Fast radio bursts (FRBs) are brief, luminous pulses with unknown physical origin. The repetition pattern of FRBs contains essential information about their physical nature and emission mechanisms. Using the two largest samples of FRB 20121102A and FRB 20201124, we report that the sources of the two FRBs reveal memory over a large range of timescales, from a few minutes to about an hour. This suggests that bright bursts are very likely to be followed by bursts of similar or larger magnitude, allowing for the prediction of intense bursts. The memory is detected from the coherent growths in burst-rate structures and the Hurst exponent. The waiting time distribution displays a power-law tail, which is consistent with a Poisson model with a time-varying rate. From cellular automaton simulations, we find that these characteristics can be well understood within the physical framework of a self-organized criticality system driven in a correlation way, such as random walk functions. These properties indicate that the triggers of bursts are correlated, preferring the crustal failure mechanism of neutron stars.