The geomagnetic field is the result of the superposition of different magnetic substances and their dynamic processes within the Earth, as well as the magnetic field generated by the current systems both inside and outside the Earth. Study on the geomagnetic field is not only crucial for revealing the Earth’s spatial electromagnetic environment, exploring the Earth’s internal structure, and understanding the geodynamo of the Earth’s core, but also plays an extremely important role in monitoring earthquake and volcanic activity, exploring mineral and energy resources, as well as positioning and navigating carrier. The magnetic field of the Earth’s core, also known as the main magnetic field, is widely believed to be generated by the geodynamo of the Earth’s core, accounting for over 95% of the total magnetic field. The wavelength of the main magnetic field is relatively long, and its spatial distribution is predominantly characterized by dipole fields. In terms of temporal variation, it exhibits long-term changes on a scale ranging from hundreds to thousands of years and polarity reversal on a scale of millions of years. The main magnetic field and its secular variation have long been important research topics in geomagnetism.

Machine learning can extract features from large amounts of data, and can also perform learning and iterative operations to discover the data patterns and features that meet our requirements. As an important branch of machine learning, deep learning undertakes learning and mining of data features through deep neural networks. Deep learning can handle non-linear data without relying on the spectral characteristics of temporal data, and has good performance. Long short-term memory （LSTM） incorporates a gate mechanism into the traditional recurrent neural network （RNN） structure, which can effectively solve the problems of gradient explosion and vanishing during RNN training. Therefore, LSTM has more complex temporal information memory units and is widely used in temporal data analysis and modeling.

Thus, we apply deep neural network, specifically LSTM, to the research on predicting secular variation of geomagnetic field. We select the time averaged data of the horizontal component H, magnetic declination D, and vertical component Z of the geomagnetic field from 32 geomagnetic stations in Chinese mainland and its neighboring regions. By leveraging local time conditions and geomagnetic index conditions, we select and compute the daily means of the time-averaged data. Subsequently, based on the geomagnetic quiet days released by the World Geomagnetic Data Center we conduct further data filtering. Linear fitting is then applied to the filtered data to eliminate outliers and calculate the monthly means. And we further obtain the secular variation time-series of the main magnetic field through the annual difference of the monthly means. Finally, the secular variation time-series of the main magnetic field is input into the LSTM model for training, and the predicted results of the model are compared and analyzed with those obtained by conventional methods.

The prediction results show that for the D element the average RMSE and NRMSE from LSTM model are 1.139′ and 0.040, for the H element the average RMSE and NRMSE from LSTM model are 11.85 nT and 0.086, for the Z element the average RMSE and NRMSE from LSTM model are 15.10 nT and 0.026, suggesting the LSTM model has the highest prediction accuracy for Z element, followed by D element, while the accuracy is the lowest for H element. There are two main reasons accounting for the poor accuracy of the model in predicting H elements. Firstly, during the geomagnetic quiet period, the distribution of S_q current system and equatorial current directly affects the recording of H elements at ground stations, especially in low latitude areas where H elements undergo significant changes. Secondly, the training set contains limited sample data and lacks comprehensive secular variation information. As a consequence, the model can fit well on the training set but exhibits poor prediction accuracy on the testing set. Expanding the sample size of the training set as much as possible can improve this situation.

We calculate the annual rate error for various elements of the station derived from LSTM model, linear extrapolation, and quadratic extrapolation. For the D element, the average RMSE of the LSTM prediction results is 0.361′ /a, which is 54% higher than that of linear extrapolation and 59% higher than that of quadratic extrapolation. For the H element, the average RMSE of the LSTM prediction results is 3.921 nT/a, which is 58% higher than that of linear extrapolation and 76% higher than that of quadratic extrapolation. For the Z element, the average RMSE of the LSTM prediction results is 4.339 nT/a, which is 47% higher than the value obtained from linear extrapolation and 57% higher than that from quadratic extrapolation.