Magnetic crystal generation with site-resolved magnetic moments via Gaussian-mixture retargeting and finite-budget yield prediction

Technical Summary: GMMag – Magnetic Crystal Generation with Site-Resolved Moments

Problem Statement
The inverse design of magnetic materials requires simultaneous control over chemical composition, crystal structure, and local spin configuration. Existing generative models often rely on net magnetic moment labels, which fail to distinguish between weak local magnetism and large, antiparallel local moments that cancel out. Furthermore, a high screening pass rate in generative workflows does not guarantee the discovery of novel chemical formulas; repeated generation of the same known formulas can inflate success metrics while limiting the accumulation of distinct, reference-unseen candidates. There is a need for a workflow that generates site-resolved magnetic moments, allows for post-training retargeting without retraining, and accurately predicts the yield of distinct novel formulas under a finite generation budget.

Methodology
The authors introduce GMMag, a conditional variational autoencoder (CVAE) workflow designed to generate crystal structures alongside site-resolved signed magnetic moments.

• Data and Labeling: The training dataset combines 3.5 million structures from public repositories (Materials Project, NOMAD, OQMD) and in-house Density Functional Theory (DFT) calculations. For structures lacking site-resolved magnetic labels, an atomistic line graph neural network (ALIGNN) predictor, trained on 232,506 labeled structures, supplies estimated signed collinear magnetic moments. DFT-derived labels take precedence.
• Architecture: The CVAE employs separate encoder branches for elemental, space-group, Wyckoff-site-group, and signed magnetic-moment features. The encoder maps structures to a 128-dimensional latent space. The decoder is an autoregressive transformer that predicts discrete crystal descriptors sequentially.
  • Conditioning: Magnetic conditions are integrated via two paths: (1) Cross-attention, where token representations read memory vectors projected from the latent vector $z$; and (2) Feature-wise Linear Modulation (FiLM), which scales and shifts decoder hidden representations based on a global magnetic-condition bin (ten bins representing net moment magnitude).
  • Loss Function: The training objective includes token cross-entropy, plus auxiliary losses for magnetic-moment signs, signed values, and structure-level magnetic summaries (net moment, absolute moment, and alignment ratio).
• Retargeting via Gaussian Mixture Models (GMM): Instead of retraining, the workflow redirects generation by fitting a GMM to the latent vectors of selected reference structures (the "latent bank"). Sampling from this fitted GMM (rather than the standard prior) steers the generator toward specific chemical and magnetic targets.
• Finite-Budget Yield Modeling: To address the issue of formula repetition, the authors model the accumulation of distinct qualifying reference-unseen formulas ($D_n$) as a function of generation attempts ($n$). They use a statistical occupancy model where the probability of a qualifying draw ($Q$) and the accumulation curve parameters (crossover point $a_0$ and late-window exponent $\beta_{late}$) are predicted based on sampling parameters (GMM spread $\alpha$, decoder temperature $\tau$) and GMM complexity ($K$).

Key Results

1. Generation Performance: The model achieves high reconstruction validity (95.8% structural validity, 98.3% composition matching) and low magnetic moment error (MAE 0.01 $\mu_B$).
2. Target Adaptation: GMMag successfully generates candidates for three distinct targets: Nd-containing high-moment crystals, rare-earth-free Fe–Co high-moment crystals, and Mn-containing compensated-moment crystals. A "Joint GMM" (fitting multiple mixture components) outperformed a single joint Gaussian in balancing target qualification with the exploration of distinct formulas.
3. Yield Prediction: The finite-budget model accurately predicts the number of distinct qualifying formulas found in long runs (up to 10 million draws). The study demonstrates that screening pass rates alone are insufficient; the accumulation of distinct formulas is limited by the repetition of common formulas, a phenomenon captured by the occupancy model.
4. Ablation Studies:
   • Decoder Depth: Increasing transformer depth from 2 to 8 layers improved reconstruction and screening pass rates but did not increase the number of distinct novel formulas; a 4-layer architecture was found to be the optimal balance.
   • Conditioning: Removing the latent conditioning path (FiLM only) drastically reduced targeted discovery, confirming the necessity of the latent route.
   • Magnetic Supervision: Including additional magnetic losses (Signed-CE&Aux) improved the exact agreement of generated magnetic-moment signs with predictions compared to models without these losses, though it did not always maximize screening pass rates.
5. Physical Validation: A subset of generated Nd-containing candidates underwent relaxation with a machine-learning interatomic potential (eSEN) and DFT. Of 38 candidates with accepted DFT results, 34 had negative formation energies, and 26 exceeded an estimated saturation magnetization of 1.5 T. Specific candidates like Nd$_2$Fe$_{14}$Co$_3$ were identified with negative formation energies and high magnetization.
6. Site Occupancy: Using ab initio Configuration Sampling (abICS), the authors examined atomic site-occupancy disorder in promising candidates, finding that while magnetization estimates remained stable, relative energies were sensitive to sublattice occupations.

Significance and Claims
The paper claims that GMMag provides a robust framework for the inverse design of magnetic materials by explicitly representing site-resolved magnetic moments, a feature often lost in net-moment approaches. The workflow demonstrates that:

• Post-training retargeting via GMM sampling is an effective strategy for exploring specific chemical and magnetic subspaces without the computational cost of retraining.
• Distinct-formula accumulation is a critical metric that differs from screening pass rates; high pass rates can mask a lack of novelty due to formula repetition. The proposed statistical model allows for the prediction of discovery yields under finite computational budgets.
• Model design choices (such as decoder depth and magnetic supervision) involve trade-offs between reconstruction accuracy, screening success, and the diversity of discovered formulas.
• The workflow successfully identified Nd-containing candidates with chemical formulas absent from the reference dataset, negative formation energies, and high estimated saturation magnetization, validating the potential of the approach for discovering new magnetic materials.

The authors maintain a modest stance, noting that the magnetic ground states were not fully established (due to single-initialization collinear protocols) and that the yield response requires recalibration if the generator, reference structures, or screening rules change. The work establishes a complementary route to conditional training for magnetic material discovery.