Are there any Universities that accept PhD students based on Research Proposals? I am firm on the problem I want to work (with publications and arcticles to support, for which I have already done significant work)) and I'd like to know if there are any PhD programmes that accept students primarily based on research proposals? Especially in the fields of mathematics or machine learning theory. A google search couldn't give much results. I am interested to know any in the US and Europe.

Most of the universities in US do not seem to be interested in research proposals and they don't ask for it.
If you are asking whether some university will accept you solely based on a research proposal, then I think the answer would be no. You might be able to find an exception of some university-proposal combination that would make it work, but that isn't how it works. Universities accept students into doctoral programs based on the evaluation of a set of criteria that suggest that the student has both the appropriate background and the likelihood of success in the program and thereafter. 

Having a proposal is probably a plus, but, in the US, at least, it isn't required, as you note. It isn't likely even evaluated. 

However, assuming that you have an appropriate background, then you might be able to find a professor somewhere, certainly in the EU and possibly in the US, who is willing to work with you to advance your entrance into a program. In some places, the professor is even an essential link. But the proposal alone isn't going to be sufficient. You will need to convince people, at a minimum, that you can succeed with that proposal and the result is likely to be sufficiently significant. 

However, I think that being too committed to a particular proposal is likely to work against you, rather than in your favor. You need to find an advisor who is either hugely interested in that topic or willing to spend time advising and evaluating something that has less importance to them. Both of those reduce your chances. The latter is, perhaps, more likely than the former. Some professors are happy to have students who need little shepherding and who are self directed enough to get the job done. But I doubt that the majority of people work like that. 

So, the bottom line is that you have a chance, but you probably need to be a bit more flexible and you need to attend to the qualities in your background that suggest success.