ReGDiff: Guided Diffusion in Regulated Latent Space for Metamaterial Discovery

Generative models for metamaterial design often face a dilemma: they either reproduce training samples (lacking novelty) or produce invalid structures such as disconnected fragments or empty voxels when exploration is encouraged. ReGDiff addresses this by first organizing the latent space so that valid and invalid structures are well separated and valid structures are evenly distributed, then guiding diffusion sampling with a short-range repulsion that pushes generated samples away from known training examples only in their immediate vicinity.

Voxel Representation and Geometric Constraints

Metamaterials are periodic structures whose unusual mechanical properties arise from micro‑geometry rather than composition. The paper adopts a 48×48×48 voxel representation that can describe trusses, shells, and porous structures uniformly. Valid unit cells must satisfy three constraints: central symmetry, matching opposite faces for seamless tiling, and a single connected solid component. Existing diffusion models and GANs tend to either memorize training data or violate these constraints when pushed to explore.

Benchmark: MetaTruss Dataset and Five Evaluation Metrics

Prior to this work, only the shell‑type MetaShell dataset was available for voxel‑based generative modeling. The authors construct MetaTruss by converting 10,000 graph‑based truss samples into 48³ voxels (marking voxels within strut radii as solid) and unify MetaShell to the same resolution. Evaluation spans three dimensions with five metrics:

Geometric rationality : symmetry score (S_sym), periodicity score (S_per), connectivity score (S_con — fraction of solid voxels in the largest connected component).

Novelty (S_nov): IoU distance to the nearest training sample; higher means farther from known designs.

Diversity (S_div): number of distinct training samples covered by generated samples’ nearest neighbors, normalized by generation count; low values indicate mode collapse.

RAS: Repel‑And‑Sink Latent Space Regulation

A Transformer autoencoder compresses voxels to a 128‑dimensional latent space. Because metamaterials are highly sensitive to small geometric perturbations, the latent space must cleanly separate valid from invalid structures and spread valid structures smoothly. RAS achieves this by creating negative samples (randomly replacing 1/8 of voxels in real samples) and applying three forces during autoencoder training:

Inter‑class Repulsion (IeR) : Coulomb‑like repulsion between positive and negative latent codes to sharpen the decision boundary.

Intra‑class Repulsion (IaR) : repulsion among same‑class samples to prevent collapse and encourage coverage.

Center Sink (CS) : attraction toward the origin to keep the distribution from becoming overly sparse.

Ablation shows that without regularization, positive and negative samples intermingle; contrastive learning separates them but compresses each class into a tight cluster; RAS both separates classes and yields a smooth, well‑covered distribution, producing the most regular generated structures.

SRR: Short‑Range Repulsion During Sampling

After RAS, a standard DDPM is trained on the regulated latent space. At inference, Short‑Range Repulsion (SRR) adds a repulsive force in each denoising step that pushes the current latent code away from known training latents. The force uses a logarithmic decay so that only very nearby training samples exert noticeable influence; distant clusters are ignored to limit computation. SRR operates only at sampling time — the diffusion model itself is trained normally. Generating 1,000 structures takes 31 seconds, comparable to the strongest baseline (Yang et al., 28 s) and far faster than DiT‑3D (326 s) and 3D‑CDM (813 s).

Experimental Comparison

ReGDiff is compared against DiT‑3D, Yang et al., XCube, Trellis, and 3D‑CDM on both MetaTruss and MetaShell. On MetaTruss, ReGDiff achieves the best scores across all three key metrics: geometric rationality mean 0.725, novelty 0.296, diversity 0.420. The runner‑up, Yang et al., scores 0.626, 0.163, 0.158 respectively. The other four baselines exhibit near‑zero novelty, indicating they either replicate training samples or produce invalid structures (whose novelty is not counted). Qualitative examples confirm: DiT‑3D yields near‑solid blocks, XCube generates almost nothing, Trellis produces scattered particles, 3D‑CDM creates dense blobs, while ReGDiff preserves strut‑like features and periodic frames while differing visibly from training samples.

Ablation Study

Removing SRR (RAS + vanilla DDPM) reduces novelty from 0.296 to 0.208 and diversity from 0.420 to 0.336, while geometric rationality slightly increases (0.757 vs 0.725), showing SRR trades a small amount of rationality for substantial novelty/diversity gains. Removing RAS (no regularization + SRR) collapses novelty to 0.014, diversity to 0.011, and connectivity to 0.295, demonstrating that a well‑structured latent space is a prerequisite for SRR to be effective.

Future Directions

The authors outline three next steps: (1) extend the benchmark beyond trusses and shells to more metamaterial categories and higher resolutions; (2) integrate mechanical property prediction and physics simulation into the generation loop so evaluation shifts from geometric validity to performance targets; (3) move from unconditional to conditional generation guided by desired properties, validated with domain experts, enabling a transition from selecting near known designs to actively discovering new ones.

Paper: https://arxiv.org/abs/2609.34231 (NeurIPS 2026). Code: https://github.com/wzhan24/ReGDiff.