PGOT通过几何感知注意力，精准建模复杂边界的偏微分方程。
PGOT: A Physics-Geometry Operator Transformer for Complex PDEs
- 提出谱保持几何注意力，显式保留多尺度几何特征。
- 在4个标准基准上达到最优，大尺度工业设计任务表现突出。
- 自适应路径路由，对光滑区与激波区分别处理，精度高。
尽管Transformer在建模偏微分方程（PDEs）方面展现出巨大潜力，但在处理具有复杂几何结构的大规模非结构化网格时仍面临挑战。现有高效架构常采用特征维度压缩策略，导致几何混叠，丢失关键物理边界信息。为此，我们提出物理-几何算子Transformer（PGOT），通过显式几何感知重建物理特征学习。具体地，提出谱保持几何注意力（SpecGeo-Attention），采用“物理切片-几何注入”机制，融入多尺度几何编码，在保持线性计算复杂度O(N)的同时，显式保留多尺度几何特征。此外，PGOT根据空间坐标动态分配计算路径：光滑区域走低阶线性路径，激波与间断处走高阶非线性路径，实现空间自适应的高精度物理场建模。PGOT在四个标准基准上持续达到最先进性能，并在空气动力学翼型和汽车设计等大规模工业任务中表现优异。
原文摘要 · Abstract (English)
While Transformers have demonstrated remarkable potential in modeling Partial Differential Equations (PDEs), modeling large-scale unstructured meshes with complex geometries remains a significant challenge. Existing efficient architectures often employ feature dimensionality reduction strategies, which inadvertently induces Geometric Aliasing, resulting in the loss of critical physical boundary information. To address this, we propose the Physics-Geometry Operator Transformer (PGOT), designed to reconstruct physical feature learning through explicit geometry awareness. Specifically, we propose Spectrum-Preserving Geometric Attention (SpecGeo-Attention). Utilizing a ``physics slicing-geometry injection" mechanism, this module incorporates multi-scale geometric encodings to explicitly preserve multi-scale geometric features while maintaining linear computational complexity $O(N)$. Furthermore, PGOT dynamically routes computations to low-order linear paths for smooth regions and high-order non-linear paths for shock waves and discontinuities based on spatial coordinates, enabling spatially adaptive and high-precision physical field modeling. PGOT achieves consistent state-of-the-art performance across four standard benchmarks and excels in large-scale industrial tasks including airfoil and car designs.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。