SpiderSolver: A Geometry-Aware Transformer for Solving PDEs on Complex Geometries
Kai Qi, Fan Wang, Zhewen Dong, Jian Sun
Transformers have demonstrated effectiveness in solving partial differential equations (PDEs). However, extending them to solve PDEs on complex geometries remains a challenge. In this work, we propose SpiderSolver, a geometry-aware transformer that introduces spiderweb tokenization for handling complex domain geometry and irregularly discretized points. Our method partitions the irregular spatial domain into spiderweb-like patches, guided by the domain boundary geometry. SpiderSolver leverages a coarse-grained attention mechanism to capture global interactions across spiderweb tokens and a fine-grained attention mechanism to refine feature interactions between the domain boundary and its neighboring interior points.  We evaluate SpiderSolver on PDEs with diverse domain geometries across five datasets, including cars, airfoils, blood flow in the human thoracic aorta, as well as canonical cases governed by the Navier-Stokes and Darcy flow equations. Experimental results demonstrate that SpiderSolver consistently achieves state-of-the-art performance across different datasets and metrics, with better generalization ability in the OOD setting. The source codes will be released.