When this thread started I wondered if "frost heave" as reported was in fact what had caused whatever problems had occurred here. I knew that some contemporary "black plastic pipes" now used as culverts in some areas have very low effective "stiffness". A rough definition of stiffnesss is a unit load per unit length of pipe that will deflect the pipe ring (normally in the downward direction) a significant amount in a laboratory test. In the field load can be created by the weight and/or settlement of earth over and around the pipe AND by live loads, such as vehicles driving over the pipe. Though soil at normal burial depths normally supports flexible pipes pretty well and limits the deflection amount, a problem that I could see one getting into (particularly at very shallow very few inch cover as you report) is that dynamic live/wheel etc. loads become quite large, and additionally a larger percentage of the total load on a flexible pipe. This might cause the pipe (and whatever is over same, including a pavement) to undergoe more dynamic/repeated and magnitude of downward "flexing". At some point I wonder if even asphaltic pavement that is repeatedly flexed a certain amount might crack? While very lightweight plastic pipes with a relatively smooth exterior surface are perhaps also known to move/float in the ground more than heavier pipes, I doubt that flotation force per Archimedes is what has damaged your driveway. If too much dynamic flexing has however been the case, I suspect your situation might be improved (lesser magnitude dynamic flexing) with one or more of the following:
1. Selection of at least somewhat stiffer pipe.
2. Contruction of firmer pipe support and surrounds.
3. Burial of the conduit as deeply as is practical to satisfy hydrology/flow demands.
With regard to item 1, some of these 12" black plastic pipes are designed and manufactured with a "pipe stiffness" of only about 50 pounds per inch per inch or so. On the other hand, a minimum class ductile iron pipe culvert (e.g. in accordance with ASTM A716) would have a calculated pipe stiffness based on nominal thickness of maybe 20 times that value. Of course a ductile iron culvert would probably cost some more, unless you could snag some left over from a jobsite (as there is very little "free lunch"!) On the other hand, I'm not sure that a very rigid concrete pipe laid very shallow immediately underneath your pavment is necessarily the answer either, per reasons stated by cvg and additionally as very rigid might act as a fulcrum point if the pavement subgrade yields more on either side of the pipe than it does immediately over same (if you then get pavement flexing in a different, negative beam moment fashion?).
I guess an ideal situation would be a finite amount of flexing of the asphaltic pavement over the pipe that is precisely the same as it is anywhere else on your driveway as vehicles roll over same, and that said flexing locally damages neither the pipe nor asphalt (but this may require some quite fancy Engineering!)