Projective Geometric Algebra Graph Neural Networks: Edge-Modulated Message Passing via the Geometric Product
Abstract
Processing 3D geometric graphs in physical sciences requires neural networks that strictly preserve continuous spatial symmetries. Projective Geometric Algebra (PGA) offers an elegant alternative to heavy spherical harmonics and limited invariant scalars, yet existing PGA models rely on global attention, diluting crucial sparse topological priors. We introduce the Projective Geometric Algebra Graph Neural Network (PGA-GNN), a strictly E(3)-equivariant architecture bridging the mathematical rigor of PGA with local message-passing efficiency. PGA-GNN explicitly computes messages between neighboring nodes using an edge-modulated geometric product, natively capturing robust relative spatial trans- formations. We empirically demonstrate that this localized paradigm provides a superior structural inductive bias. PGA-GNN achieves strong results on N-body and spring dynamics at a fraction of the data required by baselines. On Revised MD17, it outperforms standard scalar (EGNN) and harmonic (SEGNN) baselines in force prediction, drastically reducing errors compared to global attention. Finally, on QM9, PGA-GNN establishes solid results for directionally dependent properties. While domain-specific models remain SOTA in quantum chemistry, our results validate explicit geometric message-passing as a highly robust, general-purpose framework for geometric deep learning.
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BibTeX
@misc{toure2026projective,
title = {{Projective Geometric Algebra Graph Neural Networks: Edge-Modulated Message Passing via the Geometric Product}},
author = {Papa Samba Khary Touré and Frédéric Jurie and Antonin Vacheret and Sylvain Le Tonquesse},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05797601v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Projective Geometric Algebra Graph Neural Networks: Edge-Modulated Message Passing via the Geometric Product
AU - Papa Samba Khary Touré
AU - Frédéric Jurie
AU - Antonin Vacheret
AU - Sylvain Le Tonquesse
PY - 2026
DA - 2026/10/13
PB - HAL
UR - https://hal.science/hal-05797601v1
AB - Processing 3D geometric graphs in physical sciences requires neural networks that strictly preserve continuous spatial symmetries. Projective Geometric Algebra (PGA) offers an elegant alternative to heavy spherical harmonics and limited invariant scalars, yet existing PGA models rely on global attention, diluting crucial sparse topological priors. We introduce the Projective Geometric Algebra Graph Neural Network (PGA-GNN), a strictly E(3)-equivariant architecture bridging the mathematical rigor of PGA with local message-passing efficiency. PGA-GNN explicitly computes messages between neighboring nodes using an edge-modulated geometric product, natively capturing robust relative spatial trans- formations. We empirically demonstrate that this localized paradigm provides a superior structural inductive bias. PGA-GNN achieves strong results on N-body and spring dynamics at a fraction of the data required by baselines. On Revised MD17, it outperforms standard scalar (EGNN) and harmonic (SEGNN) baselines in force prediction, drastically reducing errors compared to global attention. Finally, on QM9, PGA-GNN establishes solid results for directionally dependent properties. While domain-specific models remain SOTA in quantum chemistry, our results validate explicit geometric message-passing as a highly robust, general-purpose framework for geometric deep learning.
ER -