Air carrier Captain reported GPWS terrain warning while descending at a high rate; on a visual approach at night; in mountainous terrain. The Captain started a climb until the warning stopped then continued the approach to landing.

Narrative

We were approaching TYS from the South when Approach turned us to a heading of 314 and cleared us to descend to 6000 feet. I briefed the approach as a VFR approach backed up by the RNAV 5R. Weather was CLR skies and better than 10SM at altitude. I had the Terrain Mode on which was part of my brief going into TYS. We were 12NM from the FAF VOYUR on the approach when ATC asked if we had the Airport in sight. We did have the airport in sight. Then ATC cleared us for the a VFR approach 5R. I then went direct to the FAF and select 3000 ft for the altitude and FLCH. Our speed was 210. The rate of decent was 1000-1200fpm. On the MFD the TERRAIN Mode was displaying low-density green dots ahead of us. The visibility was good and we see the lighted tower on the last ridge line to cross before the valley. The highest point on the ridge line on the was 2817msl (which was a tower). After passing at around 3;700ft we got the audio and message on the MFD for terrain. I then shallowed out the rate of descent to 500-600fpm. Message went away. TERRAIN mode on the display was still showing low-density green dots and looking outside we could see we were passing over the ridge line. Then we got a TERRAIN; PULL UP audio. I kicked off the AP; and started to climb. After climbing for about 5 seconds the audio stopped. Leveled off at 3;500ft. There was no indications of any RED or AMBER dots of the MFD in TERRAIN Mode. ATC also didn't warn us that we might have been to low in that region. We contacted ATC after the level off. ATC replied that the terrain we just crossed was 2000msl. Once over the valley we then continued our approach and landing with incident.Cause: Descending to early in a high terrain area and descending at to high of a rate contributed to GND PROX warning.Solution: Delay top of descent till you know you are clear of terrain. If not sure ask ATC what's the MVA altitude in the area you're flying through. And if necessary; ask for vectors to help descending to a lower altitude for a better stabilized approach.