PA28 Flight Instructor reported receiving a low altitude alert warning from ATC during cruise due to uncontrolled descent in a downdraft.

Narrative

We had leveled off at 11;000 feet on an IFR training flight from ZZZ to fly. I am the CFII and the student was another CFII who was very new and trying to get IFR experience. On our previous leg from ZZZ1 to ZZZ; we noted strong thermal activity and strong surface winds at ZZZ. Due to the high temperature and downdrafts we struggled to climb to altitude; but eventually made it. We encountered continuous light chop; but primarily downdrafts unlike our previous leg. We began hearing turbojets reporting mountain wave in the flight levels; and further in the flight saw standing lenticulars which we reported to ATC. Near the ZZZ VOR; we encountered a downdraft that was very persistent and even at Vy with full power; we were descending. Upon reaching 10;700 I advised center that we were unable to hold altitude and would be climbing back to 11;000 as soon as able. Controller than advised of a low altitude alert. Controller then asked if i could guarantee holding altitude. I responded by explaining that I can as long as we don't hit another downdraft that bad. He vectored us heading 020 to a lower MVA. He then explained that he needs guarantee that I can hold the min IFR altitude and vectoring me 50 miles east; off course would be an option. I explained that we would rather not go 50 miles east and try to hold 11;000 since the downdrafts had only been that bad once. I also explained that if we encountered another severe downdraft that we would cancel IFR. He seemed comfortable with that plan and handed me off to ZZZ Center and said that he notified them of our situation. A few minutes after checking in with Center; we were given a warning that was forwarded on from ZZZ1 Center and given a phone number. I talked with the supervisor who explained that descending wasn't the issue. He said the issue was that we were below the MIA which triggered a low altitude alert which is a mandatory report. I asked if there is another way to handle this type of situation. He explained that I could request an altitude above the MIA and get a block; and asked if that would have been helpful. I told him that it would have been; and that we did everything we could to not bust altitude.