Polar shift forecasting method and system based on least square and Transform combined model

Abstract

The invention discloses a polar shift forecasting method based on a least square and Transform combined model, which integrates the advantages of a linear LS model and a deep learning Transform model, and aims to improve the forecasting precision and reliability of polar shift parameters by cooperating with the complementary characteristics of the two methods. The forecasting method comprises the steps of obtaining of a polar shift parameter basic sequence, data preprocessing, LS linear model construction, Transform model parameter training, forecasting result integration and precision analysis. The LS + Transform combined model provided by the invention can effectively make up for the defect that the prediction precision of a traditional linear model on the polar shift parameter random component is insufficient, and has higher prediction precision and stability compared with a single model through effective extrapolation of the LS model on the trend component and deep mining and prediction of the Transform model on the polar shift random sequence feature. According to the method, medium-and-long-term high-precision forecasting of the polar shift parameters is realized, and a new research method is provided for polar shift parameter forecasting.

Description

Technical Field

[0001] This invention belongs to the fields of Earth rotation variation prediction technology and geophysics technology, specifically involving a high-precision polar motion prediction method based on a combination of least squares (LS) and a Transformer time series prediction model. This method can adapt to different observational data and application scenarios, and by optimizing model parameter configuration and combination strategies, it can meet diverse polar motion parameter prediction needs. Background Technology

[0002] Improving the forecast accuracy of polar motion parameters is of great significance for scientific research and engineering applications. At the scientific research level, polar motion, as a comprehensive response to internal and external dynamic processes (such as mantle convection and atmospheric-oceanic angular momentum exchange), provides crucial evidence for analyzing the Earth's internal structure and quantifying the mass migration effects of surface fluids when accurately predicted. In practical applications, polar motion parameters, as a core component of Earth Orientation Parameters (EOP), directly affect the accuracy of satellite navigation and positioning and the stability of the International Earth Reference Frame (ITRF), playing an irreplaceable role in improving the reliability of navigation systems and optimizing astronomical observation benchmarks.

[0003] In recent years, the limitations of traditional polar motion prediction methods have become increasingly apparent: physical models based on geophysical hydrodynamic equations, while capable of reflecting the core excitation mechanisms, suffer from high computational complexity, weak ability to quantify local spatiotemporal heterogeneity, and difficulties in real-time fusion of multi-source observational data; convolutional neural networks, while possessing strong nonlinear fitting capabilities, have limited the effective capture of long-range temporal correlations in polar motion sequences due to their local receptive field characteristics. In contrast, the Transformer model, with its multi-head self-attention mechanism, exhibits superior temporal modeling capabilities. It can efficiently capture the complex correlation between long-term and short-term fluctuations in polar motion through parallel computation, flexibly integrate the characteristics of multi-source physical excitation factors, and effectively overcome the limitations of traditional time-series models in balancing long-term and short-term dependencies, providing stronger modeling potential for analyzing the nonlinear relationship between polar motion parameters and geophysical processes.

[0004] However, the Transformer model still has certain limitations in polar motion prediction: the accuracy of its attention mechanism in capturing local high-frequency subtle features is limited by the granular design of the attention head; and the model training is highly dependent on high-quality, long-term polar motion observation data, and its generalization ability is easily affected in scenarios with sparse data or strong noise; existing research has not yet built an efficient framework that takes into account both linear trend analysis and nonlinear time series modeling, which is difficult to meet the stringent requirements of high-precision application scenarios for polar motion prediction. Summary of the Invention

[0005] To overcome the shortcomings of the prior art, this invention provides a polar motion prediction method based on a combination of least squares and Transformer models. By leveraging the stability of collaborative linear modeling and the nonlinear representation capability of deep time series learning, a synergistic leap in polar motion prediction accuracy and generalization ability is achieved.

[0006] According to one aspect of the present invention, a polar motion prediction method based on a combined least squares and Transformer model is provided, comprising: Acquire polar motion observation data and perform multi-dimensional preprocessing on it; The preprocessed polar motion observation data is input into the trained LS+Transformer combined model, and the extrapolated sequences of the trend and periodic terms of the LS linear model and the residual prediction sequences of the Transformer model are added with equal weights to obtain the final polar motion parameter prediction values; wherein, the training of the LS+Transformer combined model includes: Acquire historical polar motion observation data and perform multi-dimensional preprocessing on it; Feature analysis was performed on the preprocessed historical polar motion observation data to identify and separate the linear trend components and nonlinear fluctuation components. Construct an LS+Transformer combined model, where the LS linear model is used for predicting the trend and periodic terms, and the Transformer model is used for predicting the residuals; The linear features in historical polar motion observation data are input into the constructed LS linear model, which is solved by the least squares method to output the predicted values ​​of the trend and periodic terms for future periods. The nonlinear features in historical polar motion observation data are input into the trained Transformer model, which outputs the residual prediction values ​​for future periods.

[0007] As a further technical solution, the multi-dimensional preprocessing includes: time consistency verification, handling of missing and outlier values, observation noise suppression, and normalization processing.

[0008] As a further technical solution, feature analysis is performed on the preprocessed historical polar motion observation data, including: Long-term trend components in polar shift sequences are identified by linear fitting, and significant periodic components are extracted based on spectral analysis. Subtracting the identified long-term trend component and the extracted significant periodic component from the original polar shift sequence yields a residual sequence that reflects nonlinear fluctuations.

[0009] As a further technical solution, the LS linear model is constructed according to the following formula: , in, For constant terms, The linear trend coefficient is... Let i be the period of the i-th periodic term. , The amplitude and phase coefficients of the periodic term, This is the fitting residual.

[0010] As a further technical solution, the training of the Transformer model includes: Obtain the polar shift residual sequence; A Transformer model is constructed with polar-shift residual sequence segments as input and future residual sequence segments to be predicted as the target. The Transformer model first preprocesses the input sequence and then embeds positional codes to preserve temporal positional information. The encoder consists of multiple stacked encoder units. Each unit captures the long-range temporal dependencies within the input sequence through a multi-head self-attention mechanism. After layer normalization and feedforward neural network processing, the encoded features are output. The decoder adopts multiple decoder units. Each layer first models the internal dependencies of the target sequence through masked multi-head self-attention, and then fuses the historical information output by the encoder through encoder-decoder attention. After layer normalization and feedforward neural network processing, the decoder result is mapped to the predicted value through a linear output layer. Using the error between the predicted and actual values ​​as the loss function, the model parameters are iteratively optimized through the backpropagation algorithm to output the trained Transformer model.

[0011] As a further technical solution, the extrapolated sequences of the trend and periodic terms of the LS linear model in the combined model and the residual prediction sequences of the Transformer model are added with equal weights to obtain the final predicted values ​​of the polar shift parameters, including: The terms and periodic terms extrapolated sequences of the LS linear model are added together with the residual prediction sequences of the Transformer model with equal weights to obtain the final predicted values ​​of the polar shift parameters.

[0012] As a further technical solution, the method also includes: The accuracy of the polar motion parameter predictions was statistically analyzed using the mean absolute error (MAE) and compared with the accuracy of the prediction sequence Bulletin A provided by the International Earth Rotation and Reference System Service (IERS). The effectiveness and reliability of the LS+Transformer combined model in improving the accuracy of polar motion predictions were verified through statistical significance testing.

[0013] According to one aspect of the present invention, a polar motion prediction system based on a combined least squares and Transformer model is provided, comprising: The input and preprocessing module is used to acquire polar motion observation data and perform multi-dimensional preprocessing on it. The polar motion prediction module is used to input preprocessed polar motion observation data into the trained LS+Transformer combined model, and to add the extrapolated sequences of the trend and periodic terms of the LS linear model and the residual prediction sequences of the Transformer model with equal weights to obtain the final polar motion parameter prediction values; wherein, the training of the LS+Transformer combined model includes: Acquire historical polar motion observation data and perform multi-dimensional preprocessing on it; Feature analysis was performed on the preprocessed historical polar motion observation data to identify and separate the linear trend components and nonlinear fluctuation components. Construct an LS+Transformer combined model, in which the LS linear model is used for predicting the trend and periodic terms, and the Transformer model is used for predicting the residuals; The linear features in historical polar motion observation data are input into the constructed LS linear model, which is solved by the least squares method to output the predicted values ​​of the trend and periodic terms for future periods. The nonlinear features in historical polar motion observation data are input into the trained Transformer model, which outputs the residual prediction values ​​for future periods.

[0014] According to one aspect of the present invention, a polar motion prediction device based on a least squares and Transformer combined model is provided, comprising a memory and a processor, wherein the memory stores program instructions that are executed by the processor, and the processor invokes the program instructions to execute the polar motion prediction method based on the least squares and Transformer combined model.

[0015] According to one aspect of the present invention, a non-transitory computer-readable storage medium is provided, the non-transitory computer-readable storage medium storing computer instructions that cause the computer to execute the polar motion prediction method based on a least squares and Transformer combined model.

[0016] This invention constructs a polar motion prediction framework based on a combination of least squares and Transformer models, fully leveraging the synergistic advantages of linear and deep time-series models to achieve high-precision prediction of polar motion parameters. Compared with existing technologies, the advantages of this invention are: Improving forecast accuracy: The Transformer model, with its multi-head self-attention mechanism, can accurately capture the long-range temporal correlation and nonlinear fluctuation characteristics of polar motion sequences. It complements the LS linear model in its stable fitting of trend periodic components, overcomes the limitations of traditional single models in nonlinear modeling, and significantly improves the forecast accuracy of polar motion parameters.

[0017] Improving forecast efficiency: Leveraging the parallel computing paradigm of Transformer, its multi-head self-attention mechanism can simultaneously extract features and learn associations from polar motion residual data, avoiding redundancy in local feature calculations; combined with the efficient fitting characteristics of least squares method for trend periodic components. The two work synergistically to achieve rapid output of polar motion parameters while ensuring forecast accuracy, significantly reducing the computational overhead of large-scale time series data processing.

[0018] Excellent adaptability and generalization ability: By optimizing hyperparameters such as the number of attention heads and sequence window length, the model can flexibly adapt to polar motion observation data at different time scales; with the help of strategies such as wavelet denoising and early stopping mechanism, it can still maintain stable performance in scenarios where there are differences in data noise level or sparsity, and the standard deviation of generalization error is controlled within 1mas.

[0019] Significant engineering application value: This method can be integrated into the polar motion correction module of a satellite navigation system, providing key inputs for dynamic optimization of the reference frame through high-precision polar motion parameter prediction, effectively improving the benchmark consistency and reliability of navigation and positioning, and meeting the application needs of high-precision surveying, deep space exploration and other fields.

[0020] In summary, this invention provides an innovative polar motion parameter prediction scheme that integrates linear modeling and deep temporal learning, which has both scientific research value and practical application prospects. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the accompanying 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.

[0022] Figure 1 This is a schematic diagram of the polar motion prediction method based on the least squares and Transformer combined model involved in the examples of the present invention; Figure 2 This is a schematic diagram of LS linear fitting and extrapolation (PMX) involved in an example of the present invention; Figure 3 This is a schematic diagram of LS linear fitting and extrapolation (PMY) involved in an example of the present invention; Figure 4This is a schematic diagram of the polar-shift PMX residual sequence and prediction sequence involved in the examples of this invention; Figure 5 This is a schematic diagram of the polar-shift PMY residual sequence and prediction sequence involved in the examples of this invention; Figure 6 This is a schematic diagram of the polar motion prediction results (PMX) of the LS+Transformer combined model involved in the example of the present invention; Figure 7 This is a schematic diagram of the polar motion prediction (PMY) results of the LS+Transformer combined model involved in the example of the present invention. Detailed Implementation

[0023] The terms “comprising” and “having”, and any variations thereof, in the specification, claims, and accompanying drawings of this invention are intended to cover a non-exclusive inclusion, such as a process, method, system, product, or apparatus that includes a series of steps or units, not necessarily limited to those explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0024] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. In addition, the technical features of the various embodiments or individual embodiments provided by the present invention can be arbitrarily combined to form new technical solutions. Such combinations are not bound by the order of steps and / or structural composition patterns, but must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0025] This invention provides a polar motion prediction method based on a combined least squares and Transformer model, such as... Figure 1 As shown, firstly, polar motion observation data is acquired and preprocessed in multiple dimensions; then, the preprocessed polar motion observation data is input into the trained LS+Transformer combined model, and the extrapolated sequences of the trend and periodic terms of the LS linear model and the residual prediction sequences of the Transformer model are added together with equal weights to obtain the final polar motion parameter prediction values.

[0026] The training of the LS+Transformer combined model includes: Data acquisition: Obtain historical polar motion observation sequences (PM); Data preprocessing: Standardization transformation is performed on multi-source observation data to eliminate differences in dimensions and numerical scale bias. Time series interpolation algorithm is used to unify the time base of multi-source data. Outliers are processed by sliding window detection and correction algorithm. Wavelet threshold denoising technology is combined to suppress observation noise interference and ensure the integrity of input sequence and data quality. Polar shift sequence feature separation: Based on time series analysis methods, the polar shift parameter (PM) is decomposed into linear trend components and nonlinear residual components; Constructing an LS linear model: Fitting historical polar shift data using the least squares method to extract linear features including long-term trends and periodic changes; Constructing a Transformer model: Using the historical polar motion residual sequence as input and the polar motion residual prediction sequence as output, construct and train this deep learning network to learn nonlinear relationships; Combined prediction: The polar shift residual prediction sequences of the LS linear model and the Transformer model are combined and linearly added to obtain the final polar shift parameter prediction value.

[0027] The following details the implementation process of the method described in this invention: Step 1: Polar Motion Observation Data Acquisition. The system collects historical polar motion observation data (PM), covering the X component (PMX) and Y component (PMY) of polar motion, forming a complete time-series dataset of polar motion parameters. The data source is preferentially selected from high-precision polar motion sequences published by the International Earth Rotation and Reference Systems Service (IERS) to ensure that the data has a globally unified observation benchmark and time scale. The sampling interval is 1 day to balance data resolution and computational efficiency.

[0028] Step 2: Deep data preprocessing. Multi-dimensional preprocessing is performed on the collected polar shift time series data to eliminate noise interference and improve data quality. Before model construction, time interpolation is first performed on the polar shift sequences to ensure consistent sampling times. Then, feature integration is performed on the interpolated polar shift data to form a unified polar shift time series matrix. Simultaneously, outlier correction and denoising are performed on the sequences to ensure the temporal consistency and integrity of the input data. Specifically, this includes: Step 2-1: Temporal consistency verification: An interpolation algorithm is used to unify the collected polar motion time series data to the same time intervals as the PM series provided by IERS. Since observation data from different sources or at different times may have inconsistent time bases, temporal consistency verification ensures that all data remain synchronized in the time dimension, providing a unified time scale for subsequent analysis and modeling; Missing value handling: Carefully check the data for missing values ​​(NaN values). For any missing values ​​found, use appropriate interpolation methods to impute them. The presence of missing values ​​can affect the model's training performance and prediction accuracy; therefore, using appropriate imputation methods ensures data integrity. Outlier handling: Outliers in the data are detected using machine learning algorithms. They are corrected or removed as needed to prevent them from interfering with model learning and to ensure data reliability.

[0029] Outlier detection and correction, and missing value imputation are performed simultaneously to ensure that the preprocessed data matches the original PM sequence in terms of time scale and numerical integrity.

[0030] Step 2-2: Noise suppression: Methods such as wavelet thresholding denoising are used to suppress observational noise interference. Polar motion observation data may be affected by various noises during acquisition; noise suppression improves the signal-to-noise ratio and enhances data quality. Normalization: The data processed above is normalized to ensure it falls within a suitable numerical range. This eliminates dimensional differences and numerical scaling biases, facilitating model learning and analysis, and improving model convergence speed and prediction accuracy.

[0031] Step 3: Polar Motion Sequence Feature Extraction. Feature analysis is performed on the preprocessed polar motion time series data to identify and separate the linear trend component and the nonlinear fluctuation component, i.e., to identify linear and nonlinear features. This separation process uses least squares fitting to extract long-term trends and periodic terms, establishing a quantitative correlation between linear and nonlinear features in polar motion observations, providing a data foundation for subsequent model input feature construction. Specifically, this includes: Step 3-1: Identify the long-term trend components in the polar shift sequence by linear fitting, and extract the significant periodic components in the polar shift sequence based on spectral analysis.

[0032] Step 3-2: Subtract the identified trend component and the extracted periodic component from the original polar shift sequence to obtain the residual sequence that reflects nonlinear fluctuations, thereby separating linear and nonlinear features and laying a reliable foundation for subsequent model analysis.

[0033] Step 4: Construction of the LS linear model, which is constructed according to the following formula: (1), in, For constant terms, The linear trend coefficient is... The period of the i-th periodic term (e.g.) Year), , The amplitude and phase coefficients of the periodic term, This is the fitting residual.

[0034] The aforementioned LS linear model is constructed using linear feature segments (including trend and periodic terms) of historical polar shift sequences as input and predicted linear feature values ​​for future periods as the objective. The parameters are then solved using the least squares method to achieve optimal fitting of the polar shift linear features. The fitting results are as follows: Figure 2 and Figure 3 As shown by the solid red line in the middle. Extrapolating the polar-shift linear characteristic term using the constructed LS linear model, the extrapolation result is as follows: Figure 2 and Figure 3 As shown by the solid blue line.

[0035] Step 5: Constructing the Transformer model. Using the nonlinear residual segments of the historical polar shift sequence as input features and the predicted residual values ​​for future periods as the target variable, a Transformer time series prediction model is constructed. Finally, the predicted residual sequence is generated through a linear output layer. During training, the mean squared error (MSE) is used as the loss function, and the model parameters are iteratively optimized using the Adam optimizer. Predicting polar shift residual sequences using the Transformer model includes the following steps: Step 5-1: Data Loading and Preprocessing: Load the polar shift residual sequence data, carefully check for NaN values, outliers, etc., and impute NaN values ​​using appropriate interpolation methods. Outliers can be processed using statistical methods or machine learning algorithms. Then, normalize the data to ensure it falls within a suitable numerical range, facilitating model learning and analysis.

[0036] Step 5-2: Construction of the Transformer Model: Taking the polar shift residual sequence segment as input and the future residual sequence segment to be predicted as the target, the input sequence is first preprocessed by standardization, and then positional encoding is embedded to retain temporal positional information. The encoder consists of multiple stacked encoder units. Each unit captures the long-range temporal dependencies within the input sequence through a multi-head self-attention mechanism. After layer normalization and feedforward neural network processing, the encoded features are output. The decoder uses multiple decoder units. Each layer first models the internal dependencies of the target sequence through masked multi-head self-attention (to avoid leaking future information), and then fuses the historical information output by the encoder through encoder-decoder attention. After layer normalization and feedforward neural network processing, the decoder result is finally mapped to the predicted value through a linear output layer. During training, the error between the predicted value and the true value (MSE) is used as the loss function to optimize the model parameters.

[0037] Step 5-3: Model Training: Input the preprocessed polar-shift residual sequence training set into the constructed Transformer model. Use the error between the predicted and true values ​​(root mean square error) as the loss function, and iteratively optimize the weight parameters of modules such as the multi-attention mechanism and feedforward network through the backpropagation algorithm. In the polar-shift residual sequence prediction process, constructing an adapted deep learning model (Transformer) is crucial, and model parameter training is the core step. The parameters are iteratively optimized using the Adam optimizer, and the model evaluation uses mean absolute error (MAE) and root mean square error (RMSE) to quantify the prediction performance.

[0038] Step 5-4: Model Prediction: Load the trained and converged Transformer model, input the time series test set data, and use the model's encoder-decoder forward computation process (including multi-head self-attention feature interactions) to output the future time series prediction results for the test set, such as... Figure 4 and 5 As shown.

[0039] Step 5-5: Model Evaluation: Compare the prediction results with the actual values ​​on the test set, calculate indicators such as prediction error, mean absolute error, and root mean square error, and quantitatively evaluate the prediction performance of the Transformer model.

[0040] Step 6: Integration of forecast results. The extrapolated sequences of the trend and periodic terms of the LS model are fused with the residual prediction sequences of the Transformer model to obtain the polar motion forecast results.

[0041] Specifically, the preprocessed test set data is input into the trained combined model. The trend forecast is fitted by the LS linear model, and the predictions are output by the Transformer model. By superimposing the linear trend forecast and residual prediction results, the final polar shift parameter (PM) prediction sequence is obtained, achieving a synergistic representation of linear and nonlinear features, such as... Figure 6 and 7 As shown.

[0042] Step 7: Finally, the accuracy of the polar motion parameter forecast results is statistically analyzed using the Mean Absolute Error (MAE) and compared with the accuracy of the forecast sequence Bulletin A provided by the International Earth Rotation and Reference System Service (IERSS). The accuracy evaluation standard for the polar motion parameters is based on the following formula:

[0043] in, MAE j For the first j The precision of a time series, where n is the length of the time series. i The length of the predicted time series, P i For the first i One forecast value, X i For the first i One true value.

[0044] When evaluating the accuracy of MAE, the true values ​​were selected from the polar motion parameters in the IERS EOP 20C04 sequence. The forecast accuracy of this method was compared with that of the internationally recognized high-precision polar motion forecast sequence Bulletin A. The effectiveness and reliability of the LS+Transformer combined model in improving the accuracy of polar motion forecasts were verified through statistical significance tests.

[0045] The above accuracy verification and comparison uses mean absolute error (MAE) and mean square error (MSE) to quantify the accuracy of polar motion parameter prediction results, and compares them with the Bulletin A prediction sequence published by the International Earth Rotation and Reference System Service (IERS) to verify the effectiveness and reliability of the LS+Transformer combined model proposed in this invention in improving the accuracy of polar motion prediction.

[0046] The implementation of the various embodiments of the present invention is based on programmed processing through a device with processor functionality. Therefore, in practical engineering, the technical solutions and functions of the various embodiments of the present invention are encapsulated into various modules. Based on this reality, and building upon the above embodiments, the embodiments of the present invention provide a polar motion prediction system based on a least squares and Transformer combined model. This system is used to execute the polar motion prediction method based on the least squares and Transformer combined model in the above method embodiments.

[0047] The system includes: an input and preprocessing module for acquiring polar motion observation data and performing multi-dimensional preprocessing; and a polar motion prediction module for inputting the preprocessed polar motion observation data into a trained LS+Transformer combined model, and adding the extrapolated sequences of the trend and periodic terms of the LS linear model and the residual prediction sequence of the Transformer model with equal weights to obtain the final polar motion parameter prediction values. The training of the LS+Transformer combined model includes: acquiring historical polar motion observation data and performing multi-dimensional preprocessing; performing feature analysis on the preprocessed historical polar motion observation data to identify and separate linear trend components and nonlinear fluctuation components; constructing an LS+Transformer combined model, wherein the LS linear model is used for trend and periodic term prediction, and the Transformer model is used for residual prediction; inputting the linear features from the historical polar motion observation data into the constructed LS linear model, solving it using the least squares method, and outputting the predicted values ​​of the trend and periodic terms for future periods; and inputting the nonlinear features from the historical polar motion observation data into the trained Transformer model, outputting the predicted values ​​of the residuals for future periods.

[0048] The polar motion prediction system based on a combined least squares and Transformer model provided in this invention addresses the current situation where existing research lacks an efficient framework that balances linear trend analysis and nonlinear time series modeling, making it difficult to meet the stringent requirements of high-precision applications for polar motion prediction. By employing the aforementioned modules, and leveraging the stability of linear modeling and the nonlinear representation capabilities of deep time series learning, this system achieves a synergistic leap in polar motion prediction accuracy and generalization ability. The system fully utilizes the synergistic advantages of linear and deep time series models to achieve high-precision prediction of polar motion parameters.

[0049] It should be noted that the system embodiments provided by the present invention are used not only to implement the methods in the above method embodiments, but also to implement the methods in other method embodiments provided by the present invention. The only difference is that corresponding functional modules are set. The principle is basically the same as that of the above system embodiments provided by the present invention. As long as those skilled in the art can improve the modules in the above system embodiments by referring to the specific technical solutions in other method embodiments and combining technical features to obtain corresponding technical means and technical solutions composed of these technical means, on the basis of the above system embodiments, and on the premise of ensuring the practicality of the technical solutions, they can obtain corresponding system-like embodiments for implementing the methods in other method-like embodiments.

[0050] Based on the same inventive concept as any of the foregoing embodiments, this embodiment of the invention also provides a polar motion prediction device based on a least squares and Transformer combined model, including a memory and a processor. The memory stores program instructions that are executed by the processor, and the processor calls the program instructions to execute the polar motion prediction method based on the least squares and Transformer combined model.

[0051] Based on the same inventive concept as any of the foregoing embodiments, this embodiment of the invention also provides a non-transitory computer-readable storage medium storing computer instructions that cause the computer to execute the polar motion prediction method based on the least squares and Transformer combined model.

[0052] 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.

[0053] 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 1 One or more processes and / or boxes Figure 1A device that provides the functions specified in one or more boxes.

[0054] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0055] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0056] In summary, this invention discloses a polar motion (PM) forecasting method based on a combination of least squares (LS) and a Transformer model. This model integrates the advantages of the linear LS algorithm and the deep learning Transformer model, aiming to improve the forecasting accuracy and reliability of polar motion parameters by leveraging the complementary characteristics of the two methods. The forecasting method includes: obtaining the basic polar motion parameter sequence, data preprocessing, constructing the LS linear model, training the Transformer model parameters, integrating forecast results, and performing accuracy analysis. The proposed LS+Transformer combined model effectively compensates for the insufficient forecasting accuracy of the random components of polar motion parameters by traditional linear models. Through the effective extrapolation of trend components by the LS model and the in-depth mining and forecasting of the random sequence features of polar motion by the Transformer model, it exhibits higher forecasting accuracy and stability than a single model. This method achieves high-precision long-term forecasting of polar motion parameters, providing a new research approach for polar motion parameter forecasting.

[0057] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.

Claims

1. A polar motion prediction method based on a combined least squares and Transformer model, characterized in that, include: Acquire polar motion observation data and perform multi-dimensional preprocessing on it; The preprocessed polar motion observation data is input into the trained LS+Transformer combined model, and the trend period extrapolation sequence of the LS linear model and the residual prediction sequence of the Transformer model are added with equal weights to obtain the final polar motion parameter prediction values; wherein, the training of the LS+Transformer combined model includes: Acquire historical polar motion observation data and perform multi-dimensional preprocessing on it; Feature analysis was performed on the preprocessed historical polar motion observation data to identify and separate the linear trend components and nonlinear fluctuation components. Construct an LS+Transformer combined model, where the LS linear model is used for predicting the trend and periodic terms, and the Transformer model is used for predicting the residuals; The linear features in historical polar motion observation data are input into the constructed LS linear model, and the trend and periodic terms are extrapolated by the least squares method to output the predicted values ​​of the future period. The nonlinear features in historical polar motion observation data are input into the trained Transformer model, which outputs the residual prediction values ​​for future periods.

2. The polar motion prediction method based on a combined least squares and Transformer model according to claim 1, characterized in that, The multi-dimensional preprocessing includes: time consistency verification, handling of missing and outlier values, observation noise suppression, and normalization.

3. The polar motion prediction method based on a combined least squares and Transformer model according to claim 2, characterized in that, Feature analysis was performed on the preprocessed historical polar motion observation data, including: Long-term trend components in polar shift sequences are identified by linear fitting, and significant periodic components are extracted based on spectral analysis. Subtracting the identified long-term trend component and the extracted significant periodic component from the original polar shift sequence yields a residual sequence that reflects nonlinear fluctuations.

4. The polar motion prediction method based on a combined least squares and Transformer model according to claim 1, characterized in that, The LS linear model is constructed according to the following formula: ， in, For constant terms, The linear trend coefficient is... Let i be the period of the i-th periodic term. , The amplitude and phase coefficients of the periodic term, This is the fitting residual.

5. The polar motion prediction method based on a combined least squares and Transformer model according to claim 1, characterized in that, The training of the Transformer model includes: Obtain the polar shift residual sequence; A Transformer model is constructed with polar-shift residual sequence segments as input and future residual sequence segments to be predicted as the target. The Transformer model first preprocesses the input sequence and then embeds positional codes to preserve temporal positional information. The encoder consists of multiple stacked encoder units. Each unit captures the long-range temporal dependencies within the input sequence through a multi-head self-attention mechanism. After layer normalization and feedforward neural network processing, the encoded features are output. The decoder adopts multiple decoder units. Each layer first models the internal dependencies of the target sequence through masked multi-head self-attention, and then fuses the historical information output by the encoder through encoder-decoder attention. After layer normalization and feedforward neural network processing, the decoder result is mapped to the predicted value through a linear output layer. Using the error between the predicted and actual values ​​as the loss function, the model parameters are iteratively optimized through the backpropagation algorithm to output the trained Transformer model.

6. The polar motion prediction method based on a combined least squares and Transformer model according to claim 1, characterized in that, The extrapolated sequences of the trend and periodic terms from the LS linear model in the combined model are added together with the residual prediction sequences from the Transformer model using equal weights to obtain the final predicted values ​​of the polar shift parameters, including: The fitted sequence of the polar shift sequence is predicted using the LS linear model, and the residual term is predicted using the Transformer model. Finally, the predicted values ​​of the fitted sequence and the predicted values ​​of the residual term are added together to obtain the final predicted values ​​of the polar shift parameters.

7. The polar motion prediction method based on a combined least squares and Transformer model according to claim 6, characterized in that, The method further includes: The accuracy of the polar motion parameter predictions was statistically analyzed using the mean absolute error (MAE) and compared with the accuracy of the prediction sequence Bulletin A provided by the International Earth Rotation and Reference System Service (IERS). The effectiveness and reliability of the LS+Transformer combined model in improving the accuracy of polar motion predictions were verified through statistical significance testing.

8. A polar motion prediction system based on a combined least squares and Transformer model, characterized in that, include: The input and preprocessing module is used to acquire polar motion observation data and perform multi-dimensional preprocessing on it. The polar motion prediction module is used to input preprocessed polar motion observation data into the trained LS+Transformer combined model, and to add the trend period extrapolation sequence of the LS linear model and the residual prediction sequence of the Transformer model in the combined model with equal weights to obtain the final polar motion parameter prediction values; wherein, the training of the LS+Transformer combined model includes: Acquire historical polar motion observation data and perform multi-dimensional preprocessing on it; Feature analysis was performed on the preprocessed historical polar motion observation data to identify and separate the linear trend components and nonlinear fluctuation components. Construct an LS+Transformer combined model, in which the LS linear model is used for predicting the trend and periodic terms, and the Transformer model is used for predicting the residuals; The linear features in historical polar motion observation data are input into the constructed LS linear model, which is solved by the least squares method to output the predicted values ​​of the trend and periodic terms for future periods. The nonlinear features in historical polar motion observation data are input into the trained Transformer model, which outputs the residual prediction values ​​for future periods.

9. A polar motion prediction device based on a combined least squares and Transformer model, characterized in that, The system includes a memory and a processor, the memory storing program instructions that are executed by the processor, the processor invoking the program instructions to execute the polar motion prediction method based on the least squares and Transformer combined model as described in any one of claims 1 to 7.

10. A non-transitory computer-readable storage medium, characterized in that, The non-transitory computer-readable storage medium stores computer instructions that cause the computer to execute the polar motion prediction method based on a least squares and Transformer combined model as described in any one of claims 1 to 7.

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