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. Researching the geomagnetic field is not only crucial for revealing the Earth’s spatial electromagnetic environment, exploring the Earth’s internal structure, and understanding the magnetohydrodynamic dynamics 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 magnetohydrodynamic generator mechanism in 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 dominated by dipole fields. The temporal variation shows long-term changes on the scale of hundreds to thousands of years and polarity reversal on the scale of millions of years. The main magnetic field and its secular variation have always been important research topics in geomagnetism.

Machine learning can extract features from large amounts of data, and can also learn and iterate to discover the data patterns and features we need. As an important branch of machine learning, deep learning learns and mines 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. LSTM （long short-term memory） adds a gate mechanism to the traditional RNN （recurrent neural network） 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 LSTM to the research of secular variation prediction 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; use local time conditions and geomagnetic index conditions to select and calculate the daily mean of the time averaged data; further filter the data based on the geomagnetic quiet days published by the World Geomagnetic Data Center, and perform linear fitting on the filtered data to remove outliers and calculate the monthly mean; further obtain the secular variation time-series of the main magnetic field through the annual difference of the monthly mean. 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 of general methods.

The prediction results shows that for the D element the average RMSE and NRMSE of LSTM are 1.139' and 0.040, for the H element the average RMSE and NRMSE of LSTM are 11.85 nT and 0.086, for the Z element the average RMSE and NRMSE of LSTM are 15.10 nT and 0.026, suggesting the LSTM model has the highest prediction accuracy for Z element, followed by D element, and the worst for H element. There are two main reasons why the model has poor accuracy 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 has limited sample data and lacks comprehensive secular variation information, resulting in the model which is able to fit well on the training set but has 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 obtained 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 linear extrapolation and 59% higher than quadratic extrapolation. For the H element, the average RMSE of the LSTM prediction results is 3.921 nT/a, which is 58% higher than linear extrapolation and 76% higher than quadratic extrapolation. For the Z element, the average RMSE of the LSTM prediction results is 4.339 nT/a, which is 47% higher than linear extrapolation and 57% higher than quadratic extrapolation.