Measuring trainable degrees of freedom in materials graph neural networks: a random-subspace intrinsic dimension analysis

Shehroz Ahmad Shoaib; Kangming Li

cond-mat.mtrl-sciSep 28, 2026

Measuring trainable degrees of freedom in materials graph neural networks: a random-subspace intrinsic dimension analysis

Authors: Shehroz Ahmad Shoaib, Kangming Li

Organizations: Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. · Department of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.

Abstract

Final predictive accuracy is the standard basis for comparing graph neural networks (GNNs) in materials-property prediction, but it does not show how strongly performance depends on access to trainable parameter-space directions. Here, we introduce trainable-degree dependence as a complementary characterization of materials GNN learning. Using random-subspace intrinsic-dimension analysis, we train CGCNN, ALIGNN, and DimeNet++ in randomly oriented parameter subspaces across six prediction tasks and measure how performance recovers as independent trainable degrees of freedom are restored. The resulting recovery curves separate endpoint accuracy from the trainable-dimensional demand required to recover it. They reveal distinctions that final errors alone miss: metallic classification and log-bulk-modulus regression recover near-reference performance from small fractional subspaces, formation-energy and band-gap prediction show stronger architecture dependence, and phonon prediction is most sensitive to dimensional restriction. Dataset-size sweeps show that band-gap models require larger fractional subspaces as training data grows, whereas formation-energy and bulk-modulus responses are more stable. A width sweep shows that fractional thresholds can remain stable while absolute threshold dimensions increase with model size. Random-subspace analysis therefore provides a targeted stress test for how materials GNNs use their optimization space.

Figures & tables

Figure 1: Schematic of random-subspace training in a toy parameter space with ambient dimension $D=3$ . A model is initialized at $\boldsymbol{\theta}_{0}$ , and optimization is restricted to an affine subspace $\boldsymbol{\theta}_{0}+\mathbf{A}\boldsymbol{\phi}$ of dimension $d\leq D$ . The blue region represents parameter configurations that satisfy the target performance criterion, with $\boldsymbol{\theta}^{*}$ denoting a representative solution. As $d$ increases, a randomly oriented subspace is more likely to intersect the acceptable-solution region; when $d=D$ , the full parameter space is available for optimization.

Task	Data source	$N$	Dummy	ALIGNN	DimeNet++	CGCNN
Phonon frequency	matbench_phonons	1,265	323.8	73.4	87.3	108.0
Refractive index	matbench_dielectric	4,764	0.809	0.324	0.611	0.512
Bulk modulus	matbench_log_kvrh	110k	0.290	0.064	0.087	0.071
Band gap	MP23	153k	1.327	0.332	0.363	0.368
Formation energy	MP23	155k	1.059	0.048	0.060	0.053
Metallicity	matbench_is_metal	106k	0.500	0.949	0.948	0.937

Table 1: Dataset sizes and

d=D

predictive performance of the compact, approximately parameter-matched models. Regression tasks are evaluated using mean absolute error (MAE; lower is better), and the metallicity classification task using ROC-AUC (higher is better). The dummy reference is the dataset mean absolute deviation for regression tasks and 0.5 for metallicity classification.

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Dual-Level Atomic and Coordination Geometry Learning for Crystal Property Prediction Using Graph Neural Networks

Sanjay Chakraborty

Department of Computer and Information Science (IDA), REAL, AIICS, Linköping University, Linköping, Sweden

Graph Neural Networks Crystal Atoms Coordination Representation Learning

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Accurate prediction of crystal properties remains a key challenge in computational materials science. While graph neural networks (GNNs) such as CGCNN, MEGNet, ALIGNN, and SchNet have shown strong performance, they primarily represent crystals at the atomic level and implicitly learn local chemical environments through message passing. However, many material properties are governed by coordination polyhedra, the fundamental structural units formed by atoms and their neighboring atoms. To address this limitation, we propose the Coordination Polyhedron Graph Network (CPGN), a multi-scale GNN that jointly learns atomic, bond, and coordination-polyhedron representations. CPGN constructs three coupled graphs: an atom graph encoding elemental and bonding information, a line graph capturing angular interactions, and a coordination polyhedron graph describing Voronoi-derived local environments through corner-, edge-, and face-sharing relationships. Physically meaningful geometric descriptors are incorporated for each polyhedron, while an interleaved message-passing mechanism with bidirectional cross-attention enables effective information exchange across structural levels. Extensive evaluations on the Materials Project, JARVIS-DFT, and QM9 benchmark datasets demonstrate that CPGN outperforms existing state-of-the-art GNN models. It achieves a formation-energy MAE of 0.060 eV/atom and a band-gap MAE of 0.292 eV on the Materials Project, while providing competitive multi-property prediction on JARVIS-DFT and superior HOMO prediction on QM9. The results highlight that explicit modeling of coordination polyhedra improves crystal representation learning and enables accurate, physically interpretable prediction of material properties.

Abstract

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