Hello Everyone, Little Background: I am a graduated Math and Computer Science Major, currently working with a professor on PINNs, more on the error analysis and convergence side.
I am currently in the process of applying to PhD in Applied Math, my research supervisor for undergraduate research will hopefully be my supervisor for the PhD and we have talked about going into SciML as focus for my PhD along with structure preserving methods. The problem is while working for him it is very overwhelming to comb through the research due to high number of publications, and I once I was able to grasp everything coming up with anything new was difficult because all the possible ideas in my grasp were already done. This is also due to my limited knowledge as an undergraduate, there is only so much math and comp sci I could’ve learnt during my undergraduate to help me come up with ideas. Also this is not to mention how terrible PINNs are as a solver. But even after this I came up with an idea and did my due diligence of looking it up thoroughly in the literature. I was starting to compile the results and found a paper with the same ideas as me, this really demotivated me and now I am questioning my decision to do a PhD in this area because if this keeps happening I would just not be graduating. There is also almost no one doing error bounds on SciML methods even though they are trying to compete with traditional methods, and their numerical simulations often seem naive at best. Usually they would do a 3-4 dimensional system as a high dimensional benchmark and say that should be enough proof of the method working. Coming from a math background I simply don’t buy that. So even though I came up with an idea and worked on it, some person who has a half-assed paper on arxiv is the reason I cannot write on a topic.
I wanted to know your thoughts on this, especially who joined PhD since 2019.
This is my first post on this subreddit please let me know if I violated any rules.
Hi I am a PhD student and work with PINNs. Technically my department is CS but I have a physics background and work in a group of physicists. I agree with your thoughts that most papers show a toy example and call it a day. I feel that more rigour is definitely needed for PINNs to be useful as replacements for numerical methods. Right now PINNs are not used for big scientific problems precisely because there are no error bounds or theoretical guarantees. Training is very finicky as well. It still seems to be early days, and the field might become more useful as it matures. Somebody with a math background and more mathematical rigour would be a welcome addition!
Finding the research gap is part of the phd process. It can be demotivating but think of your phd as an apprenticeship. You shouldn't expect grand results while you are learning to be a scientist. If there is a half-assed paper, you can still cite it and continue working on your own thing and publish it.
I would be happy to discuss more, not sure what I can add.