Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries

TL;DR

This paper introduces AMG, a multi-graph neural operator framework that effectively solves PDEs on complex, irregular geometries by leveraging multi-scale graphs and advanced graph-based techniques, outperforming existing methods.

Contribution

The paper presents a novel multi-graph neural operator architecture, AMG, that handles arbitrary geometries and complex data dependencies more efficiently than prior PDE solving approaches.

Findings

AMG outperforms previous methods on six benchmark tasks.

The multi-graph approach effectively manages complex geometries.

AMG demonstrates superior accuracy and efficiency in PDE solutions.

Abstract

Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph neural operator approach designed for efficiently solving PDEs on Arbitrary geometries. AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture, enabling precise management of diverse spatial domains and complex data interdependencies. By constructing multi-scale graphs to handle variable feature frequencies and a physics graph to encapsulate inherent physical properties, AMG significantly outperforms previous methods, which are typically limited to uniform grids. We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models. Our findings highlight the transformative potential of tailored graph neural operators in surmounting the challenges faced by conventional PDE solvers. Our code and datasets are available on https://github.com/lizhihao2022/AMG.