I have a sliding glass door off my kitchen which used to have a screen on it (the screen slid on a track just outside of the glass doors). When I moved into my house the screen that was there was a replacement screen (I do not think it was the original) which quickly ripped (the screen that was there just was not made for 4 kids 6 years and under). I tried another replacement screen from Home Depot this spring and although the screen itself did not rip, it just did not fit perfectly, would not slide smoothly, and got bent and went in the garbage in about 2 weeks.
I would try another replacement but it would need to be something better than what I got at Home Depot last. And I would go to the door manufacturer but the name on the glass is Guardian and I have not been able to find them.
UPDATE (2011-09-18): The patio door is an American Craftsman by Andersen sold at the Home Depot. See my answer for more info.
Both are very flexible and robust materials.
You can remove one side of the frame and replace the screen then.
Best Answer
Guardian is the manufacturer of the glass itself, but not the door assembly. See http://www.guardian.com/en/na/gd_002563.html
I think the key to finding the proper sliding screen is going to be finding out the manufacturer of the assembly. Have you looked all around the door frame for a nameplate?