TRACON reported an aircraft on instrument approach turned off the final approach course resulting in the aircraft flying towards a higher MVA while descending. TRACON advised the Tower who then canceled the approach clearance and issued an altitude and heading to be sequenced to the airport.

Narrative

I was training my trainee on the radar position. It was IFR weather and my trainee vectored Aircraft X onto the RNAV GPS Y RWY XXR approach. They were given vectors to and they joined the final approach course to RWY XXR at 090. Once they appeared to be established inbound my trainee switched them to the tower frequency. My trainee and I happened to observe Aircraft X turn westbound off the final as they neared the final approach fix of ZZZZZ. They were in a 084 MVA and approaching an 110 MVA a few miles to their west as they appeared to be continuing their descent. My trainee quickly called the tower and asked them what was going on with Aircraft X turning off the approach and descending. Once it was brought to the tower's attention they took action to cancel the approach clearance; climbed them back to 090; and turned them eastbound. They were switched back to us and we vectored them for the ILS to RWY XXL this time and they landed without incident. When asked why the aircraft turned westbound prematurely off the GPS approach they replied that they thought they were following the approach which doesn't make sense and is a concerning answer.Suggestion: If my trainee and I had not been paying extra attention to an aircraft we were no longer talking to; the appropriate action might have taken too long and the aircraft could have easily entered the 110 MVA while still descending on an approach he thought he was still on. I think they may have descended below the 084 MVA but I don't remember the low altitude alert going off which would have helped to get the tower's attention focused faster too. Traffic at the time was light which means attention is hard to keep focused and people are easily distracted. The aircraft needs to figure out why their equipment took them dangerously off course while flying an IFR approach. The consequences could have been really bad.