PGOT: A Physics-Geometry Operator Transformer for Complex PDEs

TL;DR

PGOT introduces a geometry-aware Transformer architecture with spectrum-preserving attention and adaptive routing, significantly improving PDE modeling on complex unstructured meshes with diverse geometries.

Contribution

The paper proposes PGOT, a novel Transformer with explicit geometric encoding and adaptive computation routing, addressing geometric aliasing and enhancing PDE modeling accuracy.

Findings

Achieves state-of-the-art results on four benchmark PDE tasks.

Excels in large-scale industrial applications like airfoil and car design.

Maintains linear computational complexity while preserving multi-scale geometric features.

Abstract

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.