These methods are really interesting for high-dimensional PDE (like HJB), but there's a ton of skepticism about the applicability of NN models for solving the more common PDE that arise in physical sciences and engineering.
The tests are rarely equivalent, in that standard PDE technology can move to new domains, boundary conditions, materials, etc., without new training phases. If one needs to solve many nearby problems, there are many established techniques for leveraging that similarity. There is active research on ML to refine these techniques, but it isn't a silver bullet.
Far more exciting, IMO, is to use known methods for representing (reference-frame invariant and entropy-compatible) constitutive relations while training their form from observations of the PDE, and to do so using multiscale modeling in which a fine-scale simulation (e.g., atomistic or grain-resolving for granular/composite media) is used to train/support multiscale constitutive relations. In this approach, the PDEs are still solved by "standard" methods such as finite element or finite volume, and thus can be designed with desired accuracy and exact conservation/compatibility properties and generalize immediately to new domains/boundary conditions, but the trained constitutive models are better able to represent real materials.
A good overview paper on ML in the context of multiscale modeling: https://arxiv.org/pdf/2006.02619.pdf
