I may have said GFCI when I actually meant "arc fault" outlet. If that makes a difference.
Yes it does. ARC fault breakers are a relatively new safety requirement to meet new electrical codes. They are not ground fault devices, they deal with arcing only. An arc is when a wire shorts out and sparks. This can be caused by failing lamps, switches, wiring, and other plug in devices and appliances, with worn cords being a common fault. It can protect against cords/extension cords that have been run under rugs, or taped to the floor, that are damaged from wear and about to ignite rugs and flooring. (this practice, BTW is an extremely dumb thing to do, and results in many home fires) They protect against damaged or loose connections and wiring concealed in walls and in junction and switch boxes. That said, they can occasionally be a real PITA to deal with. They can false trip when things like vacuum cleaners and blow dryers are used, due to arcing of the brushes in the blower motors. In your case, my opinion is that you probably had a genuine arc when the power failed the first time, but it was outside the home, when a high voltage transformer reset, or a tree hit the lines. The fact that two separate circuits tripped at once means that it's highly unlikely that you have an internal issue.
Here is a bit more from the National Electrical Manufacturers Association:
How does an AFCI work?
In essence, the detection is accomplished by the use of advanced electronic technology to monitor the circuit for the presence of “normal” and “dangerous” arcing conditions. Some equipment in the home, such as a motor driven vacuum cleaner or furnace motor, naturally create arcs. This is considered to be a normal arcing condition. Another normal arcing condition that can sometimes be seen is when a light switch is turned off and the opening of the contacts creates an arc.