Prediction of solar activities: Sunspot numbers and solar magnetic synoptic maps

The evolution of solar magnetic fields is significant for understanding and predicting solar activities. And our knowledge of solar magnetic fields largely depends on the photospheric magnetic field. In this paper, based on the spherical harmonic expansion of the photospheric magnetic field observed by Wilcox Solar Observatory, we analyze the time series of spherical harmonic coefficients and predict Sunspot Number as well as synoptic maps for Solar Cycle 25. We find that solar maximum years have complex short-period disturbances, and the time series of coefficient g70 is nearly in-phase with Sunspot Number, which may be related to solar meridional circulation. Utilizing Long Short-Term Memory networks (LSTM), our prediction suggests that the maximum of Solar Cycle 25 is likely to occur in June 2024 with an error of 8 months, the peak sunspot number may be 166.9 +/- 22.6, and the next solar minimum may occur around January 2031. By incorporating Empirical Mode Decomposition, we enhance our forecast of synoptic maps truncated to Order 5, validating their relative reliability. This prediction not only addresses a gap in forecasting the global distribution of the solar magnetic field but also holds potential reference value for forthcoming solar observation plans.