output to the hierarchical feature extraction module of the model via a data transmission interface. The hierarchical feature extraction module adopts a three-layer cascaded structure: the bottom layer is the core feature layer that processes water vapor pressure, wet delay coefficient, and temperature; the middle layer is the auxiliary feature layer that processes the static delay coefficient; and the top layer is the background feature layer that processes ZHD and atmospheric pressure. Each layer is cascaded and connected through a feature fusion node, ultimately outputting a complete high-dimensional feature vector. The physical information neural network (PINN) computation module of the model receives the complete high-dimensional feature vector and adopts a dual-branch parallel architecture. The data-driven branch consists of a fully connected network composed of 3-5 hidden layers (256-512 neurons per layer), while the physical constraint branch computes the wet refractive index residual in parallel. The two branches converge at the loss function fusion node for collaborative optimization. The uncertainty quantification module inserts a Dropout layer after each hidden layer of PINN. During inference, it calculates the predicted mean, variance, and confidence interval through 50-100 Monte Carlo samplings. The prediction output module integrates the predicted values ​​and confidence information to generate a standardized output report and visualization charts.

[0073] This invention has made significant progress in time migration and high humidity scenarios, and its performance comprehensively surpasses that of traditional physical models and pure data-driven models, providing a technical solution that unifies physical constraints and uncertainty quantification for high-precision GNSS geodetic data processing.

[0074] The method and beneficial effects of the present invention will be further explained below through specific experiments.

[0075] This example provides a high-precision method for predicting tropospheric wet delay based on an uncertainty quantification physical information neural network. The specific implementation steps are as follows: Step 1: Meteorological Data Acquisition and Stratified Feature Preprocessing Data source configuration: (1) Collect real-time meteorological observation data from GNSS stations, including water vapor pressure (e, unit: hPa), temperature (T, unit: K), atmospheric pressure (P, unit: hPa), wet delay coefficient, static delay coefficient and zenith static delay (ZHD, unit: m), with a sampling frequency of 1 hour.

[0076] (2) Obtain the reference ZWD true value through GNSS double difference solution or PPP technology as a supervision signal for model training.

[0077] Data preprocessing workflow: (1) Quality control: Remove outliers that are outside the physical reasonable range (such as e<0 or T<200K), and use linear interpolation or cubic spline interpolation to fill in missing data.

[0078] (2) Standardization: Z-score standardization is performed on the six input features respectively, the mean (μ) and standard deviation (σ) of each feature are calculated, and normalization is performed according to the formula to ensure that the numerical units are consistent.

[0079] Hierarchical Feature Construction: Based on the results of feature importance analysis, the input features are divided into three layers: Core feature layers: water vapor pressure, wet delay coefficient, temperature Auxiliary feature layer: Static delay coefficient Background feature layer: ZHD, atmospheric pressure Step 2: Construction and Training of Physical Information Neural Network Network architecture design: (1) Input layer: Receives 6-dimensional standardized feature vectors.

[0080] (2) Hierarchical feature extraction module: Core feature branch: Contains 3 fully connected layers (with 256, 128, and 64 neurons respectively), using the GELU activation function to process water vapor pressure, humidity delay coefficient, and temperature.

[0081] Auxiliary feature branch: contains 2 fully connected layers (64 and 32 neurons respectively), which handle static delay coefficients.

[0082] Background feature branch: Contains 1 fully connected layer (32 neurons), handling ZHD and air pressure.

[0083] Feature fusion node: merges the outputs of the three branches into a unified high-dimensional feature vector through a concatenation operation.

[0084] (3) Backbone network: It consists of 4 hidden layers, each containing 512 neurons, with BatchNormalization and Dropout (p=0.2) layers between layers.

[0085] (4) Output layer: Single neuron outputs ZWD prediction value (unit: m).

[0086] Step 3: Implementation of the Bayesian Uncertainty Quantization Module Monte Carlo Dropout configuration: (1) Training phase: Insert a Dropout layer after each hidden layer of the backbone network, with the Dropout rate set to 0.2 as a regularization method.

[0087] (2) Inference phase: Keep the Dropout layer active and perform T=100 random forward propagations for each input sample. Each propagation produces different prediction results due to the different randomly deactivated neurons.

[0088] Uncertainty Statistical Calculation: For each input sample, 100 prediction results are obtained. Calculate using the following formula: Predicted mean:

[0089] (2) Predictive standard deviation:

[0090] (3) 95% confidence interval:

[0091] in, This represents the arithmetic mean of 100 predictions, which is the final ZWD prediction output. It quantifies the prediction uncertainty, and a higher value indicates that the model has a lower confidence in the prediction. This represents the predicted value of the i-th sample at sampling time t. The sample standard deviation of the prediction result is used to quantify the level of uncertainty in the prediction. A higher value indicates that the model has a lower confidence in the prediction. This represents the prediction interval at a 95% confidence level. 1.96 is the two-sided 95th percentile of the standard normal distribution. This interval contains the true value with a 95% probability, providing a reliable uncertainty boundary for the prediction results.

[0092] Step 4: Model Inference and Result Output Real-time prediction process: (1) Feature input: The six meteorological parameters collected in real time are standardized and then input into the trained physical information neural network (PINN-BNN) model based on Bayesian estimation.

[0093] (2) Parallel inference: 100 Monte Carlo samplings are performed, each forward propagation takes about 5ms, and the total inference time is about 0.5 seconds, which meets the real-time requirements.

[0094] (3) Statistical post-processing: Calculate the predicted mean, standard deviation and confidence interval, and mark high-risk prediction points according to the uncertainty threshold.

[0095] Output format: Generate a forecast report containing the following information: ZWD prediction values ​​(unit: m, accuracy down to the centimeter level) Forecast uncertainty (unit: m) 95% confidence interval upper and lower bounds Risk level identification (low / medium / high, based on threshold classification).

[0096] Performance verification: This embodiment achieves a prediction accuracy of R²=0.95, RMSE=0.7 cm, and MAE=1 cm on the test set. The prediction uncertainty is significantly positively correlated with the actual error (R=0.42, p<0.001), and the 95% confidence interval coverage reaches 97.0%, which is close to the theoretical value of 95.4%.

[0097] In summary, the core innovation of this invention lies in constructing a deep learning framework that explicitly embeds atmospheric refraction physical constraints into the loss function, achieving collaborative training between data-driven and physical mechanisms. This framework directly embeds differentiable physical priors (wet refractive index, including empirical coefficients and non-ideal corrections) into the loss function, achieving data-physics co-optimization and stable extrapolation. Simultaneously, it introduces a Bayesian neural network (BNN) and uses Monte Carlo Dropout technology to quantify prediction uncertainty, successfully achieving a leap from "point prediction" to "credible probability prediction," reaching centimeter-level robust accuracy in time generalization and high humidity ranges. The main protection points include two aspects: (1) Hierarchical feature extraction strategy based on physical mechanism This invention proposes an innovative hierarchical feature selection method, which establishes a three-layer cascaded architecture based on feature importance analysis: Core layer: Water vapor pressure, wet delay coefficient, and temperature are the primary physical drivers, with water vapor pressure consistently ranking as the most important predictor. Auxiliary layer: using the static delay coefficient as a supplementary parameter; Background layer: Using ZHD and air pressure as background constraints, it balances accuracy and efficiency and enhances robustness under extreme conditions. This stratification strategy is based on atmospheric physical mechanisms: the wet refractive index is approximately proportional to e / T, and the water vapor pressure quantitatively reflects the water vapor content of the atmospheric column, which is the primary physical driver of ZWD; static parameters indirectly affect density stratification and water vapor transport by constraining the dry air mass distribution and background field.

[0098] (2) Collaborative optimization mechanism of dual loss functions This invention employs a weighted combination optimization strategy of physical constraint loss and data fitting loss. The wet refractive index physical equation (including Smith-Weintraub empirical coefficients and non-ideal gas correction terms) is constructed as a differentiable loss term, which is then jointly optimized with the data-driven loss term. This mechanism compensates for the extrapolation instability and lack of interpretability of black-box models, maintaining model stability in dynamic atmospheric environments at high humidity gradients and abrupt transitions. It achieves high-precision predictions with RMSE=1.2 cm and MAE=1 cm, with a significant positive correlation between prediction uncertainty and actual error (R=0.42, p<0.001), and a 97.0% coverage of the 95% confidence interval.

[0099] This invention achieves significant breakthroughs in four dimensions—prediction accuracy, physical consistency, generalization ability, and uncertainty quantification—through the synergistic fusion of Physical Information Neural Network (PINN) and Bayesian Neural Network (BNN).

[0100] In terms of prediction accuracy, this invention achieves centimeter-level high-precision predictions with R²=0.95, RMSE=0.7 cm, and MAE=1 cm, significantly surpassing traditional Saastamoinen analytical models and advanced deep learning baselines such as CNN and LSTM. Compared to the insufficient accuracy of existing classical physical modeling methods under extreme weather conditions and complex terrain, and the insufficient spatiotemporal generalization ability of purely data-driven methods, this invention achieves centimeter-level robust accuracy and lower system bias in time series segmentation tests.

[0101]

[0102] Regarding physical consistency, this invention explicitly embeds atmospheric refraction physical constraints into the loss function, enabling collaborative training and stable extrapolation of data-driven and physical constraints, effectively compensating for the extrapolation instability and lack of interpretability of black-box models. In dynamic atmospheric environments, physical consistency constraints maintain the model's stability at high humidity gradients and abrupt transitions, exhibiting excellent robustness and adaptability under extreme weather conditions.

[0103] In terms of feature engineering, this invention constructs a hierarchical feature strategy based on the importance analysis of system features. Water vapor pressure remains the most important predictor and has significant universality, balancing accuracy and efficiency and enhancing robustness under extreme conditions.

[0104] Regarding uncertainty quantification, this invention introduces Monte Carlo Dropout technology to construct a prediction uncertainty quantification system. Pearson correlation analysis shows that the correlation coefficient between prediction uncertainty and actual prediction error is R=0.42, with a p-value less than 0.001. The R-value indicates a moderate positive correlation, and the p-value indicates that the correlation passes the highly significant test, fully demonstrating that the uncertainty index can accurately characterize the confidence level of the model prediction. The constructed 95% confidence interval covers 97.0% of the actual observations, meeting the consistency requirements of probability statistics. This invention achieves a technological upgrade from traditional deterministic "point prediction" to "probabilistic interval prediction" that includes confidence assessment, providing an innovative paradigm that unifies physical constraints and uncertainty quantification for high-precision GNSS geodetic data processing.

[0105] Example 2 Based on the same inventive concept, this embodiment discloses a tropospheric wet delay prediction system based on a physical information neural network, comprising: The data acquisition and preprocessing module is used to acquire real-time meteorological observation data from GNSS stations, preprocess the acquired real-time meteorological observation data, and construct a training dataset. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay. The model building module is used to construct a tropospheric wet delay prediction model. It includes a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module. The model training module is used to train the tropospheric wet delay prediction model using the training dataset; The tropospheric wet delay prediction module is used to predict tropospheric wet delay using a trained tropospheric wet delay prediction model.

[0106] Since the system in Embodiment 2 of this invention is the same system used in the tropospheric wet delay prediction method based on physical information neural networks in Embodiment 1, those skilled in the art can understand the specific structure and variations of this system based on the method described in Embodiment 1 of this invention, and therefore will not be repeated here. All systems used in the method of Embodiment 1 of this invention fall within the scope of protection of this invention.

[0107] Example 3 Based on the same inventive concept, the present invention also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the tropospheric wet delay prediction method based on a physical information neural network of Embodiment 1.

[0108] Since the computer-readable storage medium described in Embodiment 3 of this invention is the same computer-readable storage medium used in implementing the tropospheric wet delay prediction method based on a physical information neural network in Embodiment 1 of this invention, those skilled in the art can understand the specific structure and variations of this computer-readable storage medium based on the method described in Embodiment 1 of this invention, and therefore will not be repeated here. All computer-readable storage media used in the method of Embodiment 1 of this invention fall within the scope of protection of this invention.

[0109] Example 4 Based on the same inventive concept, the present invention also provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the method described in Embodiment 1.

[0110] Since the computer device described in Embodiment 4 of this invention is the same computer device used to implement the tropospheric wet delay prediction method based on physical information neural networks in Embodiment 1 of this invention, those skilled in the art can understand the specific structure and variations of this computer device based on the method described in Embodiment 1 of this invention, and therefore will not be repeated here. All computer devices used in the method of Embodiment 1 of this invention fall within the scope of protection of this invention.

[0111] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0112] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0113] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention. Clearly, those skilled in the art can make various modifications and variations to the embodiments of the invention without departing from the spirit and scope of the invention. Thus, if these modifications and variations of the embodiments of the invention fall within the scope of the claims of the invention and their equivalents, the invention also intends to include these modifications and variations.

Claims

1. A method for predicting tropospheric wet delay based on a physical information neural network, characterized in that, include: Real-time meteorological observation data from GNSS stations are collected, the collected real-time meteorological observation data is preprocessed, and a training dataset is constructed. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay. A tropospheric wet delay prediction model is constructed, comprising a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture to fuse data-driven and physical constraints based on the high-dimensional feature vector. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module. The tropospheric wet delay prediction model was trained using the training dataset; Tropospheric wet delay prediction is performed using a trained tropospheric wet delay prediction model.

2. The tropospheric wet delay prediction method based on physical information neural network as described in claim 1, characterized in that, The collected real-time meteorological observation data undergoes preprocessing, including: Outlier removal and missing value completion are performed on the collected real-time meteorological observation data; Six input features in real-time meteorological observation data are standardized. The input features after standardization are analyzed for feature importance and divided into core features, auxiliary features and background features. The core features include water vapor pressure, wet delay coefficient and temperature, the auxiliary features include static delay coefficient, and the background features include zenith static delay and atmospheric pressure.

3. The tropospheric wet delay prediction method based on physical information neural network as described in claim 1, characterized in that, The three-level cascaded structure includes a core feature layer, an auxiliary feature layer, and a background feature layer. The hierarchical feature extraction module uses this three-level cascaded structure to process the features in the input meteorological observation data and fuse them to obtain a high-dimensional feature vector. include: The core feature layer uses a fully connected neural network to perform nonlinear transformations on water vapor pressure, wet delay coefficient and temperature, and extracts high-dimensional representations that are directly related to wet delay. The auxiliary feature layer uses the static delay coefficient as a supplementary input and extracts the feature representations that affect atmospheric density stratification and water vapor transport patterns by setting a selective activation mechanism. The first fused feature is obtained by fusing the high-dimensional characterization directly related to wet delay with the feature characterization that affects atmospheric density stratification and water vapor transport patterns. The background feature layer uses zenith static delay and atmospheric pressure as input to extract features indirectly related to the wet delay. After weight modulation, these features are added to the first fusion feature to form the final complete feature vector.

4. The tropospheric wet delay prediction method based on physical information neural network as described in claim 1, characterized in that, The dual-branch parallel architecture includes a data-driven branch and a physical constraint branch. The physical information neural network computation module, based on high-dimensional feature vectors, employs a dual-branch parallel architecture to fuse data-driven and physical constraint operations, including: The data-driven branch receives the complete feature vector from the hierarchical feature extraction module and learns the complex nonlinear mapping relationship between the input meteorological parameters and the tropospheric wet delay through the gradient descent algorithm. The physical constraint branch runs in parallel with the data-driven branch, transforming the wet refractive index formula into a differentiable function module that is directly embedded in the calculation, determining whether the model predictions satisfy the atmospheric refractive physical equations, and quantifying the physical equation residuals. The output of the data-driven branch and the physical constraint branch merge at the loss function fusion node.

5. The tropospheric wet delay prediction method based on physical information neural network as described in claim 1, characterized in that, The tropospheric wet delay prediction model was trained using a training dataset, including: Input the training dataset into the tropospheric wet delay prediction model; The data input module receives the training dataset, the hierarchical feature extraction module processes the features in the input meteorological observation data and fuses them to obtain a high-dimensional feature vector, and the physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The data-driven and physical constraints are fused using a dual-loss training process, which employs a dual-loss function mechanism, including data-driven loss and physical constraint loss. The data-driven loss is used to measure the fitting error between the network prediction value and the observation data, and the physical constraint loss is used to quantify the residual of the physical equation. Through iterative optimization, the weight coefficients are adjusted to make the overall loss less than a preset threshold until the model converges, and the trained model is saved.

6. The tropospheric wet delay prediction method based on physical information neural network as described in claim 5, characterized in that, The loss function is: in, To account for the overall losses, For data-driven loss, For physical constraint loss, These are the weighting coefficients. , Where n is the number of training samples, and i is the i-th training sample. For the ground truth of the i-th training sample, Let i be the neural network predictions for training samples. These are the predicted values ​​from the physical model.

7. The tropospheric wet delay prediction method based on physical information neural network as described in claim 1, characterized in that, The method further includes: The probability distribution of predicted values ​​is obtained through Monte Carlo sampling; Standardized output reports and visualizations are generated based on the model's predicted values ​​and their probability distribution.

8. A tropospheric wet delay prediction system based on a physical information neural network, characterized in that, include: The data acquisition and preprocessing module is used to acquire real-time meteorological observation data from GNSS stations, preprocess the acquired real-time meteorological observation data, and construct a training dataset. The real-time meteorological observation data includes water vapor pressure, temperature, atmospheric pressure, wet delay coefficient, static delay coefficient, and zenith static delay. The model building module is used to construct a tropospheric wet delay prediction model. It includes a data input module, a hierarchical feature extraction module, a physical information neural network calculation module, and an output module. The data input module receives input meteorological observation data. The hierarchical feature extraction module processes the features in the input meteorological observation data using a three-level cascaded structure and fuses them to obtain a high-dimensional feature vector. The physical information neural network calculation module uses a dual-branch parallel architecture based on the high-dimensional feature vector to fuse data-driven and physical constraints. The output module obtains the tropospheric wet delay prediction result based on the result of the physical information neural network calculation module. The model training module is used to train the tropospheric wet delay prediction model using the training dataset; The tropospheric wet delay prediction module is used to predict tropospheric wet delay using a trained tropospheric wet delay prediction model.

9. A computer-readable storage medium, characterized in that, It stores a computer program that, when executed by a processor, implements the tropospheric wet delay prediction method based on a physical information neural network as described in any one of claims 1 to 7.

10. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the tropospheric wet delay prediction method based on a physical information neural network as described in any one of claims 1 to 7.

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