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Article(id=1322570899933475615, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322570898712937423, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.07.248, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1752422400000, receivedDateStr=2025-07-14, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793255074723, onlineDateStr=2026-10-29, pubDate=1778774400000, pubDateStr=2026-05-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793255074723, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793255074723, creator=13701087609, updateTime=1793255074723, updator=13701087609, issue=Issue{id=1322570898712937423, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='5', pageStart='522', pageEnd='661', issueExtLink='null', onlineDate='null', pubDate='1778774400000', pubDateStr='2026-05-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793255074432, creator='13701087609', updateTime=1793255074432, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=610, endPage=617, ext={EN=ArticleExt(id=1322570900109636384, articleId=1322570899933475615, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Medium- and Long-Term ERP Prediction Based on Machine Learning, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Currently, the accuracy of medium- and long-term predictions of Earth rotation parameters (ERP) (prediction duration exceeding 30 days) can only reach the milliarcsecond level. This level of accuracy is significantly lower than the microarcsecond level required by astrometry and the millimeter level needed by space geodesy. This paper systematically evaluated the performance of four deep learning models (BP, RNN, LSTM, and Transformer) in short-, medium-, and long-term ERP forecasts, with a focus on long-term predictions. The experimental results show that in 180-day predictions, the precision of polar motion forecasts by the Transformer model is 43.8% higher than that of traditional methods, and the prediction error of length of day (LOD) is reduced to 0.27 ms. In UT1-TAI predictions, the BP model achieved the best accuracy with an RMSE of 7.83 ms. This study can provide reference for solving the medium- and long-term ERP forecasting challenges in the autonomous operation of the Beidou system.

, authors=Sichen ZHANG1, 2, Shanshi ZHOU1, Rui GUO3, Xiaojie LI3, authorsList=Sichen ZHANG, Shanshi ZHOU, Rui GUO, Xiaojie LI, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322570901464396589, articleId=1322570899933475615, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于机器学习的ERP中长期预报方法研究, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

目前, 地球自转参数(ERP)的中长期预测(预测时长超过30 d)精度仅能达到毫角秒量级, 与天体测量学所要求的微角秒量级以及空间大地测量学所需的毫米量级存在显著差距, 难以满足实际需求。本文系统评估了4种深度学习模型(BP、RNN、LSTM和Transformer)在ERP短中长期预报中的性能, 重点关注长期预测。实验结果表明, 在180 d预测中, Transformer模型极移预测精度较传统方法提升43.8%, 日长变化(LOD)预测误差也降至0.27 ms; UT1-TAI预测中, BP模型精度最佳, RMSE为7.83 ms。该研究可为解决北斗系统自主运行中的ERP中长期预报难题提供参考。

, authors=张思晨1, 2, 周善石1, 郭睿3, 李晓杰3, authorsList=张思晨, 周善石, 郭睿, 李晓杰, authorCompany=null, correspAuthors=null, authorNote=

张思晨, 硕士生, 主要从事天文地球动力学和北斗导航研究, E-mail:

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张思晨, 硕士生, 主要从事天文地球动力学和北斗导航研究, E-mail:

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张思晨, 硕士生, 主要从事天文地球动力学和北斗导航研究, E-mail:

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Improvement in the Medium-and Long-Term Prediction Accuracy of Polar Motion Using Singular Spectrum Analysis[J]. 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journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Fig. 6, caption=Four models' UT1-TAI prediction results with data before and after 2020 versus IERS benchmark, figureFileSmall=9coGRnnFceDEcKnrzTREGw==, figureFileBig=+Vfha30K8HUmb2h+4WsT3A==, tableContent=null), ArticleFig(id=1322570908028482401, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=CN, label=图6, caption=4种模型使用2020年前后数据的预测结果与IERS的UT1-TAI误差对比, figureFileSmall=9coGRnnFceDEcKnrzTREGw==, figureFileBig=+Vfha30K8HUmb2h+4WsT3A==, tableContent=null), ArticleFig(id=1322570908095591266, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Tab. 1, caption=

Prediction results of polar motion and LOD at different time periods by the four models

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预测天数/dBPRNNLSTMTransformer
XP/masYP/masLOD/msXP/masYP/masLOD/msXP/masYP/masLOD/msXP/masYP/masLOD/ms
10.721.030.121.071.410.060.200.180.040.460.290.04
51.821.920.221.891.880.181.171.330.100.830.890.09
103.792.740.283.212.570.273.402.410.152.431.490.13
302.5110.730.1212.925.860.306.773.840.258.023.770.19
9010.1232.830.6121.1620.710.5124.529.050.4512.339.600.23
18047.9746.410.57梯度爆炸49.0527.320.4116.8916.300.27
), ArticleFig(id=1322570908187865955, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=CN, label=表1, caption=

4种模型的极移与日长不同时段预测结果

, figureFileSmall=null, figureFileBig=null, tableContent=
预测天数/dBPRNNLSTMTransformer
XP/masYP/masLOD/msXP/masYP/masLOD/msXP/masYP/masLOD/msXP/masYP/masLOD/ms
10.721.030.121.071.410.060.200.180.040.460.290.04
51.821.920.221.891.880.181.171.330.100.830.890.09
103.792.740.283.212.570.273.402.410.152.431.490.13
302.5110.730.1212.925.860.306.773.840.258.023.770.19
9010.1232.830.6121.1620.710.5124.529.050.4512.339.600.23
18047.9746.410.57梯度爆炸49.0527.320.4116.8916.300.27
), ArticleFig(id=1322570908292723556, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Tab. 2, caption=

Comparison of MAE in polar motion predictions

, figureFileSmall=null, figureFileBig=null, tableContent=
预测天数/dLSTMTransformerLS+ARSSA+LS+AR
XP/masYP/masXP/masYP/masXP/masYP/masXP/masYP/mas
10.430.160.460.290.260.210.260.21
51.050.870.760.801.641.181.581.13
102.021.392.311.363.232.153.031.99
305.593.276.823.278.176.296.535.38
9017.4712.0111.818.6020.8320.8613.9816.09
18035.8920.1816.7414.4724.5925.7517.0720.40
), ArticleFig(id=1322570908401775461, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=CN, label=表2, caption=

极移预测MAE对比

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预测天数/dLSTMTransformerLS+ARSSA+LS+AR
XP/masYP/masXP/masYP/masXP/masYP/masXP/masYP/mas
10.430.160.460.290.260.210.260.21
51.050.870.760.801.641.181.581.13
102.021.392.311.363.232.153.031.99
305.593.276.823.278.176.296.535.38
9017.4712.0111.818.6020.8320.8613.9816.09
18035.8920.1816.7414.4724.5925.7517.0720.40
), ArticleFig(id=1322570908498244454, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Tab. 3, caption=

The MAE improvement rates of LSTM, Transformer, and SSA+LS+AR over LS+AR for polar motion

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预测天数/dLSTMTransformerSSA+LS+AR
XP/%YP/%XP/%YP/%XP/%YP/%
1-65.423.8-76.9-38.10.00.0
536.026.353.732.23.74.2
1037.535.328.536.76.27.4
3031.648.016.548.020.114.5
9016.142.443.358.832.922.9
180-46.021.631.943.830.620.8
), ArticleFig(id=1322570908565353319, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=CN, label=表3, caption=

LSTM、Transformer和SSA+LS+AR相对于LS+AR的极移MAE提升率

, figureFileSmall=null, figureFileBig=null, tableContent=
预测天数/dLSTMTransformerSSA+LS+AR
XP/%YP/%XP/%YP/%XP/%YP/%
1-65.423.8-76.9-38.10.00.0
536.026.353.732.23.74.2
1037.535.328.536.76.27.4
3031.648.016.548.020.114.5
9016.142.443.358.832.922.9
180-46.021.631.943.830.620.8
), ArticleFig(id=1322570908670210921, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Tab. 4, caption=

Prediction results of UT1-TAI values at different time periods by the four models

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模型预测结果/ms
1 d5 d10 d30 d90 d180 d
BP0.360.391.385.3121.93143.61
LSTM3.345.907.5815.0985.26122.47
RNN4.862.806.8110.8932.8057.85
Transformer超过30超过200
), ArticleFig(id=1322570908812817258, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=CN, label=表4, caption=

4种模型的UT1-TAI值在不同时段的预测结果

, figureFileSmall=null, figureFileBig=null, tableContent=
模型预测结果/ms
1 d5 d10 d30 d90 d180 d
BP0.360.391.385.3121.93143.61
LSTM3.345.907.5815.0985.26122.47
RNN4.862.806.8110.8932.8057.85
Transformer超过30超过200
), ArticleFig(id=1322570908888314731, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570899933475615, language=EN, label=Tab. 5, caption=

RMSE values of UT1-TAI predictions at different periods before and after 2020 by the four models

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预测天数/dRMSE/ms
BPRNNLSTMTransformer
前后前后前后前后
129.533.711.151.381.290.4920.383.87
542.174.672.412.432.481.7128.714.34
1052.684.703.683.494.462.1931.264.47
3061.426.43/6.7414.735.2541.867.78
9069.836.79//20.607.2555.6214.79
18091.097.83//42.398.3065.7713.61
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4种模型预测UT1-TAI在2020年前后不同时段的RMSE值

, figureFileSmall=null, figureFileBig=null, tableContent=
预测天数/dRMSE/ms
BPRNNLSTMTransformer
前后前后前后前后
129.533.711.151.381.290.4920.383.87
542.174.672.412.432.481.7128.714.34
1052.684.703.683.494.462.1931.264.47
3061.426.43/6.7414.735.2541.867.78
9069.836.79//20.607.2555.6214.79
18091.097.83//42.398.3065.7713.61
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基于机器学习的ERP中长期预报方法研究
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张思晨 1, 2 , 周善石 1 , 郭睿 3 , 李晓杰 3
大地测量与地球动力学 | 大地测量学 2026,46(5): 610-617
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大地测量与地球动力学 |大地测量学 2026 , 46 (5) : 610 -617
基于机器学习的ERP中长期预报方法研究
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张思晨1, 2 , 周善石1, 郭睿3, 李晓杰3
作者信息
  • 1 中国科学院上海天文台, 上海, 200030
  • 2 中国科学院大学天文与空间科学学院, 北京, 100049
  • 3 北京卫星导航定位中心, 北京, 100094
作者简介:

张思晨, 硕士生, 主要从事天文地球动力学和北斗导航研究, E-mail:

Medium- and Long-Term ERP Prediction Based on Machine Learning
Sichen ZHANG1, 2 , Shanshi ZHOU1, Rui GUO3, Xiaojie LI3
Affiliations
  • 1 Shanghai Astronomical Observatory, CAS, Shanghai 200030, China
  • 2 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3 Beijing Satellite Navigation Center, Beijing 100094, China
出版时间: 2026-05-15 doi: 10.14075/j.jgg.2025.07.248
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摘要
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目前, 地球自转参数(ERP)的中长期预测(预测时长超过30 d)精度仅能达到毫角秒量级, 与天体测量学所要求的微角秒量级以及空间大地测量学所需的毫米量级存在显著差距, 难以满足实际需求。本文系统评估了4种深度学习模型(BP、RNN、LSTM和Transformer)在ERP短中长期预报中的性能, 重点关注长期预测。实验结果表明, 在180 d预测中, Transformer模型极移预测精度较传统方法提升43.8%, 日长变化(LOD)预测误差也降至0.27 ms; UT1-TAI预测中, BP模型精度最佳, RMSE为7.83 ms。该研究可为解决北斗系统自主运行中的ERP中长期预报难题提供参考。

地球自转参数(ERP)  /  深度学习  /  长期预报  /  极移  /  UT1-UTC

Currently, the accuracy of medium- and long-term predictions of Earth rotation parameters (ERP) (prediction duration exceeding 30 days) can only reach the milliarcsecond level. This level of accuracy is significantly lower than the microarcsecond level required by astrometry and the millimeter level needed by space geodesy. This paper systematically evaluated the performance of four deep learning models (BP, RNN, LSTM, and Transformer) in short-, medium-, and long-term ERP forecasts, with a focus on long-term predictions. The experimental results show that in 180-day predictions, the precision of polar motion forecasts by the Transformer model is 43.8% higher than that of traditional methods, and the prediction error of length of day (LOD) is reduced to 0.27 ms. In UT1-TAI predictions, the BP model achieved the best accuracy with an RMSE of 7.83 ms. This study can provide reference for solving the medium- and long-term ERP forecasting challenges in the autonomous operation of the Beidou system.

Earth rotation parameters (ERP)  /  deep learning  /  long-term prediction  /  polar motion  /  UT1-UTC
张思晨, 周善石, 郭睿, 李晓杰. 基于机器学习的ERP中长期预报方法研究. 大地测量与地球动力学, 2026 , 46 (5) : 610 -617 . DOI: 10.14075/j.jgg.2025.07.248
Sichen ZHANG, Shanshi ZHOU, Rui GUO, Xiaojie LI. Medium- and Long-Term ERP Prediction Based on Machine Learning[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (5) : 610 -617 . DOI: 10.14075/j.jgg.2025.07.248
正文
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地球定向参数(EOP)在地球科学、空间导航和深空探测等领域具有重要的应用价值，也在北斗全球卫星导航系统(BDS)的性能保障中起到决定性作用。最重要的是，在BDS自主运行期间, EOP参数无法实时更新，其长期预报误差将是卫星导航自主运行的主要误差源[1-2]。其中，极移误差主要影响天球参考框架与地球参考框架之间的转换精度，虽然其对整体转换的影响相对较小，但仍会间接导致卫星轨道确定和广播星历的精度有所下降；日长变化误差则可能引起时间同步问题，削弱北斗授时服务的可靠性。因此，EOP的长期预报精度尤为重要。为缓解这些影响，需探寻更高精度的长期预报方法。
近年来，诸多学者[3-13]针对EOP中的地球自转参数(ERP)预测方法展开了广泛研究，提出多种基于统计模型、动力学模型与机器学习模型的预测方法。传统统计模型与动力学模型的精度研究主要集中于0~30 d的短期预报范围。根据IERS 2019年年度报告[3]，评估的预测方法可实现长达90 d的预报，其中，极移X与Y方向的预报精度分别达到17.25 mas与9.28 mas，UT1-UTC的预报精度则为9.748 ms。在众多预测方法中，神经网络凭借其强大的非线性映射能力和自学习特性，在时间序列预测中表现出显著优势。例如，国际大地测量协会(IAG)旗下的全球大地测量观测系统(GGOS)[6-7]正积极推进人工智能与机器学习在ERP预报中的应用研究，旨在提升预报精度。王宇谱等[7]与王宗建[8]分别采用小波神经网络和误差反向传播神经网络(BP神经网络)进行日长变化预报与极移预报，在0~5 d的短期预测中，极移误差最低可达2.3 mas与3.2 mas，日长预报误差最低为0.35 ms。IERS第2次EOP预报比较的结果显示，在0~30 d的预报范围内，UT1-UTC在30 d时的最低预报误差为1.987 ms；而在10 d极移预报中，表现最佳的机器学习方法精度达到了2.3 mas[9]。尽管国内外已有大量基于机器学习的ERP参数预测研究，但其焦点几乎全部集中于短期(0~30 d)预报精度[7-11]，针对长期预报的研究仍相对匮乏。
本文针对北斗自主导航系统对ERP预测的特殊需求，选择常用于时间序列预测任务的BP神经网络、RNN神经网络、LSTM神经网络和Transformer神经网络4种深度学习模型，系统比较ERP参数在不同时间尺度(短期、中期和长期)的预测性能，并与IERS发布的结果及其他学者给出的结果进行对比，重点关注各模型在长期预测中的精度，旨在为北斗系统自主运行提供最优的预测模型。
1 数据源与研究方法
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本文使用的ERP数据来源于国际地球自转与参考系统服务组织(IERS)提供的事后精密EOP产品finals2000A.all数据集(1973-01-02—2024-08-01)，该数据集包含了标定时间范围内的极移、日长变化、UT1-UTC和岁差章动校正等地球定向参数(EOP)的精密测量值。
本文所使用的神经网络模型中，输入、输出项均为需预测的各ERP参数(极移/日长/UT1-UTC)，各参数序列如图 1所示。
由图 1可见，UT1-UTC的数据不连续，存在由于地球自转速率变化而产生的跳秒，本文去除跳秒生成UT1-TAI(地球自转与国际原子时之间的差异)数据。
1.1 BP神经网络
反向传播神经网络(back propagation neural network，BPNN)是一种典型的多层前馈网络，通过误差反向传播算法进行训练。该网络由输入层、隐藏层和输出层构成，其核心优势在于强大的非线性逼近能力，具有全局优化性，对UT1-TAI这种无明显周期项的时间序列处理上具有优势。
在网络训练过程中，输入信号经由隐藏层逐层传递，最终在输出层产生预测结果。网络通过最小化预测误差来调整连接权值，其数学表达为：
$W_{t-1}=W_t-\eta\left(\frac{\partial L}{\partial W}\right)$
式中，η为学习率；L为损失函数。
在长期预报实验中，BP神经网络(图 2)输入层节点为180、隐藏层节点为128+64，使用ReLU激活函数和Dropout正则化(比率分别为0.5和0.3)以防止过拟合。模型训练采用Adam优化器(学习率0.001)，配合早停机制(耐心值20轮)。
1.2 RNN神经网络
循环神经网络(recurrent neural network，RNN)通过引入隐状态机制对序列数据的时序依赖性建模。将上一时刻的隐状态Ht-1和当前输入Xt结合，生成当前隐状态Ht。网络的隐状态更新方程为：
$\boldsymbol{H}_t=f\left(\boldsymbol{W}_{X H} \boldsymbol{X}_t+\boldsymbol{W}_{H H} \boldsymbol{H}_{t-1}+b_H\right)$
式中，f(·)为激活函数(如tanh或ReLU)；WXH和WHH分别为输入和隐状态的权重矩阵；bH为偏置项。RNN的循环结构使其能够处理变长序列，但存在梯度消失问题，难以学习长期依赖关系。
在长期预报实验中，RNN模型采用单层RNN结构(隐藏层128节点)，配合Dropout正则化(0.3)和Adam优化器(初始学习率0.001)，并引入学习率衰减策略(ReduceLROnPlateau)与梯度裁剪(阈值1.0)，训练过程采用早停机制(耐心值20轮)。
1.3 LSTM神经网络
长短期记忆网络(long short-term memory, LSTM)作为一种特殊的循环神经网络，通过设计门控机制解决了传统RNN神经网络的长期依赖问题，理论上可以较好地满足ERP参数长期预测的需求。LSTM单元包含3个关键门控结构。
遗忘门：
$\boldsymbol{f}_t=\sigma\left(\boldsymbol{W}_f \cdot\left[\boldsymbol{H}_{t-1}, \boldsymbol{X}_t\right]+b_f\right)$
输入门：
$\begin{gathered}\boldsymbol{i}_t=\sigma\left(\boldsymbol{W}_i \cdot\left[\boldsymbol{H}_{t-1}, \boldsymbol{X}_t\right]+b_i\right) \\\widetilde{\boldsymbol{C}}_t=\tanh \left(\boldsymbol{W}_C \cdot\left[\boldsymbol{H}_{t-1}, \boldsymbol{X}_t\right]+b_C\right)\end{gathered}$
细胞更新状态：
$\boldsymbol{C}_t=\boldsymbol{f}_t \odot \boldsymbol{C}_{t-1}+\boldsymbol{i}_t \odot \widetilde{\boldsymbol{C}}_t$
输出门：
$\begin{gathered}\boldsymbol{O}_t=\sigma\left(\boldsymbol{W}_O \cdot\left[\boldsymbol{H}_{t-1}, \boldsymbol{X}_t\right]+b_O\right) \\\boldsymbol{H}_t=\boldsymbol{O}_t \odot \tanh \boldsymbol{C}_t\end{gathered}$
式中，σ(·)为激活函数；Wf、Wi、WC、WO为权重矩阵；bf、bi、bC、bO为偏置项；Ht－1为上一时间步的隐状态；Xt为当前输入。
这种门控机制使LSTM能够通过遗忘门、输入门和输出门来控制信息的流动，选择性地保留或遗忘信息，从而适应不同时间尺度的依赖关系，有效适用于地球自转参数数据短中长期预报的3种需求，也能较好地顾及到极移参数的明显周期性(图 3)。
在长期预报实验中，LSTM模型采用单层LSTM结构(隐藏层32节点)，配合Dropout正则化(0.5)和Adam优化器(初始学习率0.000 1)，训练过程采用早停机制(耐心值20轮)。
1.4 Transformer神经网络
Transformer模型基于自注意力机制，完全摒弃了循环结构。其核心是缩放点积注意力：
$\operatorname{Attention}(\boldsymbol{Q}, \boldsymbol{K}, \boldsymbol{V})=\operatorname{softmax}\left(\frac{\boldsymbol{Q} \boldsymbol{K}^{\mathrm{T}}}{\sqrt{d_k}}\right) \boldsymbol{V}$
式中，Q、K、V分别为查询向量、键向量、值向量矩阵；dk为维度缩放因子。Transformer通过多头注意力机制从不同子空间捕获多样的时序模式：
$\quad \operatorname{MultiHead}(\boldsymbol{Q}, \boldsymbol{K}, \boldsymbol{V})= \\\operatorname{Concat} \left( \operatorname{head}_1, \ldots, \operatorname{head}_h\right) \boldsymbol{W}^o$
这种架构基于自注意力机制，使Transformer能够直接对任意2个时间步之间的依赖关系建模，无需像循环神经网络那样依赖隐状态传递。该特性使其能够显式地捕捉极移序列中存在的长周期项(如钱德勒摆动和周年摆动)与不规则变化之间的全局动态关联，从而更准确地刻画其非线性演变规律。因此，Transformer模型在地球自转参数的长期预测中预计将表现出明显优势，尤其在捕捉跨数百天的周期性特征和趋势性变化方面，有望显著优于传统时序预测方法。
在长期预报实验中，Transformer模型采用3层Encoder结构(d_model=64，nhead=8)，配合位置编码(PositionalEncoding)和Dropout正则化(0.1)。优化器选用Adam(初始学习率0.001)，并引入学习率动态衰减(ReduceLROnPlateau)与梯度裁剪(阈值0.5)。训练过程采用早停机制(耐心值20轮)。
2 实验结果与分析
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2.1 评价指标
使用均方根误差(RMSE)和平均绝对误差(MAE)评估模型的预测性能，衡量模型预测值与真实值之间的平均偏差程度。
$\text {RMSE}=\sqrt{\frac{1}{n} \sum\limits_{i=1}^n\left(y_i-\hat{y}_i\right)^2}$
$\mathrm{MAE}=\frac{1}{n} \sum\limits_{i=1}^n\left|y_i-\hat{y}_i\right|$
其中，预测值由神经网络模型用数据集的前70%(1973-01-02—2009-02-09)进行训练得出，真实值则来源于后30%(2009-02-10—2024-08-01)的实际观测数据。前一部分用于模型训练，后一部分用于比较模型预报结果与实际观测值之间的差异，从而评估模型在未知数据上的预测能力。
2.2 实验结果
使用1973-01-02—2024-08-01期间ERP参数数据，分别用BP、RNN、LSTM和Transformer四种神经网络模型对ERP各参数进行1~180 d的训练预测，记录多次实验预报值的平均RMSE值，结果见表 1。
2.2.1 极移与日长结果
表 1、图 4为4种模型(LSTM、RNN、Transformer、BP)与IERS[3]进行极移(包括XP和YP)和LOD日长参数预测的误差对比。
表 1详细对比了4种模型在不同预报期内的极移(XP，YP)与日长变化(LOD)的预测精度，其中每个时段内表现最优的模型误差值以黑体表示。分析结果表明，LSTM与Transformer模型在绝大多数预测任务中均展现出更高的精度优势，尤其在长周期预测中表现更为稳健。相比之下，RNN模型因梯度更新步长过大，难以捕捉序列中长期依赖关系，导致预测误差呈指数级增长并出现剧烈波动，也即梯度爆炸现象，严重限制了其实际应用价值。
由图 4可见，在极移预测中，BP模型在长时间预测时的稳定性较差；LSTM和RNN模型虽然在短时间预测时表现较好，但随着预测时间的增加，其误差增长速度也较快；而Transformer模型则表现出与IERS相近的误差增长速度，尤其是在长时间预测时，其误差增长较慢，显示出较好的预测稳定性和准确性。在日长预测中，Transformer模型的误差增长较慢且相对较低，较为稳定。
2.2.2 与LS+AR和SSA+LS+AR模型极移结果对比
为验证各模型在不同参数、不同时段对极移预报的改进情况，选取预报性能较好的LSTM与Transformer模型进行1~180 d的极移预报，并与传统LS+AR模型及基于奇异谱分析改进的SSA+LS+AR模型的预报结果进行对比。其中, SSA+LS+AR模型首先通过设置窗口长度为730 d的奇异谱分析，将极移序列分解为低频和高频组分，并选取前5阶重构分量作为低频信号，继而建立包含线性趋势、钱德勒摆动和周年摆动项的LS模型对低频组分进行外推，同时对高频组分及LS残差建立基于AIC准则定阶的AR模型进行预测，最终叠加2个部分预测值获得极移预报结果。各模型在1~180 d预报跨度内的MAE对比结果见表 2与表 3。
分析表 2、表 3可以看出, LSTM和Transformer模型在极移预测中表现优越，尤其在中长期预测方面表现突出，而传统方法(如LS+AR)仅在超短期预测中保持优势。
在短期预测(1~10 d)方面，LS+AR和SSA+LS+AR模型在1 d预测中表现最佳(XP和YP的MAE分别为0.26 mas和0.21 mas)，但LSTM和Transformer模型在5~10 d的预测中迅速展现出更强的适应性，体现出其捕捉序列中非线性特征和复杂时间依赖关系的强大能力。例如，LSTM模型在5 d预测时的XP误差(1.05 mas)比LS+AR模型(1.64 mas)降低了36.0%，而Transformer模型在10 d预测时的YP误差(1.36 mas)优于LSTM模型(1.39 mas)，表明其在方向性预测上的稳定性更强。
在中长期预测(30~180 d)方面，Transformer模型展现出显著优势，其自注意力机制能更有效地捕捉极移的长周期特征。例如，在90 d预测中，Transformer模型的YP误差(8.60 mas)比LS+AR模型(20.86 mas)降低58.8%；在180 d预测时，其XP误差(16.74 mas)仍比LS+AR模型(24.59 mas)低31.9%，表明其凭借全局依赖建模能力能够更有效地学习地球自转参数中的长期周期项与趋势性变化。相比之下，LSTM模型在30 d内的预测表现较优(如30 d的XP误差5.59 mas，比LS+AR模型低31.6%)，但在更长预测时性能下降明显(180 d的XP误差35.89 mas，比LS+AR模型高46.0%)，表明其在超长期预测中存在局限性。
2.2.3 UT1-TAI结果
表 4、图 5为4种模型在UT1-TAI数列上的预测结果。其中，BP模型的误差增长较快，尤其是在预测时间超过90 d后，误差显著增加；LSTM和RNN模型的误差增长速度适中，但仍然高于IERS；在极移与日长预测任务中表现较好的Transformer模型，其预测误差出现了显著增大且不稳定的现象。
对于UT1-TAI出现的问题，本文研究发现，由于2020年地球自转速率发生变化，跳秒消失，可能会对预测结果产生较大影响。将数据分为2015-06-01—2019-12-31和2020-01-01—2024-08-01两段，保持相同长度，对4个模型再次进行训练预测，结果如表 5、图 6所示。
根据表 5和图 6所示，在使用2020年之后的数据时，所有模型的预测误差均有所降低。推测其原因可能与地球自转速率加快所引起的数据趋于平缓有关(如图 1下面2个分图末端所示)。数据平稳性的提高有助于提升神经网络模型的学习效果，因此本文后续分析将集中于2020年之后数据的预测结果。在短中期预测(1~30 d)中，LSTM模型表现较好，30 d预报误差仅为5.25 ms，说明其能够有效学习该序列中的中短周期特征；在长期预测(90~180 d)中，BP网络模型表现出较高的精度及平稳度，180 d预报误差只有7.83 ms；Transformer模型相对于其他模型精度较差，但180 d误差也未超过15 ms。
3 结语
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针对目前ERP参数长期预报精度不高的问题，本文系统评估了4种神经网络模型(BP、RNN、LSTM和Transformer)在ERP短中长期预报中的性能，重点关注长期预测，主要得出以下结论。
1) 在极移和LOD预测任务中，不同时间尺度下的模型性能表现出显著差异。在短期预测(1~10 d)中，传统方法与深度学习模型各有优势：超短期(1 d)预测中，基于最小二乘拟合与自回归(LS+AR)的传统方法仍保持最优性能。当预测时长扩展至5~10 d时，LSTM与Transformer模型的预测误差显著低于LS+AR模型，体现出其捕捉序列中非线性特征和复杂时间依赖关系的强大能力。在中长期预测(30~180 d)中，模型性能差异进一步增大，Transformer模型表现出明显优势，尤其在180 d预测中，X和Y极移分量的误差分别仅为16.74 mas和14.47 mas，LOD预测误差也稳定在0.27 ms以内，表明其凭借全局依赖建模能力能够更有效地学习地球自转参数中的长期周期项与趋势性变化。
2) 在UT1-TAI预测任务中，本文重点关注基于2020年后的数据开展的实验。结果表明，在短中期预测(1~30 d)中，LSTM模型表现最优，说明其能够有效学习该序列中的中短周期特征；在长期预测(90 d和180 d)中，结构简单的BP模型反而取得最佳性能。推测其原因，复杂模型(如LSTM和Transformer)容易过度拟合短期波动，而BP模型凭借较强的泛化能力和简洁的非线性映射结构，更适用于描述长期尺度下的总体变化趋势。
实验结果充分证明了相对于短期预报表现良好的传统LS+AR模型，机器学习在长期预报中表现出较高的预报性能，更符合目前北斗导航系统的长期预报需求，为提升北斗系统的自主运行能力提供了技术支撑。
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  • 中国科学技术委员会基础强化计划技术领域基金(2024-JCJQ-JJ-0336)
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2026年第46卷第5期
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doi: 10.14075/j.jgg.2025.07.248
  • 接收时间：2025-07-14
  • 首发时间：2026-10-29
  • 出版时间：2026-05-15
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  • 收稿日期：2025-07-14
基金
中国科学技术委员会基础强化计划技术领域基金(2024-JCJQ-JJ-0336)
作者信息
    1 中国科学院上海天文台, 上海, 200030
    2 中国科学院大学天文与空间科学学院, 北京, 100049
    3 北京卫星导航定位中心, 北京, 100094
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2种不同金属材料的力学参数
科
Family		 属数
Number of
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Number of
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Percentage of
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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