The weather institute is now on Mt.TeraJuma now
Get their from Kakori Village's Rail lift thing? (Top right of Village or going this path
From the Braviary Statue at beginning of Deep terajuma Jungle>Top Left of map>Deep Terajuma Jungle again>Mirimura Cave
Go up lift>Exit back out to Deep Terajuma Jungle (Area with the Binoculars and the Chatots sitting on it)
Go down on this tile here
Go all the way to the right to 2nd lift, take the lift up, Exit cave to Mt.Terajuma
Cross the bridge and go up the stairs to top, go right
If you see this statue here, The Weather Institute is right here