We present principal components analysis (PCA) of temporal magnetic field variations over the solar cycles 21-24 and their classication with symbolic regression analysis using Hamiltonian method. PCA reveals 4 pairs of magnetic waves with a signicant variance and the two principal
components with the highest eigen values covering about 40% of this variance. The PC waves
are found to have close frequencies while travelling from the opposite hemispheres with an
increasing phase shift. Extrapolation of these PCs through their summary curve backward for
5000 years reveals a repeated number of 350-400 year grand cycles superimposed on 22 year-
cycles with the features showing a remarkable resemblance to sunspot activity reported in the
past including Maunder, Wolf and other minima, as well as the modern, medieval and roman
warmth periods and discuss some puzzling details of the supposed Sporer minimum and is link to external extra-galactical processes.The summary curve calculated forward for the next millennium predicts
further three grand cycles with the closest grand minimum (Maunder minimum) occurring in
the forthcoming cycles 25-27 when the two magnetic field waves approach the phase shift of 11 years. We also note a super-grand cycle of about 2000 years which reveal the 5 repeated grand cycles of 350 years with the similar patterns. We discuss a role of other 3 pairs of magnetic waves in shaping the solar activity and compare our predicted curve with the previous predictions of the solar activity on a long timescale based on the pure terrestrial proxies. These grand cycle variations are probed by Parker's two layer dynamo model with meridional circulation
revealing two dynamo waves generated with close frequencies that produces the beating effect and grand and super-grand cycles.