OK, that is weird. And you are sure it's powering down, rather than the screen not waking again? It's easy to tell: if it was just the screen you'd stlll hear notifications, and to restart it you'd have to hold the power key down for more than the couple of seconds you need if it has shut down.
To be honest I've never heard of this before. It may be a faulty device, so making sure it's backed-up in case you need to replace it is probably sensible. I'm hoping someone will come along with some other ideas, because all I can think of is a few very generic things to try (which are in my opinion unlikely to work).
First thing, has it always done this since it was new, or did it start at some point?
You could try booting into safe mode: this requires rebooting, then pressing the "volume down" key when you see the words "Galaxy Note 8" on the screen and keeping it pressed until you see the words "safe mode" in the bottom-left corner. What this does is disable all user-installed apps temporarily, so if it works normally in this state it's one of your apps that's causing it (then it's a matter of working out which).
You could try booting into recovery mode. Rather than type instructions, just follow
this link (with pictures). I'd do it with the hardware buttons and ignore the other stuff, which requires installing software on a computer. When in recovery you should have an option to wipe the system cache (along with other options). This really doesn't sound like the sort of thing that a cache wipe will fix, but as long as you only wipe cache you lose nothing and there's a very remote chance.
If all else fails, and especially if this wasn't this way from the start, you could back up and do a factory reset. If you get to the point where it's that or replace it then you've nothing to lose by trying this.