Feeling unprepared for PhD because of poor educational system in my home country. Should I leave PhD to retake courses somewhere else? I will give some background. I am from a third world country, and I am doing my math PhD in Europe.  During high school and university years I participated in many maths competitions, and even got to the pre-national team. (I mention this just to show how interested I have been over the years with the topic.)

Here is my problem: my education is not the best. I would say it is mediocre at best. By education I mean the whole system of the country. I pretty much had to learn everything on my own during those years, starting from Analysis. So, even though I took courses, they were meaningless, because the level was very low and in many cases the professors were not experts in the field. On top of that, the interaction with other students was almost null, I was very ahead of all of them.

Despite that, as I said, I studied very hard to get to where I am now. However, I have to admit that due to this personal training, there are many topics that should be basic with which I don't feel comfortable about: ODEs, PDEs, Numerical Linear Algebra, etc. This is due to time constraints because I feel capable of learn anything.

To summarize so far: I have been a lonely man in a cave with a book for the last 7 years. That being said, I managed to get some decent knowledge in the area I am currently working in.

Fast forward to the PhD program. As usual in Europe, there are no courses involved: only research. And there is were my conflict lies: I am very unsure of myself because I think that everyone has a better education than me and hence they are better prepared. Another issue is that the institute were I am is very small and hence there is no ‘big purpose’: no one seems to care about you. Also, the institute is not well known, so that I am also afraid that the ‘brand’ issue will affect me later.

I feel like I don't belong there, and that I could do much better if I go to another Program that starts the courses from the beginning. However, the way I am doing it right now has got me very demotivated and it is starting to affect my research: I have been here for a year already and so far I am only at the middle stage of a first paper. The paper is not that strong, because after I got the results I found that someone has done it before (I lost months on this).

I guess that what troubles me most is the fact that, due to lack of data, I can not compare myself (main source of motivation, in a healthy sense of course) with the rest of my peers and that is the main reason for my insecurities. The comparison is in the sense of me being sure that I am roughly at the same level of my colleagues. I know no other way to do this other than taking the same courses and tests. This comes naturally with a program in which one take courses on the first year but, as explained earlier, I didn't have this interaction with really good students. Due to this, I am thinking of quitting and starting all over again somewhere else. Also, I don't think this is a case of the Impostor Syndrome: I am sure that with proper training I could get far, I just haven't had the opportunity to take a bunch of graduate courses.

Should I proceed?
In the European programs I have been involved in, there is no (or a very minimal) fixed curriculum students need to go through. However, students do follow specific courses to learn specific techniques. Typically/ideally the student discusses with her/his advisor what is needed for the project, what is needed for her/his future career, and where the gaps in her/his knowledge is, and how to fill those gaps. This is the kind of discussion you need to have with your advisor.