Choosing a surrogate-model architecture to predict steady-state thermal fields from CAD geometry — FNO vs PINN vs GNN?

I work on thermal and CFD simulation for electronics and battery systems, and I’m trying to stand up an ML surrogate that predicts steady-state temperature fields from a parameterized geometry plus boundary conditions, so we can explore design variants without running a full solver each time.

I’ve narrowed it to three candidate approaches and would value input from anyone who’s shipped something similar:

1. Fourier Neural Operators (FNO) — attractive for resolution independence, but most examples I’ve seen are on structured grids. Has anyone made FNO work cleanly on irregular engineering geometries without heavy remeshing onto a regular grid?

2. Physics-Informed Neural Networks (PINN) — appealing because they bake in the governing equations, but I’ve found training stability poor on stiff problems with sharp gradients (which thermal hot-spots are). Have people gotten PINNs reliable enough for engineering-grade accuracy, or are they still mostly demonstrative?

3. Graph Neural Networks on the mesh directly — feels the most natural fit for unstructured meshes, but I’m unsure about generalization to geometries outside the training distribution.

My constraints: training data is expensive (each label is a full CFD/thermal solve), so sample efficiency matters a lot, and I need errors low enough to trust for early design screening, not just qualitative trends.

For those who’ve built PDE surrogates in practice — which of these held up, and where did the accuracy actually break down?