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

Abstract

Current approaches for steady partial differential equations surrogate models on unstructured meshes apply a single mixing mechanism to the whole solution. This paper proposes an architecture that partitions the solution among vanilla transformer experts, each focused on a different spectral scale. The partition is built from a discrete Hodge Laplacian $L_0=\star_0^{-1} d_0^{\top}\star_1 d_0$, that serves as a fitting layer measuring high-frequency content. A gating mechanism takes the mesh as input and produces the per-vertex weights that recombine the expert outputs. AirFRANS is a steady Reynolds-averaged Navier--Stokes (RANS) airfoil benchmark. We evaluate our approach in a reduced remeshed version of the benchmark with each unstructured mesh counting 9.6K faces and 4.9K vertices. We compare our approach with the Transolver on the same dataset and dataset slit. Our approach reduces relative $L_2$ error by 31% and interior mean squared error by 47%.