MatterGen: a generative model for inorganic materials design

• Existing screening methods explore only 10^6–10^7 previously unknown crystals, a tiny fraction of the 10^10 potential stable inorganic compounds, and cannot efficiently target desired properties.Screening benefits from high-throughput screening, open databases, property predictors, and machine-learning force fields but remains limited by known materials.Source: §1 · p. 2
• Current generative models often produce DFT-unstable materials, cover narrow element subsets, or optimize mainly formation energy.These limitations motivate inverse design methods that directly generate structures under rare or conflicting property constraints.Source: §1 · p. 2
• MatterGen more than doubles stable, unique, and novel materials versus the previous state-of-the-art model and generates structures more than 15 times closer to DFT local energy minima.Fine-tuned MatterGen also generates materials satisfying chemical, symmetry, mechanical, electronic, and magnetic constraints.Source: §1 · pp. 2–3
• MatterGen can generate materials with desired chemical composition, symmetry, and scalar properties such as band gap, bulk modulus, and magnetic density.Its fine-tuned model can generate highly symmetric structures and directly satisfy target mechanical, electronic, and magnetic property constraints.Source: §1 · pp. 2–3
• MatterGen uses diffusion to iteratively denoise atom types, coordinates, and lattice parameters, with an equivariant score network pretrained on stable structures.Adapter modules incorporate encoded properties during fine-tuning to steer generation.Source: Figure 1 · p. 3
• MatterGen demonstrates multi-property design by generating materials with both high magnetic density and low chemical supply-chain risk.Source: §1 · p. 3

2.1 Diffusion process for crystalline material generation

• MatterGen’s crystal-specific diffusion process jointly refines atom types, periodic coordinates, and lattices using geometry-aware corruption and equivariant scores.Coordinates use wrapped Normal diffusion, lattices approach a cubic-lattice distribution, and atom types diffuse through a masked categorical state; adapters and classifier-free guidance enable property steering.Source: §2.1 · pp. 3–4

2.2 Generating stable, diverse materials

• MatterGen-MP yields 1.8 times more S.U.N. structures and 3.1 times lower RMSD than CDVAE, while full MatterGen adds 1.6 times more S.U.N. structures and 5.5 times lower RMSD.The benchmark measures S.U.N. success rate and average RMSD to DFT-relaxed structures; scaling from MP-20 to Alex-MP-20 drives the further improvement.Source: Figure 2 · p. 5, §2.2 · p. 6

2.3 Generating materials with target chemistry

• MatterGen generates the highest percentage of S.U.N. structures across all evaluated chemical-system types and complexities, outperforming substitution particularly for quinary systems.For quinary systems, MatterGen uses 10,240 samples versus approximately 70,000 for substitution and 600,000 for RSS, and finds five novel V-Sr-O hull structures versus four and two.Source: §2.3 · pp. 6–8, Figure 3 · p. 7

2.4 Designing materials with target symmetry

• Fine-tuning MatterGen to 14 target space groups produces, on average, 20% S.U.N. structures belonging to the target group, exceeding 10% for selected highly symmetric groups.The evaluated groups span all seven crystal systems, including P63/mmc and Im¯3.Source: §2.4 · p. 8, Figure 4 · p. 9

2.5 Designing materials with target magnetic, electronic, and mechanical properties

• MatterGen shifts generated S.U.N. property distributions toward targets for magnetic density, band gap, and bulk modulus, even with substantially limited labeled data.Targets are 0.20 Å^-3 magnetic density, 3.0 eV band gap, and 400 GPa bulk modulus; with 500 DFT calculations, it finds 277 high-bulk-modulus S.U.N. structures versus 149 for screening.Source: §2.5 · pp. 8–11

2.6 Designing low-supply-chain-risk magnets

• Jointly targeting magnetic density and HHI generates S.U.N. structures narrowly distributed around both targets while reducing supply-chain-risk elements such as Co and Gd.The requested targets are 0.2 Å^-3 magnetic density and an HHI score of 1250; compared with magnetic-density-only generation, joint conditioning shifts HHI toward the target while retaining high magnetic density.Source: §2.6 · p. 11

Contributions

• MatterGen combines joint diffusion of atom types, coordinates, and lattice with adapter modules to improve generated-material stability, uniqueness, and novelty while supporting diverse property constraints.The approach builds on the larger Alex-MP-20 dataset and improves over prior methods including MLFF-assisted RSS, substitution, and ML-assisted screening.Source: §3 · p. 11

Applications

• MatterGen demonstrates low-supply-chain-risk magnet design by jointly optimizing HHI score and magnetic density, producing S.U.N. structures on a Pareto front.Figure 6 presents the generated structures, element frequencies, chemical compositions, space groups, magnetic densities, and HHI scores for single- and joint-property fine-tuning.Source: Figure 6 · p. 12

Limitations

• MatterGen disproportionately generates P1 structures, particularly for larger crystals, indicating a tendency toward less symmetric outputs than the training data.Potential remedies include improving the denoising process and backbone architecture and expanding the training dataset; experimental validation and characterization remain the ultimate tests of real-world applicability.Source: §3 · p. 12

Significance

• The authors characterize MatterGen’s breadth and generated-material quality as a major advancement toward a universal generative model for materials with high real-world impact.They anticipate that generative models will substantially influence materials design in the coming years.Source: §3 · pp. 12–13

Future directions

• Future extensions could cover catalyst surfaces and metal-organic frameworks and condition generation on non-scalar targets such as band structures or X-ray diffraction spectra.These extensions could support applications including nitrogen fixation, carbon capture, band engineering, and reconstructing atomic structures from experimentally measured XRD spectra.Source: §3 · p. 13

A.1 Representation of periodic materials

• MatterGen represents a material as atom species A, fractional coordinates X, and a 3 × 3 lattice matrix L defining the repeating unit cell.The lattice must be non-singular, with volume Vol(L) = |det L|, while fractional coordinates are periodic modulo integer translations.Source: §A.1 · pp. 15–16

A.4 Joint diffusion process

• The model’s forward process independently diffuses atom types, fractional coordinates, and lattices, while the reverse process need not factorize across these components.The joint transition kernel factorizes into q(A_{t+1}|A_t)q(X_{t+1}|X_t)q(L_{t+1}|L_t), and separate appendices specify each component and the score network.Source: §A.4 · p. 18

A.5 Atom type diffusion

• Atom species use masked D3PM diffusion, where each species remains unchanged or transitions to an absorbing [MASK] state, yielding a point-mass limit distribution.The reverse model predicts distributions over original atom types, and marginalization has complexity O(K), with K ≃100 for atomic species.Source: §A.5 · pp. 18, 20

A.6 Coordinate diffusion

• Fractional coordinates diffuse on the flat torus T3 = R3/Z3 using wrapped normal distributions and variance-exploding noise, whose prior is uniform on [0, 1)3.The wrapped normal accommodates periodic boundaries, and its score is approximated by truncating the otherwise infinite likelihood sum.Source: §A.6 · p. 21, §A.6.2 · p. 22

A.6.1 Variance adjustment for atomic density

• Fractional-coordinate noise is scaled with unit-cell volume under an approximately constant atomic-density assumption, mitigating lattice-dependent variance in Cartesian space.The scaling is chosen using Vol(L_t) ∝ n so the effective noise reflects the number of atoms and cell size.Source: §A.6.1 · p. 21

A.6.3 Fractional vs Cartesian coordinate diffusion

• Fractional rather than Cartesian diffusion preserves forward-process factorization while avoiding lattice-trajectory sampling and numerical instability caused by simultaneously diffusing the lattice.Cartesian covariance and periodic boundaries depend on the evolving lattice, requiring the entire noisy lattice trajectory and making long trajectories slow and unstable.Source: §A.6.3 · pp. 22–23

A.7 Lattice diffusion

• Lattice diffusion uses variance-preserving noise because variance-exploding noise would create extremely large noisy unit cells that challenge fixed-cutoff graph neural networks.The variance-preserving choice limits the range of lattice length scales over which the model must learn scores.Source: §A.7 · p. 23

A.7.1 Fixed-rotation lattice diffusion

• To handle rotational invariance, the lattice is canonically aligned and represented as a symmetric matrix obtained through polar decomposition based on singular value decomposition.This canonical alignment is chosen instead of a rotation-invariant prior to provide greater flexibility in designing the lattice diffusion process.Source: §A.7.1 · p. 23
• Restricting lattice diffusion and noise to symmetric matrices fixes the rotation and leaves six lattice degrees of freedom.The method models only the upper-triangular entries and mirrors them to the lower triangle.Source: §A.7.1 · p. 23

A.7.2 Lattice diffusion with custom limit mean and variance

• Naive independent variance-preserving diffusion produces noisy unit cells with very small volumes and steep angles, densely packing atoms.Source: §A.7.2 · p. 23
• The modified limit distribution favors cubic lattices, concentrates vector angles between 60° and 120°, and aligns with Niggli-reduced cells and RSS initialization.These properties provide a suitable starting prior for generation.Source: §A.7.2 · p. 24
• Scaling the prior mean as µ(n) = 3√nc and standard deviation as σ(n) = 3√nν keeps atomic density and signal-to-noise approximately independent of atom count.The coefficient c can be set to the inverse average dataset density.Source: §A.7.2 · p. 24

A.8 Architecture of the score network

• MatterGen predicts lattice, coordinate, and atom-type scores with a four-layer SE(3)-equivariant GemNet-dT network using 7 Å neighborhoods and 512-dimensional node and edge representations.Coordinate scores behave as non-conservative forces: they are rotation- and permutation-equivariant, translation-invariant, and transformed into fractional scores.Source: §A.8 · p. 24

A.8.1 Computation of the predicted lattice scores

• Lattice scores aggregate edge-length predictions transformed by the chain rule, then normalize and symmetrize them so the resulting score is scale-invariant and rotation-equivariant like a stress tensor.The initial edge-based lattice predictions were generally nonsymmetric despite diffusion occurring on symmetric lattice matrices, motivating the symmetrization modification.Source: §A.8.1 · pp. 25–26

A.8.2 Augmenting the input with lattice information

• Injecting lattice-angle cosines into edge representations lets the model distinguish equivalent unit-cell choices, while Niggli reduction removes the resulting dependence from training data.The augmented representations retain rotation and translation invariance, and the added information distinguishes the previously indistinguishable cases shown in Fig. A1.Source: §A.8.2 · pp. 26–27, Figure A1 · p. 27

A.9 Training loss

• The training objective sums score-matching losses for coordinates and the cell with a discrete atom-type loss, aggregating coordinate and species contributions over all atoms.The atom-type term follows the D3PM objective, while the coordinate and cell terms use score matching.Source: §A.9 · pp. 27–28

B Fine-tuning the score network for property-guided generation

• Property-guided generation fine-tunes the pretrained unconditional score network rather than learning conditional scores from sparse labeled data alone.The unconditional model’s broad stable-material distribution provides the initialization for classifier-free conditional generation.Source: B Fine-tuning the score network for… · p. 29, §B.1 · p. 29

B.1 Fine-tuning the score network with adapter modules

• Zero-initialized adapter modules add property embeddings before each of four message-passing layers, preserving unconditional outputs at initialization and enabling conditional and unconditional scores after fine-tuning.This initialization retains the stability-optimized pretrained behavior and makes sparse-label fine-tuning more computation- and sample-efficient than retraining from scratch, without a major sacrifice in stability and novelty.Source: §B.1 · p. 29

B.2 Classifier-free guidance; B.2.1 Continuous case

• Classifier-free guidance replaces unconditional reverse-process scores with a guided combination of conditional and unconditional predictions, using guidance factor γ = 2 in all conditional experiments.Continuous variables use condition embeddings concatenated to the score model, and multiple properties are represented by multiple condition embeddings.Source: §B.2 · p. 30, §B.2.1 · p. 30

B.2.2 Discrete case

• For discrete atom types, classifier-free guidance combines conditional and unconditional predictions of pθ(a0|at, c) to approximate the guided reverse distribution.The discrete denoising task first approximates the clean atom-type distribution conditioned on the noisy type and property condition.Source: §B.2.2 · pp. 30–31

C.1 Data sources

• The source data combine MP and Alexandria open-access DFT-relaxed structures with selected ordered experimental ICSD structures not already present in MP.MP structures use existing Materials Project API calculations, whereas other sources receive new calculations under MP settings.Source: §C.1 · p. 31
• 1,081,850 unique structures comprise the Alex-MP-ICSD reference dataset used to evaluate generated-structure stability and uniqueness.The dataset combines validated, de-duplicated calculations, applies a PBE correction scheme, and constructs convex-hull phase diagrams.Source: §C.1 · p. 31
• MatterGen training uses Alex-MP-20 structures with at most 20 atoms and energy above hull below 0.1 eV/atom, excluding well-explored systems and ICSD-only test structures.The MatterGen-MP model further uses the corresponding MP-20 subset below 0.1 eV/atom from the reference convex hull.Source: §C.1 · p. 32

C.2 DFT details

• DFT calculations use VASP with projector-augmented waves and Materials Project-consistent PBE, Hubbard U, relaxation, band-gap, and elastic-tensor workflows.Elastic tensors use the computationally efficient minimal preset with -0.01% and 0.01% deformations.Source: §C.2 · pp. 32–33

D Results

• The results section provides supplementary information for the paper’s main results, with experiment-specific details deferred to later appendices.The common-details section covers MatterGen training and sampling settings and additional machine-learning force-field information.Source: D Results · p. 34, §D.1 · p. 34

D.1 Common experimental details

• Common experimental details document settings shared across experiments, including MatterGen training, sampling, and the employed machine-learning force field.Details specific to individual experiments are deferred to Appendices D.3–D.7.Source: §D.1 · p. 34

D.1.1 Hyperparameters for training base model

• The base unconditional model was trained for 1.74 million steps with batch size 64 per GPU across eight A100 GPUs using Adam.The learning rate started at 0.0001 and used ReduceLROnPlateau with decay factor 0.6, patience 100, and minimum learning rate 10−6.Source: §D.1.1 · p. 34
• The base-model loss used λcoord = 0.1, λcell = λtypes = 1, and λCE = 0.01 for the D3PM cross-entropy term.Source: §D.1.1 · p. 34

D.1.2 Hyperparameters for fine-tuning models

• Fine-tuning used global batch size 128, Adam, gradient clipping at 0.5, and an initial learning rate of 6 × 10−5 with the base-model scheduler.Source: §D.1.2 · p. 34
• Fine-tuning stopped after validation loss failed to improve for 100 epochs, requiring 32,000–1.1 million steps depending on the dataset.Source: §D.1.2 · p. 34

D.1.3 MatterGen sampling parameters

• MatterGen discretized unconditional and conditional reverse diffusion into T = 1000 steps, using ancestral sampling followed by one corrector step at each timestep.Langevin correctors were applied to coordinates and lattice variables with signal-to-noise ratios 0.4 and 0.2, respectively.Source: §D.1.3 · p. 34

D.1.4 Machine learning force field (MatterSim) details

• MatterSim used an M3GNet force field trained on 1.08 million crystalline structures with three graph-convolution layers and 890,000 parameters.Source: §D.1.4 · p. 34
• Energy-above-hull calculations used the MaterialsProject2020Compatibility correction scheme from pymatgen.Further machine-learning force-field details are deferred to a separate publication.Source: §D.1.4 · p. 34

D.2 Qualitative analysis of generated structures

• Generated-crystal quality is assessed computationally and supplemented by human review to identify failure modes that metrics may miss.The authors acknowledge that synthesizability cannot be conclusively established from theory and computation alone.Source: §D.2 · pp. 35, 37
• Energy above the convex hull at 0 K and 0 GPa is used as a stability signal, but its interpretation depends on chemistry, DFT accuracy, and reference-set completeness.Metastable materials may be synthesized, while on-hull materials may remain unsynthesized; thresholds vary from 0.1–1 eV for some carbides and nitrides to meV for some intermetallics.Source: §D.2 · p. 35
• Human-assisted assessment considers symmetry, defects, local environments, and charge balance, while recognizing that empirical priors are biased and incomplete.Charge balance is treated cautiously because only around 40% of newly discovered materials are charge-balanced, compared with over 80% discovered a hundred years ago.Source: §D.2 · pp. 35–36
• MatterGen’s ordered, periodic-material definition can admit non-3D, amorphous, or spurious structures and cannot directly represent disordered alloys or solid solutions.Multiple ordered approximations of one disordered parent may be generated, making novelty assessments incorrect and energy-above-hull values misleading.Source: §D.2 · pp. 36–37

D.2.1 Visualization

• Crystal visualizations use Crystal Toolkit with CrystalNN bonding, uniform atomic radii, heuristic formal valences, and 2 × 2 × 2 supercells except for Fig. 1.Structures are shown as calculated rather than in conventional settings, so axes are not labelled.Source: §D.2.1 · p. 37

D.3 Generating stable and diverse materials

• The supplementary evaluation of stable and diverse generation uses RMSD and the fraction of S.U.N. structures as its two key metrics.RMSD measures local stability, while the S.U.N. fraction captures global stability and, to some extent, diversity.Source: §D.3 · p. 37, §D.3.1 · p. 37

D.3.1 RMSD, stability, uniqueness and novelty

• Lower RMSD indicates that generated structures are closer to their DFT-relaxed counterparts, reducing the costly DFT-relaxation workload.The RMSD compares generated and relaxed structures using an element-aware atom permutation.Source: §D.3.1 · pp. 37–38
• Novelty and uniqueness are computed with pymatgen’s StructureMatcher for structures sharing reduced formula and space group, but ordered approximations of disordered materials may be falsely labelled novel.The reported Fig. 2 plots and structures use 1024 generated samples that were also relaxed with DFT.Source: §D.3.1 · p. 38

D.3.2 Additional qualitative analysis of structures

• 43 unique on-hull structures emerged from 1,024 unconditional samples, including 11 binaries, 22 ternaries, and 10 quaternaries.Three structures had P1 symmetry, and three contained molecules or were molecular crystals.Source: §D.3.2 · p. 38, Table D2 · p. 39
• Qualitative examples span plausible, uncertain, known, and rocksalt-derived structures, including K3AlCl6, NaNiIO6, BaLa2Ir, and NaSmTm2Te4.NaNiIO6 illustrates that novelty before relaxation can be lost after relaxation, while BaLa2Ir’s existence remains unclear.Source: §D.3.2 · p. 38

D.4 Generating materials with target chemistry

• The chemistry benchmark covers 27 ternary, quaternary, and quinary systems classified as well explored, partially explored, or not explored.Well-explored systems test recovery of unseen stable structures, partially explored systems test convex-hull expansion, and unexplored systems contain no reference hull structures.Source: §D.4.1 · pp. 39–40, Table D3 · p. 41

D.4.1 Additional experimental details

• Nine systems of each stoichiometric order were randomly selected, using 0.0–0.1 eV/atom above-hull structures to define exploration categories and promote chemical diversity.Systems overlapping by more than two elements were randomly replaced.Source: §D.4.1 · p. 40
• MatterGen was fine-tuned on chemical system and energy above hull, then sampled at 0.0 eV/atom above hull for the target system.Both conditioning properties were available throughout the base-model training set, so the full set was used for fine-tuning.Source: §D.4.1 · p. 40
• Evaluation generated, MLFF-relaxed, deduplicated, and DFT-evaluated the 100 lowest predicted-energy structures, with 10,240 MatterGen samples per chemical system.RSS used 600,000 samples per system, while substitution enumerated 15,000–70,000 structures per system.Source: §D.4.1 · p. 40

Random structure search details

• RSS used two 300,000-structure rounds across non-overlapping unit-cell atom-count ranges, with MLFF relaxation and automatic MINSEP tuning before DFT evaluation.The final RSS pool combined 600,000 MLFF-relaxed structures, from which 100 unique low-predicted-energy structures were selected.Source: Random structure search details · pp. 40–41

Substitution details

• Substitution enumerated all unique prototype substitutions from 5,143 ICSD structures, then MLFF-relaxed and DFT-evaluated the 100 lowest predicted-energy candidates per system.The prototypes comprised 2,695 ternary, 1,875 quaternary, and 573 quinary structures with fewer than 100 atoms per unit cell.Source: Substitution details · p. 41

D.4.2 Additional qualitative analysis of structures

• Four new on-hull V-Sr-O structures were generated, including known-related compositions such as perovskite SrVO3 and chemically plausible vanadates.The structures are SrV2O6, SrVO3, Sr3V2O8, and SrV2O4; related literature includes SrVO3, Sr2VO4, and Sr3(VO4)2.Source: §D.4.2 · p. 41
• All generated vanadates have plausible atomic environments, with ideal or distorted VO4 units and oxygen-coordinated Sr atoms consistent with oxidation-state-dependent coordination.Vanadate frameworks can vary substantially with oxidation state, from ideal tetrahedral V5+ coordination to other environments.Source: §D.4.2 · p. 41
• The P¯1 SrV2O4 structure forms edge-sharing VO4 tetrahedral layers, triangular-prismatic Sr coordination, and a one-dimensional void channel.The void channel occurs within the Sr layer separating the tetrahedral sheets.Source: §D.4.2 · p. 41

D.5 Designing materials with target symmetry

• The target-symmetry analysis supplements the paper’s main symmetry-design results, while the target magnetic, electronic, and mechanical-property section provides supplementary information for its corresponding Section 2.5 results.These subsections define their role as supplementary analyses rather than presenting additional standalone results in the supplied passages.Source: §D.5 · p. 42, §D.6 · p. 42

D.5.1 Additional experimental details

• Target-space-group generation fine-tunes the base model on the full training set and conditions the latent crystal representation using one-hot space-group encodings.Evaluation samples target groups across seven lattice systems, uses 256 relaxed structures per group, and separately tests reproduction of the training space-group distribution with 10,000 generated structures while matching atom-count distributions to avoid impossible constraints.Source: §D.5.1 · p. 42

D.5.2 Additional Analysis of Structures

• Highlighted target-symmetry examples show high novelty, uncommon elemental combinations such as DyScNiPd and CeAsRh, partial formal-valence assignability, and robustness to higher symmetry-finding tolerances.Few examples matched known prototypes; half could be assigned formal valences, and the majority remained robust for their target symmetry under the stricter symmetry analysis.Source: §D.5.2 · p. 42

Fine-tuning

• MatterGen is fine-tuned on approximately 605,000 magnetic-density, 42,000 band-gap, and 5,000 bulk-modulus DFT labels using sinusoidal scalar-property embeddings.The property embedding follows a sinusoidal encoding from the Transformer architecture.Source: §D.6.1 · p. 42

Evaluation protocol

• Conditioned generations are relaxed with MLFF and DFT, filtered for stability, uniqueness, and novelty, and screened for property outliers before final DFT property computation.The procedure uses targets of 0.2 Å^-3 magnetic density, 3.0 eV band gap, and 400 GPa bulk modulus; outlier cutoffs include 600 GPa bulk modulus and 0.3 Å^-3 magnetic density.Source: §D.6.1 · p. 43
• After all filters and DFT computations, the study obtains 251 S.U.N. magnetic-density structures, 142 band-gap structures, and 22 bulk-modulus structures.Source: §D.6.1 · p. 43

Budgeted evaluation

• For budgeted magnetic-density evaluation, 15,360 generated samples yield 5,365 MLFF-filtered candidates, including 540 DFT-stable structures among 600 DFT-relaxed samples.For bulk modulus, 8,192 samples yield 801 candidates, of which 736 are DFT stable; both tasks subsample structures according to the DFT property-calculation budget and compare against screening baselines.Source: §D.6.1 · pp. 43–44

Property predictor details

• The bulk-modulus predictor uses a three-layer GemNet-dT encoder with 128-dimensional node and edge representations, a 10 Å cutoff, and mean readout.It is trained on 7,108 MP structures with an 80/10/10 split and achieves a 9.5 GPa MAE while predicting log10 bulk modulus.Source: §D.6.1 · p. 44

D.6.2 Additional qualitative analysis of structures

• Eight of ten randomly reviewed high-magnetic-density crystals were ordered Fe-Co alloy approximations, indicating plausible magnetic materials but limited novelty beyond database unit-cell arrangements.The generated Fe-Co structures may be useful, yet their novelty is likely confined to specific ordered cells absent from reference databases.Source: §D.6.2 · pp. 44–45
• The remaining high-magnetic-density examples involved Fe hydrides or Gd-containing structures, but phase segregation, uncertain crystallization, and strong Gd compositional bias limit interpretation.Gd was favored because of its large magnetic moment, while the Gd2N and Gd6H2CN3 structures exhibited layered or rocksalt-derived motifs.Source: §D.6.2 · p. 45

Band gap conditional generation

• Target-band-gap generations showed broad elemental diversity and nominal charge balance, consistent with producing more ionic insulating materials for a 3 eV PBE target.Only NaNO3 and BI3 matched previously synthesized compositions in the random sample, and both had incorrect symmetry but correct local bonding and similar calculated band gaps.Source: Band gap conditional generation · p. 45
• VBiO4 and TlNO3 reproduced plausible local chemistry despite differing substantially from experimentally known structures, while VBiO4 was calculated thermodynamically stable relative to known polymorphs.The authors attribute this either to DFT limitations or under-convergence among similar-energy polymorphs, rather than treating the result as model failure.Source: Band gap conditional generation · pp. 45–46

Bulk modulus conditional generation

• High-bulk-modulus generations were strongly biased toward refractory elements Re, W, Mo, and Ir, often with B or C, matching compositional patterns associated with superhard materials.Re-only systems resembled ordered alloy approximations, whereas B- or C-containing systems were typically anisotropic and layered; the reported bulk modulus was averaged over the full elastic tensor.Source: Bulk modulus conditional generation · p. 46
• The Re3Ir example represented an ordered approximation of a synthesized Ir-Re solid solution, whereas the unusual layered Re3B2C structure nominally satisfied charge balance.Directional bulk moduli were not examined for these highly anisotropic systems.Source: Bulk modulus conditional generation · p. 46

D.7 Designing low-supply-chain risk magnets

• The low-supply-chain-risk magnet discussion supplies supplementary information for the results presented in Section 2.6.No additional findings are provided in the supplied passage.Source: §D.7 · p. 46

D.7.1 Additional experimental details

• The model is fine-tuned on magnetic density and HHI score, then conditioned to generate structures targeting 0.2 ˚A−3 magnetic density and an HHI score of 1200.The evaluation follows Appendix D.6 and uses 512 generated samples.Source: §D.7.1 · p. 46
• 112 structures remain novel against the reference dataset after 130 samples pass stability and uniqueness filtering following DFT relaxation.The novel structures are reported in Fig. 6(a).Source: §D.7.1 · p. 46

• Jointly targeting low HHI and high magnetic density steers MatterGen away from structures containing Co, which is associated with poor HHI scores.Source: §D.7.2 · p. 46
• Example outputs include Fex Mn1–x O rocksalt alloys, defected FeO with vacancies, and body-centered-cubic Fe8Au, all described as well-known experimentally.The listed MnFe3O4, MnFe8O9, and Fe8O9 examples are specific generated structures.Source: §D.7.2 · p. 46
• The joint optimization remains imperfect because some candidates have high magnetization density only in hypothetical ferromagnetic states, not their actual antiferromagnetic ground states.Source: §D.7.2 · p. 46
• Performance could improve by penalizing expensive elements, preferring metallic systems, and treating alloy systems better to support ferromagnetic candidates and true novelty.The passage characterizes overall performance on the training labels as reasonable while identifying these extensions as potential improvements.Source: §D.7.2 · p. 47