Around a month ago there was a PINN post[1] on here and there was a healthy amount of skepticism in the comments. Even in the toxic positivity of LinkedIn, commentors say they're overhyped when a ML "Influencer" posts that one GIF with a MLP and PINN fitting to an oscillator. I would be interested to see what they're actively being used for.
I fully agree with the comments in that post. I started studying them because, well, they sound really cool, but my first impression was definitely that this sounded like a lot of effort (computationally) to solve a single equation.
But then I only tested on simple equations where traditional solvers have no difficulties.
In my previous position, we studied the behaviour of black holes with exotic geometries, and we never could make our solvers work with the added time dependence. I would be very curious to see how a PINN would have fared on this (given enough compute time of course).
[1] https://news.ycombinator.com/item?id=42769623