HGsolver: Position-Enhanced Physics Attention Informed Heterogeneous Geometries Neural Solver for PDEs

Zhonghao Li, Changhong Zhong, Zhang Qian

Keywords: Neural PDE Solver, Physics Attention, Heterogeneous Geometries, Positional Encoding, Sparse learning

TL;DR: Position-Enhansced Physics Attention Informed Heterogeneous Geometry Neural Solver for PDEs

Abstract: Partial differential equations (PDEs) provide a fundamental framework for modeling complex physical phenomena. However, modeling PDEs on heterogeneous geometries remains a significant challenge for both traditional numerical solvers and neural operator methods, as sparse observations, multiphysics interactions, and distinct discretizations often produce heterogeneous geometries between the observation and output spaces. In this work, we introduce a unified perspective on physics attention, formulating physical states as projections of observation embeddings onto learnable functional bases in Hilbert space. Building on this formulation, we introduce a position-enhanced physics attention mechanism that incorporates coordinate representations of these bases via rotary position embeddings, thereby enabling more effective modeling of heterogeneous interactions. Leveraging this mechanism, we develop HGsolver, an encoder–decoder framework designed for PDE tasks on heterogeneous domains. Extensive experiments demonstrate that HGsolver achieves state-of-the-art performance across forward, inverse, and reconstruction benchmarks under heterogeneous geometries, while a minimally modified variant, TransolverXP, also delivers competitive results on standard homogeneous benchmarks. These findings highlight the importance of effective interactions among physical states in advancing neural PDE solvers and their potential to address the complexity of the heterogeneous real-world geometries.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 9544