A Method and System for Predicting Total Ionospheric Electron Content Based on Dual-Track Joint Prediction

Abstract

This invention belongs to the field of space weather forecasting technology and provides a method and system for predicting the total electron content of the ionosphere based on dual-track joint prediction. First, it integrates an extreme gradient boosting machine learning model with Shapley additive interpretation analysis to quantify the contribution of physical driving factors to spherical harmonic coefficients and construct an adaptive feature selection mechanism with physical interpretability. Then, it designs a prediction model that applies flow attention between spherical harmonic coefficients and spatiotemporal features, and achieves accurate prediction by fusing historical physical information with spherical harmonic coefficients. It fully considers the physical significance of low-order spherical harmonic coefficients and their cross-order correlation, and models low-order coefficients and other high-order coefficients separately using different prediction modes, further improving the prediction accuracy of the final total electron content of the ionosphere.

Description

Technical Field

[0001] This invention belongs to the field of space weather forecasting technology, and in particular relates to a method and system for predicting the total electron content of the ionosphere based on dual-track joint prediction. Background Technology

[0002] In ionospheric forecasting, grid forecasting and spherical harmonic coefficient forecasting are two of the most common forecasting paradigms. Grid forecasting directly extrapolates the global ionospheric map at future moments based on the historical total vertical electron content of global grid points; while spherical harmonic coefficient forecasting aims to fit the temporal patterns of historical spherical harmonic coefficients by predicting future coefficients and reconstructing them through integration at global grid points.

[0003] The grid-based point forecasting paradigm, which employs a point-by-point independent modeling approach, can preserve the high-dimensional spatial structure. However, as the dimensionality increases, it leads to the dimensionality explosion problem, necessitating complex dimensionality reduction or reduced sampling resolution to mitigate this issue. Ultimately, this results in the loss of grid information and limited accuracy. In contrast, the spherical harmonic coefficient forecasting paradigm achieves dimensionality compression. However, it treats the spherical harmonic coefficients as independent single variables or as a single vector for prediction. The former ignores the inherent physical correlation and coupling mechanism between coefficients of different orders, while the latter fails to highlight the physical significance of low-order coefficients. Consequently, the advantage of low dimensionality fails to translate into an advantage in accuracy, and the forecast accuracy is on par with or even worse than that of grid-based forecasting. Summary of the Invention

[0004] To address the aforementioned problems, this invention proposes a method and system for predicting the total electron content of the ionosphere based on dual-track joint prediction. First, this invention integrates an extreme gradient boosting machine learning model with Shapley additive interpretation analysis to quantify the contribution of physical driving factors to spherical harmonic coefficients. Then, it applies flow attention to the relationship between spherical harmonic coefficients and spatiotemporal features, fusing historical physical information with spherical harmonic coefficients to achieve accurate prediction. Furthermore, it fully considers the physical significance of low-order spherical harmonic coefficients and their cross-order correlations, employing different prediction modes to model low-order coefficients and other higher-order coefficients separately, further improving the prediction accuracy of the final total electron content of the ionosphere.

[0005] To achieve the above objectives, the present invention is implemented through the following technical solution: In a first aspect, the present invention provides a method for predicting the total electron content of the ionosphere based on dual-track joint prediction, comprising: Obtain space weather parameters, temporal characteristic parameters, and spherical harmonic coefficients; Based on the spatial weather parameters and the spherical harmonic coefficients, a pre-defined extreme gradient boosting machine learning model and Shapley additive interpretation analysis are used to determine the spatiotemporal characteristic parameters, which include spatial weather parameters and temporal characteristic parameters. For a predetermined number of spherical harmonic coefficients, a low-order prediction sequence is obtained using a predetermined first prediction model with the spherical harmonic coefficients and the spatiotemporal feature parameters as input. A second prediction model is then used with all spherical harmonic coefficients as input to obtain the remaining prediction sequences. The prediction results of the first prediction model are overlaid on the corresponding predictions of the second prediction model to obtain a low-order enhanced complete spherical harmonic coefficient prediction sequence. Then, a predicted global ionospheric total electron content map is generated through global grid integration. Both the first and second prediction models adopt the iTransformer architecture, replacing the multi-head self-attention mechanism in iTransformer with a flow attention mechanism for modeling spherical harmonic order coupling, applying flow attention between the spherical harmonic coefficients and spatiotemporal features.

[0006] Furthermore, the determination of the space weather parameters and the time feature parameters includes: dividing the original training set, which includes space weather parameters, time feature parameters, and spherical harmonic coefficients, into a sub-training set and a validation set; for each spherical harmonic coefficient, constructing an extreme gradient boosting machine learning model using the space weather parameter set and the time feature parameter set as inputs; performing model training and hyperparameter tuning using cross-validation on the sub-training set, and conducting Shapley additive interpretation analysis on the validation set, calculating the average absolute Shapley additive interpretation value of each feature on the validation set and performing global ranking, and selecting a preset number of space weather parameters and a preset number of time feature parameters that contribute the most to the model.

[0007] Furthermore, in the first and second prediction models, the attention mechanism is applied between different variables, and the feedforward neural network is used to learn the sequence representation of each variable at all time points; an invertible instance normalization module is added to handle non-stationary sequences.

[0008] Furthermore, the predictions of the first and second prediction models include: normalizing the data to determine the mean and variance over time; performing normalization with affine parameters based on the mean and variance to obtain a normalized matrix; transposing the normalized matrix and encoding the complete time series of each variable through an embedding layer; and inputting the encoded set into the prediction model.

[0009] Furthermore, after the encoded set is input into the prediction model, a flow attention mechanism and a feedforward neural network are executed sequentially, supplemented by residual connections and layer normalization. The flow attention mechanism first performs a linear projection on the input, generating a query matrix, a key matrix, and a value matrix from the input set through three independent linear transformations. Activation functions are applied to the query matrix and the key matrix. Based on the activated query matrix and the key matrix, the row normalization factor and column normalization factor corresponding to the source competition and sink allocation in the network are determined and refined. Information aggregation with linear complexity is achieved through matrix multiplication to obtain an output with the same shape as the input.

[0010] Furthermore, residual connections and normalization are performed on the output: ; in, For the first l -1 hidden dimension; , Let D be the sequence length and D be the hidden dimension; for Independent operation of feedforward neural networks: ; in, It is a row feedforward neural network; This is the weight matrix. For bias terms, Linear unit with Gaussian error: ; After executing the feedforward neural network, residual connections and layer normalization are then performed: ; Repeat the preset subordination to determine the final feature. The standardized length is S Predicted values: ; in, Let be the projection weight matrix. For bias terms; in the matrix For variables in the future S Standardized prediction sequence at time steps.

[0011] Furthermore, the refined row normalization factor Normalization factor for: ; ; in, This is a normalized weighted aggregate vector for queries; This is a column-normalized weighted key aggregation vector; For query matrix; The key matrix; For column normalization factors; The normalization factor is denoted as .

[0012] Secondly, the present invention also provides a system for predicting the total electron content of the ionosphere based on dual-track joint prediction, comprising: The data acquisition module is configured to acquire space weather parameters, temporal characteristic parameters, and spherical harmonic coefficients. The feature extraction module is configured to: determine spatiotemporal feature parameters, including space weather parameters and time feature parameters, based on the space weather parameter time feature parameters and the spherical harmonic coefficients, using a preset extreme gradient boosting machine learning model and Shapley additive interpretation analysis; The prediction module is configured to: for a preset number of spherical harmonic coefficients, use a preset first prediction model to obtain a low-order prediction sequence with the spherical harmonic coefficients and the spatiotemporal feature parameters as input; use a preset second prediction model to obtain the remaining prediction sequences with only all spherical harmonic coefficients as input; overwrite the corresponding prediction of the second prediction model with the prediction results of the first prediction model to obtain a low-order enhanced complete spherical harmonic coefficient prediction sequence, and then generate a predicted global ionospheric total electron content map through global grid integration; wherein, both the first prediction model and the second prediction model adopt the iTransformer architecture, replacing the multi-head self-attention mechanism in iTransformer with a flow attention mechanism for spherical harmonic order coupling modeling, and applying flow attention between spherical harmonic coefficients and spatiotemporal features.

[0013] Thirdly, the present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in the first aspect.

[0014] Fourthly, the present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor, wherein the processor executes the program to implement the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in the first aspect.

[0015] Fifthly, the present invention also provides a computer program product, the computer program product comprising a computer program, which, when executed by a processor, implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in the first aspect.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention first integrates an extreme gradient boosting machine learning model with Shapley additive interpretation analysis to quantify the contribution of physical driving factors to spherical harmonic coefficients and constructs a physically interpretable adaptive feature selection mechanism. Then, a prediction model is designed, applying flow attention between spherical harmonic coefficients and spatiotemporal features, and fusing historical physical information with spherical harmonic coefficients to achieve accurate prediction. Taking full account of the physical significance of low-order spherical harmonic coefficients and their cross-order correlation, different prediction modes are used to model low-order coefficients and other high-order coefficients separately, further improving the prediction accuracy of the final total electron content of the ionosphere. Attached Figure Description

[0017] The accompanying drawings, which form part of this embodiment, are used to provide a further understanding of this embodiment. The illustrative embodiments and their descriptions are used to explain this embodiment and do not constitute an improper limitation of this embodiment.

[0018] Figure 1 This is a flowchart of the method in Embodiment 1 of the present invention; Figure 2 This is a diagram of the SHCformer model architecture of Embodiment 1 of the present invention. Detailed Implementation

[0019] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0020] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0021] Example 1: Total Electron Content (TEC) is a key parameter describing the ionospheric delay effect. Detailed observation and in-depth analysis of TEC variations can quantitatively reveal the ionospheric variation patterns under different space environments and multi-scale spatiotemporal conditions. For radio waves of a fixed frequency, their propagation delay in the ionosphere is directly proportional to the TEC along the propagation path. Especially in Single-Frequency Precise Point Positioning (SF-PPP), the accuracy of TEC directly determines the correction effect of ionospheric delay errors. In recent years, with the rapid development of deep learning technology, its powerful nonlinear fitting capabilities have made it a research hotspot in ionospheric TEC prediction and analysis. However, current deep learning-based ionospheric TEC prediction models still face bottlenecks in modeling strategies, feature engineering, and computational efficiency that urgently need to be overcome.

[0022] In ionospheric TEC forecasting, grid-based forecasting and spherical harmonic coefficient (SHC) forecasting are two of the most common forecasting paradigms. Grid-based forecasting directly extrapolates the future global ionospheric map (GIM) based on the historical vertical total electron content (VTEC) of global grid points; while spherical harmonic coefficient forecasting aims to fit the temporal patterns of historical spherical harmonic coefficients (SHC), reconstructing the GIM by forecasting future coefficients and integrating them at global grid points. Given that ionospheric TEC is significantly modulated by solar and geomagnetic activity and has complex nonlinear spatiotemporal evolution characteristics, introducing multidimensional spatiotemporal auxiliary features has become a key approach to improving the model's generalization ability.

[0023] One approach involves constructing an LSTM model that integrates historical spherical harmonic coefficients, temporal features, and F10.7 / DST space weather parameters to predict future TEC (Temperature and Temperature) patterns. This model accurately captures structures such as equatorial ionization anomalies and storm enhancement density, but suffers from long-term forecast error accumulation. Another approach utilizes a pure Transformer encoder model, optimizing the multi-dimensional spatiotemporal input flattening method to achieve multi-step forecasts. Its accuracy is comparable to the Center for Orbit Determination in Europe (CODE)'s 1-day Predicted Global Ionospheric Map (C1PG). A further approach combines random forest feature quantization with an improved Transformer for feature fusion, selecting the five most important space weather parameters as additional inputs. This method significantly outperforms traditional models in years of high solar activity. Finally, a global ionospheric TEC forecasting method based on spatiotemporal feature fusion constructs physically meaningful day-night maps to represent temporal features. A multilayer perceptron maps one-dimensional physical parameters are mapped to two-dimensional features matching the TEC grid, and the input is fed into a separable 3D spatiotemporal convolutional network to achieve high-precision forecasts.

[0024] As described in the background section, existing ionospheric TEC forecasting paradigms suffer from a trade-off between high accuracy and low complexity at the modeling strategy level. The grid-based forecasting paradigm, employing independent point-by-point modeling, preserves the high-dimensional spatial structure, but suffers from dimensionality explosion as dimensions increase. This necessitates complex dimensionality reduction or lower sampling resolution to mitigate the problem, ultimately resulting in loss of grid information and limited accuracy. On the other hand, while the spherical harmonic coefficient forecasting paradigm achieves dimensionality compression, it treats spherical harmonic coefficients as either independent single variables or as a single vector for prediction. The former ignores the inherent physical correlation and coupling mechanism between coefficients of different orders, while the latter fails to highlight the physical significance of lower-order coefficients. Consequently, the advantage of low dimensionality fails to translate into an advantage in accuracy, and the forecast accuracy is sometimes on par with or even worse than that of grid-based forecasting.

[0025] In summary, neither of the existing forecasting paradigms has achieved a balance between dimensionality compression and information fidelity: the grid paradigm preserves information but sacrifices efficiency, while the spherical harmonic paradigm preserves efficiency but loses correlation. Existing spatiotemporal feature selection largely relies on empirical choices. Although some studies have begun to quantify the importance of features, they generally face the following limitations: First, there is a scale difference between physical information at the station or global scale and the high-resolution TEC grid, and direct correlation analysis inevitably leads to the loss of physical information; second, the selected combinations of physical information parameters are usually static and fixed, lacking the ability to dynamically adjust and adapt; third, feature selection for the TEC grid makes it difficult to intuitively explain the specific contribution of feature introduction to the forecast results, significantly weakening the interpretability of deep learning models in ionospheric physics research. At the same time, existing methods generally neglect the computational efficiency and operational potential of the models. For global TEC grid forecasting tasks, the currently used Transformer-based models typically take hours or even days to converge to the optimum and are highly dependent on expensive computing hardware. This not only fails to meet the low-latency requirements of near-real-time space weather warnings but also poses a serious challenge to the practical operational deployment of deep learning models.

[0026] To address at least one of the aforementioned problems, this embodiment provides a method for predicting total ionospheric electron content based on dual-track joint prediction. First, it integrates an Extreme Gradient Boosting (XGBoost) machine learning model with a Shapley Additive exPlanations (SHAP) attribution algorithm to quantify the contribution of physical driving factors to spherical harmonic coefficients and construct a physically interpretable adaptive feature selection mechanism. Then, it designs a lightweight prediction model, SHCformer, applying flow attention between spherical harmonic coefficients and spatiotemporal features, fusing historical physical information with spherical harmonic coefficients to achieve accurate prediction. This embodiment fully considers the physical significance of low-order spherical harmonic coefficients and their cross-order correlations, employing different prediction modes to model low-order and high-order coefficients separately, ultimately achieving accurate prediction of total ionospheric electron content.

[0027] Specifically, the top five spherical harmonic coefficients (which carry the most important physical energy and large-scale structure of global TEC) with significant magnitude and physical meaning are separated from the remaining coefficients and used for dual-track prediction using different prediction models. Considering the inherent physical correlation and coupling mechanism between coefficients of different orders, the physical significance of lower-order coefficients is highlighted. The XGBoost machine learning model and SHAP analysis are used to screen spatiotemporal features at the spherical harmonic coefficient level, thereby increasing the interpretability of external features for ionospheric TEC forecasts: the improvement in overall grid accuracy is attributed to the increase in the order of spherical harmonic coefficients. The designed lightweight SHCformer model allows for training time control within minutes on consumer-grade graphics cards, meeting the requirements of near-real-time space weather forecasting. In the typical task of 1-day forecasting, SHCformer outperforms deep learning models such as Transformer and iTransformer, with RMSE and MAE reductions of over 15% compared to C1PG in both low and high solar activity years, and over 20% in mid-to-high latitudes. Compared with direct multivariate prediction, the dual-track prediction strategy has comparable accuracy in years of low solar activity and significantly improved accuracy in years of high solar activity. In addition, the dual-track prediction strategy reduces the model error during geomagnetic storms in all periods.

[0028] like Figure 1 As shown, the detailed content of the method in this embodiment is as follows: S1. Data preprocessing and dataset partitioning: Space weather parameters and temporal characteristic parameters were collected. Optionally, the final spherical harmonic coefficient product (15th order, 256 spherical harmonic coefficients) provided by the European Orbit Determination Centre (CODE) was used for training and feature engineering. The final GIM and C1PG products were used as evaluation benchmarks, and the OMNI2 dataset provided by NASA's Space Physics Data Facility was used as the source of space weather parameters. To address the issue of missing values ​​in space weather parameters and ensure data quality, parameters with many missing values ​​and no obvious physical meaning were removed. Parameters with fewer missing values ​​were linearly interpolated to align them with the spherical harmonic coefficients on the time axis, ultimately constructing a space weather parameter set containing 38 solar geomagnetic indices. In addition, a temporal characteristic parameter set was constructed, selecting 8 temporal characteristic parameters closely related to TEC. Based on the solar activity intensity during the test period, training and test sets were divided by year, with no overlap between the training and test sets to prevent information leakage.

[0029] S2, Feature Engineering: The original training set was further divided into a sub-training set and a validation set. For each of the first five spherical harmonic coefficients, an XGBoost machine learning model was constructed using the space weather parameter set and the temporal feature parameter set as inputs, with the corresponding spherical harmonic coefficient as the model output. Five-fold cross-validation was used for model training and hyperparameter tuning on the sub-training set, and SHAP analysis was performed on the validation set. The average absolute SHAP value of each feature on the validation set was calculated and globally ranked. Based on this, the five space weather parameters and five temporal feature parameters that contributed the most to the model were selected.

[0030] S3, Deep Learning Model SHCformer: like Figure 2 As shown, the deep learning model SHCformer largely adopts the overall architecture of iTransformer, applying the attention mechanism to different variables, while the feedforward neural network (FFN) is dedicated to learning the sequence representation of each variable at all time points. The overall structure includes an embedding layer and L stackable Transformer encoder blocks. Specifically, to address the non-stationarity of spherical harmonic coefficient sequences, a Reversible Instance Normalization (ReVIN) module is added to handle non-stationary sequences. The multi-head self-attention mechanism in the iTransformer module is replaced with a flowing attention mechanism suitable for modeling the coupling of spherical harmonic orders. This mechanism achieves variable dependency modeling through source competition and sink allocation, and its linear complexity is compatible with the physical coupling characteristics between spherical harmonic orders, avoiding the quadratic complexity and attention degradation problems of standard attention in long sequences. The overall data flow predicted by the model is given below.

[0031] Let the input historical multivariate sequence be... ,in, T For historical time steps, N The number of variables. First, perform ReVIN forward normalization, and then... n Variables Calculate its mean over the time dimension. and variance : ; ; in, ε This is a very small constant. Then, normalization with affine parameters is performed: ; in, These are learnable parameters. Initialization. The normalized matrix is ​​obtained. Transpose the normalized matrix, inverting the time and variable dimensions: ; The complete time series of each variable is then encoded into a token using an embedding layer (MLP, Multilayer Perceptron): ; Obtain the Token set, where, , D To hide the dimension. Let the first dimension be... l The input of each TrmBlock is ,in Each TrmBlock sequentially executes the flow attention mechanism and the feedforward neural network, supplemented by residual connections and layer normalization. The flow attention mechanism first performs a linear projection on the input: the input... The query matrix is ​​generated through three independent linear transformations. Key matrix Sum matrix For each attention point: ; in, , For each attention head, to ensure the non-negativity of the streaming network capacity, the query matrix... Bond matrix Apply the Sigmoid activation function: ; in, Calculate the row normalization factor Normalization factor These correspond to source contention and sink allocation in a flow network, respectively: ; ; in, To prevent division by zero by extremely small constants, the denominator in the normalization factor formula is... With global keys and vectors The dot product, the denominator in the normalization factor formula is With global queries and vectors The dot product. Further calculation of the refined row normalization factor. Normalization factor : ; ; Inner Summation The inner layer summation is a column-normalized weighted key aggregation vector. This is a query aggregation vector that has undergone normalization and weighting. The value matrix is ​​reweighted using a refined column normalization factor, and information aggregation with linear complexity is achieved through matrix multiplication. ; ; ; in, This represents element-wise multiplication. and Used for broadcasting The last dimension. The computational complexity is , with sequence length The relationship is linear. (All) The outputs of each size are concatenated along the feature dimensions to obtain the result that is the same as the input. Output of the same shape :

[0032] ; Output is processed via residual connection and normalization: ; right Each of them Execute FFN independently: ; in, This is the weight matrix. For bias terms, The Gaussian Error Linear Unit (GLU) has the following approximate form: ; After executing FFN, residual joins and layer normalization are performed: ; Repeat the above process L Second, final feature Input it into a fully connected projection layer to generate a normalized length of... S Predicted values: ; in, Let be the projection weight matrix. This is the bias term. In the matrix... Representing the n Variables in the future S Standardized prediction sequences for each time step. Ensure alignment with the input sequence dimensions. Take the transpose to get To further restore to the original scale, the statistics stored in the corresponding formula are used. and learnable parameters Perform an inverse transformation on the prediction result. Specifically, for the first... n The variable in the first... S The predicted value at each time step is calculated using the following formula: ; in, This is the square root of the variance. Combining the results of all variables yields the final prediction matrix with physical dimensions. The model uses mean squared error as the loss function during training, employs the Adaptive Moment Estimation (Adam) optimizer, and sets an early stopping strategy during training.

[0033] S4, Dual-track prediction strategy: For each of the first five spherical harmonics, a dedicated SHCformer model (first prediction model) is trained. The SHCformer model takes all spherical harmonics and previously selected spatiotemporal feature parameters corresponding to that coefficient as input, and outputs that specific spherical harmonic (i.e., the loss function is the error of that specific spherical harmonic). Simultaneously, a shared SHCformer model (second prediction model) is trained, taking only all spherical harmonics as input and outputting all spherical harmonics at once (the loss function is the error of all spherical harmonics). The prediction results of the dedicated SHCformer model are overlaid with the corresponding predictions of the shared SHCformer model to obtain a low-order enhanced complete spherical harmonic prediction sequence. Then, through global grid integration, a predicted global ionospheric total electron content map is generated.

[0034] This embodiment utilizes a deep learning model to perform dual-track joint prediction of spherical harmonic coefficients of different orders, solving the problems of excessive computational complexity, insufficient accuracy, and lack of interpretability under a single forecast paradigm, thereby improving the accuracy and computational efficiency of global TEC forecasts.

[0035] At the spherical harmonic coefficient order level, using XGBoost and SHAP attribution algorithms, for the top five coefficients that carry the large-scale structure of the global TEC, the spatiotemporal feature subsets with the strongest coupling are independently selected to realize the quantitative contribution analysis of physical driving factors and adaptive feature input.

[0036] Based on the iTransformer dimension inversion strategy, Reversible Instance Normalization (ReVIN) is introduced to handle the non-stationarity of spherical harmonic coefficient sequences. The traditional multi-head self-attention is replaced with a linearly complex flow attention mechanism. While maintaining the ability to model global dependencies, the computational overhead is significantly reduced, enabling minute-level training on consumer-grade GPUs and meeting the needs of real-time business applications.

[0037] For the first five spherical harmonic coefficients that support large-scale structures, dedicated SHCformer models are trained separately. Each model takes all spherical harmonic coefficients and spatiotemporal features specifically selected for that coefficient as input to accurately capture the evolutionary patterns of large-scale structures. For the remaining mid-to-high-order spherical harmonic coefficients, a shared SHCformer model is trained, taking only all spherical harmonic coefficients as input and outputting all remaining coefficients at once. The prediction results from the two orbitals are fused to reconstruct the complete spherical harmonic coefficient sequence, and integration is used to generate a global ionospheric map.

[0038] In another embodiment, using spherical harmonic coefficients as the object, the principal period and residual period of the coefficients are separated by Fast Fourier Transform. The principal period is incorporated into a trend function, and the residual period is incorporated into a random signal to construct a semi-parametric model. The trend function is fitted using kernel estimation, and the random signal is solved before extrapolation in the time domain to obtain the spherical harmonic coefficients at the forecast time. These coefficients are then substituted into the spherical harmonic model to output the global TEC grid value. A semi-parametric rule-based ionospheric TEC forecasting technique is also employed. This technique constructs a set of periodic feature rules and constraints based on prior ionospheric knowledge. Periodic terms of the spherical harmonic coefficients are extracted through rule learning. A self-attention mechanism and pruning layers are introduced to optimize feature weights and remove redundant periods. The estimated periodic terms are combined with a window width parameter to construct a semi-parametric variable-coefficient spherical harmonic model. After solving the model, the TEC forecast value is extrapolated in the time domain.

[0039] Example 2: This embodiment provides a system for predicting the total electron content of the ionosphere based on dual-track joint prediction, including: The data acquisition module is configured to acquire space weather parameters, temporal characteristic parameters, and spherical harmonic coefficients. The feature extraction module is configured to: determine spatiotemporal feature parameters, including space weather parameters and time feature parameters, based on the space weather parameter time feature parameters and the spherical harmonic coefficients, using a preset extreme gradient boosting machine learning model and Shapley additive interpretation analysis; The prediction module is configured to: for a preset number of spherical harmonic coefficients, use a preset first prediction model to obtain a low-order prediction sequence with the spherical harmonic coefficients and the spatiotemporal feature parameters as input; use a preset second prediction model to obtain the remaining prediction sequences with only all spherical harmonic coefficients as input; overwrite the corresponding prediction of the second prediction model with the prediction results of the first prediction model to obtain a low-order enhanced complete spherical harmonic coefficient prediction sequence, and then generate a predicted global ionospheric total electron content map through global grid integration; wherein, both the first prediction model and the second prediction model adopt the iTransformer architecture, replacing the multi-head self-attention mechanism in iTransformer with a flow attention mechanism for spherical harmonic order coupling modeling, and applying flow attention between spherical harmonic coefficients and spatiotemporal features.

[0040] The working method of the system is the same as that of the ionospheric total electron content prediction method based on dual-track joint prediction in Example 1, and will not be repeated here.

[0041] Example 3: This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in Embodiment 1.

[0042] Example 4: This embodiment provides an electronic device, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor. When the processor executes the program, it implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in Embodiment 1.

[0043] Example 5: This embodiment provides a computer program product, which includes a computer program. When the computer program is executed by a processor, it implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in Embodiment 1.

[0044] The above description is merely a preferred embodiment of this practice and is not intended to limit the scope of this practice. Various modifications and variations can be made to this practice by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this practice should be included within the protection scope of this practice.

Claims

1. A method for predicting the total electron content of the ionosphere based on dual-track joint prediction, characterized in that, include: Acquire space weather parameters, temporal characteristic parameters, and spherical harmonic coefficients; Based on the spatial weather parameters and the spherical harmonic coefficients, a pre-defined extreme gradient boosting machine learning model and Shapley additive interpretation analysis are used to determine the spatiotemporal characteristic parameters, which include spatial weather parameters and temporal characteristic parameters. For a predetermined number of spherical harmonic coefficients, a low-order prediction sequence is obtained using a predetermined first prediction model with the spherical harmonic coefficients and the spatiotemporal feature parameters as input. A second prediction model is then used with all spherical harmonic coefficients as input to obtain the remaining prediction sequences. The prediction results of the first prediction model are overlaid on the corresponding predictions of the second prediction model to obtain a low-order enhanced complete spherical harmonic coefficient prediction sequence. Then, a predicted global ionospheric total electron content map is generated through global grid integration. Both the first and second prediction models adopt the iTransformer architecture, replacing the multi-head self-attention mechanism in iTransformer with a flow attention mechanism for modeling spherical harmonic order coupling, applying flow attention between the spherical harmonic coefficients and spatiotemporal features.

2. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 1, characterized in that, The determination of the space weather parameters and the time feature parameters includes: dividing the original training set, which includes space weather parameters, time feature parameters, and spherical harmonic coefficients, into a sub-training set and a validation set; for each spherical harmonic coefficient, constructing an extreme gradient boosting machine learning model using the space weather parameter set and the time feature parameter set as inputs; performing model training and hyperparameter tuning using cross-validation on the sub-training set, and conducting Shapley additive interpretation analysis on the validation set, calculating the average absolute Shapley additive interpretation value of each feature on the validation set and performing global ranking, and selecting a preset number of space weather parameters and a preset number of time feature parameters that contribute the most to the model.

3. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 1, characterized in that, In the first and second prediction models, the attention mechanism is applied between different variables, and the feedforward neural network is used to learn the sequence representation of each variable at all time points; an invertible instance normalization module is added to handle non-stationary sequences.

4. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 1, characterized in that, The predictions of the first and second prediction models include: normalizing the data to determine the mean and variance over time; performing normalization with affine parameters based on the mean and variance to obtain a normalized matrix; transposing the normalized matrix and encoding the complete time series of each variable through an embedding layer; and inputting the encoded set into the prediction model.

5. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 4, characterized in that, After the encoded set is input into the prediction model, the flow attention mechanism and the feedforward neural network are executed sequentially, supplemented by residual connections and layer normalization. The flow attention mechanism first performs a linear projection on the input, generating a query matrix, a key matrix, and a value matrix from the input set through three independent linear transformations. Activation functions are applied to the query matrix and the key matrix. Based on the activated query matrix and the key matrix, the row normalization factor and column normalization factor corresponding to the source competition and sink allocation in the network are determined and refined. Information aggregation with linear complexity is achieved through matrix multiplication to obtain an output with the same shape as the input.

6. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 5, characterized in that, Perform residual connection and normalization on the output: ； in, For the first l -1 hidden dimension; , Let D be the sequence length and D be the hidden dimension; for Independent operation of feedforward neural networks: ； in, It is a row feedforward neural network; This is the weight matrix. For bias terms, Linear unit with Gaussian error: ； After executing the feedforward neural network, residual connections and layer normalization are then performed: ； Repeat the preset subordination to determine the final feature. The standardized length is S Predicted values: ； in, The projection weight matrix is... For bias terms; in the matrix For variables in the future S Standardized prediction sequence at time steps.

7. The method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in claim 5, characterized in that, Refined normalization factor Normalization factor for: ； ； in, This is a normalized weighted aggregate vector for queries; This is a column-normalized weighted key aggregation vector; For query matrix; The key matrix; For column normalization factors; The normalization factor is denoted as .

8. A prediction system for total ionospheric electron content based on dual-track joint prediction, characterized in that, include: The data acquisition module is configured to acquire space weather parameters, temporal characteristic parameters, and spherical harmonic coefficients. The feature extraction module is configured to: determine spatiotemporal feature parameters, including space weather parameters and time feature parameters, based on the space weather parameter time feature parameters and the spherical harmonic coefficients, using a preset extreme gradient boosting machine learning model and Shapley additive interpretation analysis; The prediction module is configured to: for a preset number of spherical harmonic coefficients, use a preset first prediction model to obtain a low-order prediction sequence with the spherical harmonic coefficients and the spatiotemporal feature parameters as input; use a preset second prediction model to obtain the remaining prediction sequences with only all spherical harmonic coefficients as input; overwrite the corresponding prediction of the second prediction model with the prediction result of the first prediction model to obtain a low-order enhanced complete spherical harmonic coefficient prediction sequence, and then generate a predicted global ionospheric total electron content map through global grid integration; wherein, both the first prediction model and the second prediction model adopt the iTransformer architecture, replacing the multi-head self-attention mechanism in iTransformer with a flow attention mechanism for spherical harmonic order coupling modeling, and applying flow attention between spherical harmonic coefficients and spatiotemporal features.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that, When the processor executes the program, it implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in any one of claims 1-7.

10. A computer program product, characterized in that, The computer program product includes a computer program that, when executed by a processor, implements the steps of the method for predicting the total electron content of the ionosphere based on dual-track joint prediction as described in any one of claims 1-7.

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