Can MeshCNN be adapted to train a PDE forward emulator?

[lanzithinking]

Hi, this is very nice work! I understand MeshCNN is designed to represent 3d objects. I am wondering whether it could be adapted to train a PDE forward emulator where the inputs are discrete parameter (e.g. PDE coefficient) values over (2d/3d) finite element mesh and outputs are the PDE solution evaluated at certain points. It is more like a regression problem defined on FEM space. Particularly, if the physical domain is 2d, is it possible to adapt MeshCNN for such application? Thanks!

[ranahanocka]

Hi @lanzithinking ,

I think it should be possible, but I am not clear on all the details. Can you give more information about what exactly is the input and outputs? Are all the meshes with the same connectivity? What do you expect to use it for in inference time? I will also suggest you take a look at our follow-up works: Point2Mesh and Deep Geometric Texture Synthesis, which use MeshCNN for regression tasks. Basically, if you have the same connectivity you can use L2-loss for regression (like in geometric textures), but if you don't have correspondence, then you should use something like the bi-directional chamfer distance we used in Point2Mesh.

[lanzithinking]

Hi @ranahanocka ,

Thanks for your reply. I want to build a meshCNN for the forward map G: X -> R^d, u -> G(u), where X is a Hilbert space of functions defined on irregular shaped (e.g. circular disc) domain on R^2. So the inputs are these nodal values of functions {u} on a given mesh and the outputs are PDE solutions (given u) observed at d locations. Repeated PDE solutions are expensive for different u. I want to build a meshCNN emulator G*: X -> R^d, u' -> G*(u') by training on pairs {u, G(u)} solved already. u is defined on a irregular shaped domain so I cannot train standard CNN with rectangle domain. It would be nice if you could give me more hints on whether this is doable by MeshCNN. Thanks for your references. I will check them out!