PINNs certainly do not fulfill their promises, the idea of outperforming a traditional PDE (or just ODE) solver by training a neural network, is on its face rid

PINNs certainly do not fulfill their promises, the idea of outperforming a traditional PDE (or just ODE) solver by training a neural network, is on its face ridiculous. It is very telling that the motivating example in https://physicsbaseddeeplearning.org/intro-teaser.html https://physicsbaseddeeplearning.org/intro-teaser.html totally fails. If you read further you will notice that all other examples also fail and I mean "fail" as in the result is literally useless. (Reading this was genuinely upsetting) The one legitimate area where I can see them being used is for interpolation in complex engineering/physics applications. Surrogate models actually can be a worthwhile endeavor if you need to solve a complex problem over a big parameter space. But of course this is fundamentally different than training a neutral network to solve a single PDE. Even the idea is fundamentally flawed, ODE solvers are well developed and are based on sound mathematical theory. The idea that you can outperform them, by a neural network is on its face pretty silly.