Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes

Transformer surrogates for steady partial differential equations on unstructured meshes apply a single mixing mechanism to the whole solution. This paper proposes an architecture that partitions the solution among experts, each focused on a different spectral scale. The partition is built from a discrete Hodge Laplacian $L_0=\star_0^{-1}d_0^{T}\star_1d_0$, a Dirichlet-energy operator that has two uses, since as a fitting layer it imposes smoothness, while applied to a field it measures high-frequency content. The architecture employs two routes, one containing vanilla transformer experts and the other learning spectral components that direct the experts toward specific scales. The routes are fused at a gate that reads the whole mesh and produces per-vertex weights to recombine the experts' outputs. We evaluate the architecture on AirFRANS, a steady Reynolds-averaged Navier–Stokes (RANS) airfoil benchmark. Velocity and pressure are almost entirely smooth, while the turbulent viscosity concentrates at the wall and in the wake and is the least accurate field in every monolithic surrogate we trained. On the official split, our approach reduces relative $L_2$ error by 31\% and interior mean squared error by 47\% relative to a Transolver baseline at half the parameter count, though it trails the baseline on the two wall quantities.