# ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks

arXiv: https://arxiv.org/abs/2206.05437 · HTML: https://mlsanity.com/paper/2206.05437/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 63 references that are in the map.
- First posted on arXiv 11 Jun 2022.

## Builds on (most cited first, 30 of 37)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)

## Built on by (most cited first, 30 of 31)

- [DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion](https://mlsanity.com/paper/2301.09474/) (2023)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [A Fractional Graph Laplacian Approach to Oversmoothing](https://mlsanity.com/paper/2305.13084/) (2023)
- [Graph Neural Convection-Diffusion with Heterophily](https://mlsanity.com/paper/2305.16780/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theory](https://mlsanity.com/paper/2402.15326/) (2024)
- [Distributed-Order Fractional Graph Operating Network](https://mlsanity.com/paper/2411.05274/) (2024)
- [From Cluster Assumption to Graph Convolution: Graph-based Semi-Supervised Learning Revisited](https://mlsanity.com/paper/2309.13599/) (2023)
- [Design Your Own Universe: A Physics-Informed Agnostic Method for Enhancing Graph Neural Networks](https://mlsanity.com/paper/2401.14580/) (2024)
- [Are We Measuring Oversmoothing in Graph Neural Networks Correctly?](https://mlsanity.com/paper/2502.04591/) (2025)
- [Oversmoothing Alleviation in Graph Neural Networks: A Survey and Unified View](https://mlsanity.com/paper/2405.01663/) (2024)
- [Domain-informed graph neural networks: a quantum chemistry case study](https://mlsanity.com/paper/2208.11934/) (2022)
- [Foundations and Frontiers of Graph Learning Theory](https://mlsanity.com/paper/2407.03125/) (2024)
- [When Graph Neural Networks Meet Dynamic Mode Decomposition](https://mlsanity.com/paper/2410.05593/) (2024)
- [Improving Graph Neural Networks by Learning Continuous Edge Directions](https://mlsanity.com/paper/2410.14109/) (2024)
- [A Signed Graph Approach to Understanding and Mitigating Oversmoothing in GNNs](https://mlsanity.com/paper/2502.11394/) (2025)
- [Early-Exit Graph Neural Networks](https://mlsanity.com/paper/2505.18088/) (2025)
- [Beyond ReLU: Bifurcation, Oversmoothing, and Topological Priors](https://mlsanity.com/paper/2602.15634/) (2026)
- [A Unified View on Neural Message Passing with Opinion Dynamics for Social Networks](https://mlsanity.com/paper/2310.01272/) (2023)
- [Graph Neural Networks with a Distribution of Parametrized Graphs](https://mlsanity.com/paper/2310.16401/) (2023)
- [TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows](https://mlsanity.com/paper/2508.05070/) (2025)
- [Graph Neural Regularizers for PDE Inverse Problems](https://mlsanity.com/paper/2510.21012/) (2025)
- [Measuring Over-smoothing beyond Dirichlet energy](https://mlsanity.com/paper/2512.06782/) (2025)
- [Geometric Mixture-of-Experts with Curvature-Guided Adaptive Routing for Graph Representation Learning](https://mlsanity.com/paper/2603.22317/) (2026)
- [DFGET: Displacement-Field Assisted Graph Energy Transmitter for Gland Instance Segmentation](https://mlsanity.com/paper/2312.07584/) (2023)
- [Message Detouring: A Simple Yet Effective Cycle Representation for Expressive Graph Learning](https://mlsanity.com/paper/2402.08085/) (2024)
- [Heterophily-informed Message Passing](https://mlsanity.com/paper/2504.19785/) (2025)
- [How Particle System Theory Enhances Hypergraph Message Passing](https://mlsanity.com/paper/2505.18505/) (2025)
- [Modeling Whole-Slide Images as Dynamic Tumor Microenvironment Fields](https://mlsanity.com/paper/2609.34451/) (2026)

## Abstract (arXiv)

Neural message passing is a basic feature extraction unit for graph-structured data considering neighboring node features in network propagation from one layer to the next. We model such process by an interacting particle system with attractive and repulsive forces and the Allen-Cahn force arising in the modeling of phase transition. The dynamics of the system is a reaction-diffusion process which can separate particles without blowing up. This induces an Allen-Cahn message passing (ACMP) for graph neural networks where the numerical iteration for the particle system solution constitutes the message passing propagation. ACMP which has a simple implementation with a neural ODE solver can propel the network depth up to one hundred of layers with theoretically proven strictly positive lower bound of the Dirichlet energy. It thus provides a deep model of GNNs circumventing the common GNN problem of oversmoothing. GNNs with ACMP achieve state of the art performance for real-world node classification tasks on both homophilic and heterophilic datasets. Codes are available at https://github.com/ykiiiiii/ACMP.