[PontifexCipher]

This is not just PR and is very interesting. However, in my view, (and from a quick read of the paper) this is actually a very classical method in applied math work:

- Build a complex intractable mathematical model (here, Navier-Stokes)

- Approximate it with a function approximator (here, a Physics Informed Neural Network)

- Use the some property of function approximator to search for more solutions to the original model (here, using Gauss-Newton)

In a sense, this is actually just the process of model-based science anyway: use a model for the physical world and exploit the mathematics of the model for real-world effects.

This is very very good work, but this heritage goes back to polynomial approximation even from Taylor series, and has been the foundation of engineering for literal centuries. Throughout history, the approximator keeps getting better and better and hungrier and hungrier for data (Taylor series, Chebyshev + other orthogonal bases for polynomials, neural networks, RNNs, LSTMs, PINNs, <the future>).

You didn't say anything to the contrary, and neither did the original video, but it's very different than what some other people are talking about in this thread ("run an LLM in a loop to do science the way a person does it"). Maybe I'm just ranting at the overloading of the term AI to mean "anything on a GPU".

[hodgehog11]

This is absolutely true, but it still makes use of the advantages and biases of neural networks in a clever way. It has to, because computationally-assisted proofs for PDEs with singularities is incredibly difficult. To me, this is not too similar from using them as heuristics to find counterexamples, or other approaches where the implicit biases pay off. I think we do ourselves a disservice to say that "LLMs replacing people" = "applications of AI in science".

I also wouldn't say this is entirely "classical". Old, yes, but still unfamiliar and controversial to a surprising number of people. But I get your point :-).