This work introduces point-group symmetry aware equivariant graph neural networks (PGEqNN) for materials science, with filter functions aligned with symmetry-aware indices for greater granularity in predictive tasks.
Abstract
Equivariant graph neural networks have proven effective tools for inference of material's properties directly from their structure. Traditionally, these have been applied such that they respect full $O(3)$ equivariance, so that any rotation or reflection of the input structure is respected in the model's output. While this works for general arrangements of atoms, additional symmetries of atomistic systems are left unleveraged. Furthermore, any symmetries of the filter functions are implicitly learned from the full dataset and not strictly enforced. In this work, we introduce point-group symmetry aware equivariant graph neural networks (PGEqNN) for materials science, with filter functions aligned with symmetry-aware indices for greater granularity in predictive tasks. With this architecture, we show that most of the predictive power of equivariant networks for tensorial elastic and dielectric datasets lies in the trivial subspaces of the point-group adapted bases. Exploiting this, an $A_1$-restricted variant matches or improves on its full point-group and $SO(3)$-partitioned counterparts while training fewer active parameters, yielding leaner models of equal accuracy.
ESNN is introduced, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance.
Alessio Borgi, M. Severino, Fabrizio Silvestri et al.· 0 citations
Mandala is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks that connects electronic-structure learning and observable-guided modeling while retaining a representation tied to quantum-mechanical operators rather than only scalar or vector targets as in MLIPs.
B. Brzoza, Wiktoria Szopa, Z. Elabid et al.· 0 citations
Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.
Ning Lin, Jiacheng Cen, Anyi Li et al.· 1 citation
This work develops this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow-matching minimizers, including for categorical endpoint prediction.
This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory.
