materials-chemistry/ai produced the result/arXiv 2024 · v2
Testing whether symmetry-aware neural networks can tell apart atomic arrangements in perovskites
Researchers built a large database of calculated energies for perovskite oxides and used it to compare two families of graph neural network. The networks were the object of study: how well each could rank different arrangements of the same atoms.
spectrum · one line per step, placed by what the step does · bright lines used AI
Data and models for: Learning Ordering in Crystalline Materials with Symmetry-Aware Graph Neural Networks
arXiv, 2024
doi:10.48550/arxiv.2409.13851 · record aix-00147 v2 · checked 2026-10-09
- AI was for
- Property prediction, Simulation surrogate
- Model family
- Graph neural network
- Checked by
- Held-out1261 tested
- Code
- available
The finding the paper is about came from the AI.
What this research was about
Perovskite oxides are a family of crystals built on a simple repeating cube of metal and oxygen atoms. Mix several different metals into the same crystal and a new question appears: which metal sits on which site? The chemical recipe stays the same, but the atoms can be shuffled between positions in many distinct ways, and each arrangement, or ordering, has a slightly different energy. Energy matters because the low-energy arrangements are the ones a real material is likely to adopt. Working out those energies from first principles means quantum-mechanical calculations that are slow and costly, and the number of possible arrangements grows quickly.
The researchers generated idealised cubic cells containing 40 atoms, drawing on 72 elements and enumerating the arrangements that are genuinely different rather than mere rotations of one another. They then calculated the energies with density functional theory, a standard quantum-mechanical method, recording for each structure its energy above the convex hull, a measure of how far it sits from the most stable mixture of that composition. That dataset became the testing ground for the machine learning comparison.
Where AI came in
Machine learning appears at several points. A pre-trained model of interatomic forces, M3GNet, was used to tidy up each structure before the quantum calculations began, saving some of that expensive work. The main study then trained three networks from scratch on the dataset to predict energy above the hull directly from a structure. These networks treat a crystal as a graph of atoms joined by bonds. CGCNN is symmetry-invariant, meaning it ignores the directions in which bonds point; e3nn and PaiNN are equivariant, meaning they track direction and rotate their internal description along with the crystal.
The networks stood in for the quantum calculations. On a test set of 1,261 structures with compositions the models had never seen, both kinds did about as well, with average errors of 21.7 and 19.0 millielectronvolts per atom for CGCNN and e3nn on unrelaxed inputs. The harder test was ranking six distinct arrangements of the same composition. There CGCNN scored an R2 of -0.15 against the calculated values and e3nn 0.37. Inspecting what CGCNN had learned showed its internal descriptions of the six arrangements were nearly identical. Among ready-made force models, the equivariant MACE did better on this task than M3GNet and CHGNet.
Written by AIxSci from the checked record below, to give context for readers outside the field. It is not part of the record.
The work
Technical · from the record
The authors built a density functional theory dataset of more than 10,000 multicomponent perovskite oxide structures and energies, covering many compositions and symmetrically inequivalent cation orderings, and used it to compare symmetry-invariant and symmetry-equivariant graph neural networks predicting energy above the convex hull. On a composition-based test set of 1,261 structures both model types performed similarly, with mean absolute errors of 21.7 meV/atom for CGCNN and 19.0 meV/atom for e3nn on unrelaxed inputs. On a holdout set of 100 AB0.5B'0.5O3 compositions, each with six symmetrically distinct arrangements, CGCNN gave an R2 of -0.15 on ordering-relative energies with unrelaxed inputs while e3nn gave 0.37, and principal component analysis of the learned embeddings showed CGCNN representations of the six orderings were close to degenerate. Among pre-trained universal interatomic potentials, the equivariant MACE scored better than the invariant M3GNet and CHGNet on the same ordering task.
How AI was used
Idealized cubic 40-atom perovskite supercells were enumerated over 72 elements and all symmetrically inequivalent cation arrangements, perturbed to break symmetry, pre-relaxed with the pre-trained M3GNet interatomic potential, and then relaxed with PBE+U density functional theory to produce structures, energies and energy above the convex hull. The dataset was split by composition into training, validation and test sets plus two ordering holdout sets, and CGCNN (symmetry-invariant), e3nn and PaiNN (symmetry-equivariant) were trained from scratch to regress Ehull from either unrelaxed or DFT-relaxed geometries, with shared CGCNN elemental node-feature initialisation, spherical harmonics truncated at l=2 for e3nn, and hyperparameters tuned with SigOpt over 50 models for 100 epochs each. The trained models were run over the test and holdout sets, and the pre-trained M3GNet, CHGNet and MACE-MP-0 potentials were applied without modification, relaxing structures and deriving Ehull from compositional phase diagrams. Node-feature embeddings from the convolution layers were pooled and projected to two dimensions by principal component analysis to compare how the models represented the distinct orderings.
The shape of the work
Structural · the record, drawn
no AI
Enumerate perovskite compositions and cation orderings
Obtaining raw data, whether by measurement, download or retrieval.
the cation orderings in structures were sampled from a complete list of all possible symmetrically inequivalent cation arrangements in the cubic supercellswhere the paper describes this · verbatim
AI
Pre-relax structures with a machine-learned potential
Numerical or physics simulation, including where a learned surrogate replaces it. The AI stood in for simulation.
M3GNet was also used as a pre-trained interatomic potential to pre-relax these structures before DFT structural relaxation.where the paper describes this · verbatim
no AI
High-throughput DFT relaxation and energy calculation
Numerical or physics simulation, including where a learned surrogate replaces it.
We used an in-house automated DFT pipeline for structural optimization and energy calculationwhere the paper describes this · verbatim
no AI
Split dataset by composition into train, validation, test and holdout sets
Cleaning, filtering, normalising or labelling data already obtained.
The training–validation–test split was conducted based on compositions, such that no specific oxide composition appeared in multiple setswhere the paper describes this · verbatim
AI
Train invariant and equivariant GCNN energy models
Fitting model parameters, including fine-tuning an existing model. The AI stood in for simulation.
hyperparameter optimization was conducted using SigOpt with a budget of 50 modelswhere the paper describes this · verbatim
AI
Predict Ehull across composition and ordering sets
Running a trained model over new data to predict, classify or score. The AI stood in for simulation.
to predict Ehull from unrelaxed or DFT-relaxed perovskite oxide structureswhere the paper describes this · verbatim
AI
Evaluate pre-trained universal interatomic potentials
Running a trained model over new data to predict, classify or score. The AI stood in for simulation.
M3GNet, CHGNet, and MACE-MP-0 were leveraged as pre-trained interatomic potentialswhere the paper describes this · verbatim
no AI
Analyse latent embeddings of inequivalent orderings
Extracting understanding from model behaviour.
we extracted the GCNN embeddings of multicomponent perovskite oxides in CGCNN and e3nn by pooling node feature vectorswhere the paper describes this · verbatim
What the record says
Technical · every part carries its own basis
+ in the paper~ our reading− not reported
How to read the quotations. A quotation shows where the paper describes something. It does not quote every value beside it: one passage locates a part of the work, and values without their own quotation are our reading of that passage.
The paper's finding is a comparison of what invariant and equivariant graph neural networks can represent and predict; the models are the object of study, so the result exists only through them.
We first selected CGCNN and e3nn as representative ML models for symmetry-invariant and symmetry-equivariant GCNNs, respectively.where the paper describes this · verbatim
leading to 6,276, 1,277, and 1,261 perovskite oxide structures in training, validation, and test sets, respectivelywhere the paper describes this · verbatim
Python computer code, DFT data, and trained models for reproducing this work are available via GitHubwhere the paper describes this · verbatim
What this paper did not report
Technical · absence is published deliberately
- Version of CGCNNWhich version of the model was used is not stated.
- Version of e3nnWhich version of the model was used is not stated.
- Version of PaiNNWhich version of the model was used is not stated.
- Version of M3GNetWhich version of the model was used is not stated.
- Version of CHGNetWhich version of the model was used is not stated.
About this article
Record aix-00147, version 2, checked by a person on 2026-10-09. The record describes the paper; it does not assess whether the paper's findings are right. The paper is published under CC-BY; quotations are at most 25 words. How we work · Report an error