[论文] Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Sur...

论文概要

研究领域: CV 作者: Stavros Nousias, Akis Nousias, Konstantinos Gkrispanis 发布时间: 2026-10-14 arXiv: 2610.20585

中文摘要

当前针对非结构网格上稳态偏微分方程的代理模型方法，对整个解施加单一的混合机制。本文提出一种架构，将解分配给多个普通 Transformer 专家，每个专家关注不同的频谱尺度。该分配基于离散 Hodge-Laplacian \(L_0=\star_0^{-1} d_0^{\top}\star_1 d_0\) 构建，它作为拟合层度量高频内容。门控机制以网格为输入，生成逐顶点的权重来重组各专家的输出。我们在 AirFRANS（稳态雷诺平均 Navier-Stokes 翼型基准）的降采样重网格版本上进行评估，每个非结构网格包含 9.6K 面和 4.9K 顶点。我们在相同数据集和数据划分上与 Transolver 比较，本文方法将相对 \(L_2\) 误差降低 31%，内部均方误差降低 47%。

原文摘要

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 co...

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