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

Zhihao Li, Haoze Song, Di Xiao, Zhilu Lai, Wei Wang

TL;DR

AMG tackles the challenge of solving PDEs on arbitrary geometries by learning operators with a multi-graph neural architecture. It introduces GraphFormer and three graph types—multi-scale, physics, and local/global connections—plus a high-frequency indicator to adaptively sample points, enabling precise modeling of multi-frequency PDE features. Across six benchmarks, AMG consistently outperforms state-of-the-art graph- and transformer-based neural operators, demonstrating strong performance on structured, unstructured, and dynamic meshes, including real-world weather data. The results suggest AMG’s potential as a foundation for scalable, mesh-agnostic PDE solvers and motivate exploring large-scale pre-training for generalized PDE learning.

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.

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

TL;DR

AMG tackles the challenge of solving PDEs on arbitrary geometries by learning operators with a multi-graph neural architecture. It introduces GraphFormer and three graph types—multi-scale, physics, and local/global connections—plus a high-frequency indicator to adaptively sample points, enabling precise modeling of multi-frequency PDE features. Across six benchmarks, AMG consistently outperforms state-of-the-art graph- and transformer-based neural operators, demonstrating strong performance on structured, unstructured, and dynamic meshes, including real-world weather data. The results suggest AMG’s potential as a foundation for scalable, mesh-agnostic PDE solvers and motivate exploring large-scale pre-training for generalized PDE learning.

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.

Paper Structure

This paper contains 53 sections, 1 theorem, 20 equations, 10 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Graph Neural Operators
  5. Methodology
  6. Overview of Model Architecture
  7. Graph Construction
  8. High-Frequency Indicator
  9. Partitioning of Point Sets
  10. Local Sampling
  11. Global Sampling
  12. Physics Graph
  13. Multi-Graph Processor
  14. Cross-Scale Graph Aggregation
  15. Physics Graph Propagation
  16. ...and 38 more sections

Key Result

Theorem 1

Given an input function $a: \Omega \to \mathbb{R}^{d}$ and a mesh point $\mathbf{x} \in \Omega$, GraphFormer seeks to approximate the integral operator $\mathcal{G}$, defined by: where $\kappa(\cdot, \cdot)$ denotes a kernel function over $\Omega \times \Omega$.

Figures (10)

  • Figure 1: Overview of the model architecture.
  • Figure 2: (a) Message passing from multi-scale graph to physics graph. (b) Three types of graphs as the input of GraphFormer. (c) Message passing from physics graph to multi-scale graph.
  • Figure 3: Left: The Architecture of GraphFormer Block. Right: An illustration of multi-head attention by node $i$ on its neighborhood $j$.
  • Figure 4: Illustration of global sampling and local sampling in Cylinder Flow (a, b) and Deforming Plate (c, d).
  • Figure 5: Visualization of a random sample from the Poisson dataset.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 1: GraphFormer as a Learnable Integral on $\Omega$