In this paper we overview the modern forecast methods of monthly sunspot numbers, such as McNish-Lincoln and Hathaway-Wilson-Reichmann standard curve-fitting. Their disadvantages are presented, leading us to the necessity of researching a new technique for the solar activity prediction. For the long-term forecast we propose to use the established nonlinear dynamo model based on negative effective magnetic pressure instability. For the short-term forecast adjustment we propose to use instead of Data Assimilation technique the Artificial Neural Networks. The implemented NAR- and NARX-nets have been trained on the 3 sunspot cycles (between 1965 and 1997), with the testing forecast into the next solar cycle up to 2009 year. The final results show plausible forecast accuracy: the misfit coefficient is only between 20–35%.