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.
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.