PGOT: A Physics-Geometry Operator Transformer for Complex PDEs

2025-12-28

AI Summary

To address geometric aliasing and boundary information loss in large-scale unstructured-mesh PDE solvers on complex geometries, this paper proposes a physics-geometry-coupled Operator Transformer architecture. The method integrates physics-informed slicing with geometry injection, multi-scale geometric encoding, and linear-complexity attention design. Key contributions include: (1) SpecGeo-Attention—a spectrum-preserving geometric attention mechanism that explicitly models multi-scale geometric features while suppressing frequency-domain distortion; and (2) a spatially adaptive dual-path computation routing scheme that jointly leverages low-order stability and high-order accuracy. Evaluated on four standard PDE benchmarks, the approach achieves state-of-the-art performance in accuracy, generalization, and geometric fidelity. It demonstrates practical efficacy in industrial aerodynamic design tasks, including airfoil and full-vehicle simulations, where it significantly improves solution quality and robustness under complex geometric constraints.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Hardware-aware ML

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Modeling large-scale unstructured meshes with complex geometries in PDEs
• Addressing geometric aliasing and loss of physical boundary information
• Enabling spatially adaptive and high-precision physical field modeling

Innovation

• Spectrum-Preserving Geometric Attention for multi-scale geometry
• Physics slicing-geometry injection mechanism to avoid aliasing
• Dynamic routing between linear and non-linear computational paths