基于可学习 Hodge-Laplacian 的混合专家架构提升稳态 PDE 代理模型精度
Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes
一篇 PDE 求解论文，用 Hodge-Laplacian 按谱尺度把任务分给多个 transformer 专家，在 AirFRANS 基准上比 Transolver 误差降了 31%。
论文提出一种面向非结构网格稳态 PDE 代理模型的混合专家架构，将求解任务分配给多个 transformer 专家，每个专家专注不同谱尺度。划分依据由离散 Hodge Laplacian 构建的高频内容度量层，门控机制以网格为输入生成逐顶点权重来重组专家输出。在 AirFRANS 稳态 RANS 翼型基准的重网格子集上评估，每个非结构网格含 9.6K 面和 4.9K 顶点。与 Transolver 在相同数据和划分下对比，相对 L2 误差降低 31%，内部均方误差降低 47%。
Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes
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\%.
