C2NP: A Benchmark for Learning Scale-Dependent Geometric Invariances in 3D Materials Generation

Technical Summary: C2NP Benchmark for Scale-Dependent Geometric Invariances

Problem Statement
Generative models for materials science have demonstrated strong performance on periodic bulk crystals, where structures are defined by infinite translational symmetry encoded in unit cells. However, their ability to generalize across scale transitions to finite nanostructures remains largely untested. Nanoparticles lack translational periodicity; instead, they are dominated by surface facets, edge sites, and reduced coordination numbers that induce structural relaxations and quantum size effects. Current benchmarks and datasets (e.g., Materials Project, CSPBench) focus primarily on bulk phases or surface slabs, failing to systematically pair periodic unit cells with size-resolved nanoparticle configurations. Consequently, it is unclear whether existing models learn scalable physical principles governing the transition from infinite lattices to finite clusters or merely memorize correlations within narrow training distributions.

Methodology
The authors introduce C2NP (Crystal-to-Nanoparticle), a systematic benchmark designed to evaluate generative models on bidirectional structural transformations between infinite unit cells and finite nanoparticles.

• Dataset Construction: The benchmark utilizes a structurally consistent subset of perovskite hydrides, a material family relevant to energy storage and catalysis. Starting from DFT-optimized crystallographic unit cells, the authors construct $20 \times 20 \times 20$ supercells. Finite nanoparticles are generated via deterministic spherical carving (truncation) at radii $R \in \{6, \dots, 30\}$ Å. This process yields over 170,000 nanoparticle configurations without further structural relaxation, isolating geometric size effects.
• Data Partitioning: To ensure rigorous evaluation of generalization, the dataset is partitioned based on particle size and orientation:
  • Size Splits: Training data covers intermediate radii. In-distribution (ID) testing uses mid-range radii, while Out-of-Distribution (OOD) testing targets extreme sizes ($R=6, 7, 29, 30$ Å) where surface-to-volume ratios are highest.
  • Orientational Augmentation: Rotational augmentation is applied over $SO(3)$ using unit quaternions. A greedy algorithm ensures geodesic separation between training, ID, and OOD sets, preventing distributional overlap and directional bias.
• Benchmark Tasks: C2NP defines two complementary tasks:
  1. Forward Generation (Unit Cell $\to$ Nanoparticle): Given a unit cell and a target radius, the model must generate a finite nanoparticle that preserves underlying periodic ordering while correctly capturing surface truncation.
  2. Inverse Reconstruction (Nanoparticle $\to$ Unit Cell): Given a finite nanoparticle configuration, the model must infer the bulk lattice parameters and space-group symmetry, despite surface disorder and broken periodicity.
• Evaluation Metrics: Performance is assessed using robust, normalized metrics. For generation, metrics include RMSD, Hausdorff distance, convex-hull volume error, and radial distribution function (RDF) error. For inverse tasks, metrics include lattice parameter RMSE, space-group accuracy, and joint recovery accuracy (simultaneous correctness of both).

Key Contributions

1. C2NP Dataset: A large-scale, reproducible dataset pairing DFT-validated unit cells with ~172,000 size-resolved nanoparticle configurations, explicitly designed to test scale-dependent generalization.
2. Bidirectional Evaluation Framework: A unified testbed for both generative (forward) and inverse (reconstruction) problems, probing whether models encode scalable structural principles or rely on template memorization.
3. Rigorous Split Strategy: A novel partitioning scheme based on particle size and geodesic orientation separation that strictly isolates interpolation from extrapolation regimes.
4. Diagnostic Insights: The benchmark reveals that minimizing training loss is a poor proxy for structural fidelity in scale-transfer tasks, exposing fundamental failure modes in current state-of-the-art models.

Experimental Results
The authors evaluated several state-of-the-art generative models, including CDVAE, DiffCSP, FlowMM, MatterGen-MP, and ADiT.

• Forward Task (Generation): Despite achieving similarly high normalized loss scores (approx. 0.61), most models (ADiT, DiffCSP, FlowMM, MatterGen) failed to produce structurally meaningful nanoparticles, exhibiting weak geometric fidelity (RMSD/Hausdorff scores between 0.34–0.54). In contrast, CDVAE achieved near-optimal geometry across all structural metrics (scores $\approx$ 1.00) despite a lower loss, suggesting its latent-variable formulation better constrains global structure. Performance for all models degraded under OOD size extrapolation, though CDVAE maintained stability.
• Inverse Task (Reconstruction): No evaluated method succeeded at jointly recovering lattice parameters and space-group symmetry. While some models achieved moderate space-group accuracy (approx. 0.61–0.66), lattice parameter recovery remained weak (RMSE scores 0.34–0.50). Crucially, joint accuracy remained fixed at 0.50 for all methods, indicating a disconnect between continuous lattice regression and discrete symmetry classification. Performance did not improve under OOD conditions, suggesting intrinsic limitations in crystallographic inference rather than overfitting.

Significance and Claims
The paper claims that C2NP provides a controlled framework for diagnosing the failure of current generative models to generalize across physical scales. The results suggest that existing methods rely heavily on template memorization rather than learning scalable physical generalizations. Specifically, the benchmark demonstrates that:

• Low training loss does not guarantee geometrically valid nanoparticle generation.
• Inferring bulk crystallographic order from finite, surface-perturbed configurations is a stringent, unresolved challenge for current architectures.
• There is a fundamental gap in the ability of models to reason about the transition between infinite periodicity and finite size.

The authors position C2NP as a foundation for developing architectures capable of reasoning about physical scale in crystalline matter, with immediate applications in nanoparticle catalyst design, nanostructured hydrides for hydrogen storage, and broader materials discovery. The dataset and code are made available to facilitate reproducible research in this domain.