I am looking for some good sci-fi movies, NOT mindless action but smart problem-solving type stuff. Like a Star Trek TNG two-parter. Any ideas?

Time Machine is a decent movie,either the Rod Taylor one or Guy Pierce's remake.

Predestination
Coherence
Triangle
Timecrimes
Source Code
Primer
ARQ
Looper
Cube

Except for the last one, those deal with all sorts of time travel paradoxes, Groundhog Day-like time loops, the concept of alternate realities/the multiverse, even basic quantum physics.

Which two-parter? Encounter at Farpoint, Best of Both Worlds or Chain of Command?

Some suggestions of "problem-solving" movies:
Andromeda Strain (the first one)
Epoch (the first one)
Contact

Just a second, I'll go open the DVD cabinet...
Hmm...tough to find something with exactly those parameters. Most scifi films that try to be smart, almost get there, but basically turn into a mindless action film in the end.
And most scifis made specifically to make the audience "think" or make a specific statement about society without using action to liven things up now and then eventually ended up with material so dry, it's unwatchable.

Blade Runner comes to mind. Not chiefly an action film, more like a futuristic detective film, themes about AI and such. Worth being in you bucket list if you haven't seen it already.

iRobot isn't too bad at first, if you don't mind Will Smith, but it fell into the same trap as I described above and got too mindless in the end.

Robocop dealt with some interesting themes, but was an action movie through and through (the sequels were franchise trash and much more mindless).

Cronenburg's The Fly was an interesting little excursion, if you like body horror and such.

A.I: Artificial Intelligence Is an interesting story, but tends to drag on in places, and is a bit sad for my liking. I've only seen it twice. Certainly not an action film that one.

I'll post more if they come to mind.

"The Martian" was very good

Solaris

Hmm. Interstellar tries to be smart, is a good movie and gets quite some physics right. (Until they do a single-stage to orbit start from a planet so deep in a gravitational well, that time dilation becomes a macroscopic effect... with a shuttle that didn't even do a single-stage to orbit start on Earth! ) But it is no problem-solving movie.

But how about Ghost in the Shell (the Anime) and it's sequel(s)?

Also I agree that Blade Runner falls into this category and is an awesome movie! The sequel didn't quite reach the same class, but was still a good movie and a worthy sequel.

And since you already mentioned Star Trek, Babylon 5 is quite smart, has lots of problem solving both social and intellectual and has a long plot-arc, that spans several seasons. Definitely a recommendation!

The Martian comes into my mind...

edit: damn, ninja'd :D
i had the reply-window open for a while to think about what.. it was too long :D

Seconded. Highly entertaining, intelligent and well written/acted.

A lot of good recommendations here. I would also suggest:
Moon
Arrival

Yes, Arrival was good too. And it has Amy Adams in it. :-)

NOt ninja

Space Pirate!

Oh and how about Gattaca? It's not so much about detective work as it is about fooling those detectives. To succeed in a world hostile to genetically non-optimized people.

I don't know if "The Man from Earth" is considered sci-fi (IMDb does) but it's a very captivating movie about a man who tells a rather outrageous story and his friends listen to it and try to decide whether he's telling the truth or lying. Saying anything more would spoil it but I recommend it.

Maybe "Interstellar" even if there isn't a lot of problem solving. As mechmouse said above, "The Martian" has a lot more of that (basically MacGyver in space). "Moon" is another very good movie, with Sam Rockwell in the lead, but it's more of a drama/mystery.

The new "Total Recall" has some twists and turns that'll keep you guessing, especially if you watch the Director's Cut.

I'll get back to you if something else comes to mind.