You might have a type of morning glory called field bindweed. It's vicious stuff! I have fought it for 15 years and just recently found out exactly what it is. My local extension office identified it and gave me some advice. They said I got this from bringing in soil or manure from a source that was infested with it. (It was probably the 2 year-old cow manure I got from a local farmer)
A patch of that stuff can create a deep root system that covers a 30 sq.ft. area in one season. That's why just digging, spraying, and pulling occasionally doesn't help. You must immediately spray or pull every sign of the stuff for months, even years, until you starve the root system. It will spread into other areas to find a way to get sunlight. Watch your lawn, bushes, etc.
This stuff is right up there with Canada thistle. In fact, one sight I found said that when a farm field is infested with bindweed it reduces its property value.
I have a pdf that shows the leaves, flowers
of the 2 main types of bindweed, but I can't seem to find a way to paste it into this forum. If anyone can help me, I will provide more pics and illustrations from web sites.