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

2026-09-28

AI Summary

This study addresses the inability of conventional accuracy metrics to characterize how material graph neural networks depend on their trainable parameter space. We propose “trainability dependence” as a complementary measure, employing random subspace intrinsic dimensionality analysis combined with stress tests across dataset sizes and model widths to quantify the independent degrees of freedom required for models such as CGCNN to recover performance. This approach effectively decouples endpoint accuracy from dimensionality requirements, revealing distinct sensitivities across tasks and architectures: metal classification is readily recoverable, whereas phonon prediction proves most sensitive. Furthermore, increasing model width preserves fractional thresholds while elevating absolute dimensionality demands, establishing a new paradigm for evaluating model optimization efficiency.

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.

Problem

• graph neural networks
• materials property prediction
• intrinsic dimension
• trainable degrees of freedom
• random-subspace analysis

Innovation

• graph neural networks
• intrinsic dimension
• random subspace
• materials property prediction
• trainable degrees of freedom