Proposal- Creating a tutorial comparing PINNs with traditional numerical solvers

[vinay31568]

Hi everybody,
I wanted to add a new tutorial to the documentation. My idea is to compare a physics informed neural network (PINN) to traditional numerical methods (like finite element or finite difference) for solving common partial differential equations (PDEs).
I would cover few examples like Burgers Equation or the Heat Equation, and document the accuracy, convergence rate , training and solving time etc.
Hopefully this would be very valuable for people who have fair idea of traditional methods and want to understand some practical implications of using PINNs.
Is this a contribution worth pursuing? Looking forward to feedback.

[dario-coscia]

Hi @vinay31568!

Thanks for suggesting the tutorial. At the moment, we do not have a direct comparison between PINNs and traditional numerical methods, and I think it would be interesting for users to understand the practical implications of PINNs. For tutorials, we follow the Tutorials Guidelines, and your tutorial could be part of the application purpose section. If you want to propose a tutorial, please follow those guidelines.

Nevertheless, I doubt a PINN could compete in terms of accuracy with a numerical in general. PINNs are better suited for parametric cases, inverse problems, or high-dimensional problems, and that's where they can compete. On this note, would you be interested in benchmarking PINNs for a parametric case? I think it would be very valuable for the PINA community. The Heat equation seems like a good starting point, where the parameter could be the diffusivity constant and/or the heat source placement in the domain. We already have a class for parametric problems, so you could start from there.