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

2024-11-18

arXiv.org

AI Summary

Solving partial differential equations (PDEs) on arbitrary complex geometries remains challenging due to the limitations of traditional mesh-based methods, which suffer from poor generalizability and scalability. To address this, we propose AMG, a multi-graph neural operator framework. Our method introduces three key innovations: (1) a dynamic attention-driven GraphFormer architecture that enables adaptive graph-structured modeling; (2) a dual-graph mechanism integrating multi-scale topological graphs with physics-constrained graphs to explicitly encode geometric irregularity and physical priors; and (3) end-to-end PDE solving on unstructured domains, eliminating reliance on predefined meshes. Evaluated on six standard PDE benchmarks, AMG consistently outperforms existing state-of-the-art methods, achieving significant improvements in both solution accuracy and cross-domain generalization. The source code and datasets are publicly available.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Multiagent Systems: Distributed Problem Solving
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 extbf{AMG} method, a extbf{M}ulti- extbf{G}raph neural operator approach designed for efficiently solving PDEs on extbf{A}rbitrary 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 url{https://github.com/lizhihao2022/AMG}.

Problem

• Complex Geometries
• Partial Differential Equations (PDEs)
• Numerical Computation

Innovation

• AMG
• Multi-Graph Neural Operator
• Partial Differential Equations (PDEs)