Dynamic sparse physics-aware network for solving partial differential equations on multiple grid types

Abstract Neural network have been widely applied for solving partial differential equations (PDEs), but they still exhibit limitations. Recently, transformer-based operators, which have demonstrated exceptional performance in fields like computer vision, have been adapted for PDE solving. We propose a dynamic sparse physics-aware network (DSPAN) based on a physics-slicing mechanism and a TopK selection architecture. First, the input features are partitioned into slices to capture the intrinsic physical states. Then, these slices are scored for importance, and only the top K most critical slices are selected for attention computation. This architecture effectively captures the underlying physical information in PDEs while improving computational efficiency, achieving dual gains in both accuracy and speed. Experiments show that DSPAN achieves a relative improvement 15.35% on eight benchmark data sets, with a particularly notable 33% enhancement in plasticity equations. This robustly validates the algorithm’s capability to deliver highly accurate and resource-efficient solutions for complex PDE systems.