I'm having this problem! My dead phone is now recharging while I search for a fix. Thought it was something I'd left running since I'm a new Android user but I turned off virtually everything yesterday and same thing - the phone drains overnight!!
Really strange because I got fairly good battery life during the day while I was using the phone and still had 61% when I last checked at 11pm. Yet at 6am the phone is dead, same as the day before.
I'm going to check the time-without-signal and try the airplane mode but sure would like to find a solution. Great phone but to have the battery drop so quickly with no use isn't something I want to be dealing with.