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

arXiv: https://arxiv.org/abs/2411.15178 · HTML: https://mlsanity.com/paper/2411.15178/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 45 references that are in the map.
- First posted on arXiv 18 Nov 2024.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [MetaFormer Is Actually What You Need for Vision](https://mlsanity.com/paper/2111.11418/) (2021)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Learning Dual Convolutional Neural Networks for Low-Level Vision](https://mlsanity.com/paper/1805.05020/) (2018)
- [MAgNet: Mesh Agnostic Neural PDE Solver](https://mlsanity.com/paper/2210.05495/) (2022)
- [M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers](https://mlsanity.com/paper/2406.04822/) (2024)

## Built on by (most cited first, 16 of 16)

- [Towards Efficient Training of Graph Neural Networks: A Multiscale Approach](https://mlsanity.com/paper/2503.19666/) (2025)
- [LaDEEP: A Deep Learning-based Surrogate Model for Large Deformation of Elastic-Plastic Solids](https://mlsanity.com/paper/2506.06001/) (2025)
- [GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2508.04463/) (2025)
- [Geometric Laplace Neural Operator](https://mlsanity.com/paper/2512.16409/) (2025)
- [From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators](https://mlsanity.com/paper/2602.21551/) (2026)
- [DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs](https://mlsanity.com/paper/2603.01762/) (2026)
- [Reformulating Neural Operators in d+1 Dimensions for Embedding Evolution](https://mlsanity.com/paper/2505.11766/) (2025)
- [MNO: Multiscale Neural Operator for 3D Computational Fluid Dynamics](https://mlsanity.com/paper/2510.16071/) (2025)
- [HGATSolver: A Heterogeneous Graph Attention Solver for Fluid-Structure Interaction](https://mlsanity.com/paper/2601.09251/) (2026)
- [SurgFormer: Scalable Learning of Organ Deformation with Resection Support and Real-Time Inference](https://mlsanity.com/paper/2603.06543/) (2026)
- [MAVEN: A Mesh-Aware Volumetric Encoding Network for Simulating 3D Flexible Deformation](https://mlsanity.com/paper/2604.04474/) (2026)
- [NPSolver: Neural Poisson Solver with Iterative Physics Supervision](https://mlsanity.com/paper/2605.25786/) (2026)
- [Factorized Neural Operators Decompose Dynamic and Persistent Responses](https://mlsanity.com/paper/2606.16900/) (2026)
- [From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics](https://mlsanity.com/paper/2608.14120/) (2026)
- [Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains](https://mlsanity.com/paper/2608.29892/) (2026)
- [AutoPDEBench: Benchmarking LLM Auto-Research for Neural PDE Solver Design](https://mlsanity.com/paper/2609.32245/) (2026)

## Abstract (arXiv)

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.