Does he ever lie down on his own without going belly up? Capturing behavior would work with him. When you see him lie down sphinx sytle or even over on one hip, toss him a treat and praise low key. Catch him as often as you can. What can really help communicate to him what you mean is clicker training. You can read up. Same thing with "come." If he just wanders over to you on his own, BEFORE he rolls over, give a treat. Have treats in your pockets all the time. Later when he starts offering these behaviors a lot, add a cue word, "down," or "come" WHILE he's doing it, not before yet. After you think he's made the association with the cue word and the action, try it out ahead of time. And then only reward the down or come WHEN you give him the cue word and stop giving treats for the behaviors he offers on his own. The exception to this is the coming to you. Always reward him for coming to you. You don't have to always reward with a treat, but praise him for coming to you no matter what.
http://www.clickersolutions.com/articles/index.htm
Some other things you can do are, when he rolls on his back when he gets to you, walk away (as heartless as that sounds). Don't rub his belly which reinforces that. Another thing you can do is put the rolling over on cue. And only reward with a belly rub the instances where you've given the cue. ("roll over") or whatever. Use your cue at the same time that he rolls over, not before. Remember, he doesn't understand English yet. You have to tie the cue word with the behavior for a while until you think he has made that association. Then try it out ahead of time and see if he rolls over.
Another thing you can do is some additional confidence building games; hide and seek, tug of war (as long as he learns the rules) Rules are no biting your hand, even by accident and he must know "give" so when you ask for the tug toy, he lets go. You teach that first. Trade him for a treat to get him started on letting go. He must wait for you to start rather than grabbing the toy out of your hand. Let him win sometimes. And more basic obedience practice, of course, just like you're doing helps. I don't recommend using any sterness with this pup. He may be rolling on his back for a belly rub, but it may be also that he is submissive and any sterness can make him more so. (not that I think you are, just a general notation)
And finally, do you have any pictures? I bet he's adorable.