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MatterGen: a generative model for inorganic materials design

Claudio Zeni, Robert Pinsler, Daniel Zügner, Andrew Fowler, Matthew Horton, Xiang Fu, Sasha Shysheya, Jonathan Crabbé, Lixin Sun, Jake Smith, Bichlien Nguyen, Hannes Schulz, Sarah Lewis, Chin-Wei Huang, Ziheng Lu, Yichi Zhou, Han Yang, Hongxia Hao, Jielan Li, Ryota Tomioka, Tian Xie

arXiv:2312.03687v2cond-mat.mtrl-scics.AI

TL;DR

Materials discovery must explore vast inorganic-composition spaces while satisfying desired property constraints, but existing generative models have limited stability and constraint coverage. MatterGen uses joint diffusion over crystal components with adapter-based property fine-tuning, yielding stable, novel materials across diverse chemistry, symmetry, and property targets, including a multi-property design example.

  • Problem

    Existing materials-discovery methods explore only a tiny fraction of potential stable inorganic compounds and cannot be efficiently steered toward target properties.

  • Method

    MatterGen jointly diffuses atom types, coordinates, and lattice structure, then uses adapter modules to fine-tune generation toward labeled property constraints.

  • Results

    MatterGen generates stable, novel materials across the periodic table and supports target chemistry, symmetry, mechanical, electronic, and magnetic properties, including combined magnetic-density and supply-chain-risk constraints.

  • Takeaways & Limitations

    MatterGen broadens generative materials design from stability and novelty toward diverse single- and multi-property constrained generation.

  • Takeaways & Limitations

    Evaluations cover only some criteria needed for real-world applications, with experimental validation and characterization remaining the ultimate test.

Abstract

from arXiv · show

The design of functional materials with desired properties is essential in driving technological advances in areas like energy storage, catalysis, and carbon capture. Generative models provide a new paradigm for materials design by directly generating entirely novel materials given desired property constraints. Despite recent progress, current generative models have low success rate in proposing stable crystals, or can only satisfy a very limited set of property constraints. Here, we present MatterGen, a model that generates stable, diverse inorganic materials across the periodic table and can further be fine-tuned to steer the generation towards a broad range of property constraints. To enable this, we introduce a new diffusion-based generative process that produces crystalline structures by gradually refining atom types, coordinates, and the periodic lattice. We further introduce adapter modules to enable fine-tuning towards any given property constraints with a labeled dataset. Compared to prior generative models, structures produced by MatterGen are more than twice as likely to be novel and stable, and more than 15 times closer to the local energy minimum. After fine-tuning, MatterGen successfully generates stable, novel materials with desired chemistry, symmetry, as well as mechanical, electronic and magnetic properties. Finally, we demonstrate multi-property materials design capabilities by proposing structures that have both high magnetic density and a chemical composition with low supply-chain risk. We believe that the quality of generated materials and the breadth of MatterGen's capabilities represent a major advancement towards creating a universal generative model for materials design.

1 Introduction

MatterGen addresses the limited scale and property steerability of materials discovery by generating stable inorganic structures and adapting them to desired constraints. Its diffusion process jointly denoises crystal components, while adapter modules enable conditioning on chemistry, symmetry, and scalar properties.

  • Contribution: MatterGen generates stable inorganic materials across the periodic table and can be fine-tuned toward chemical composition, symmetry, and scalar property constraints.Examples of scalar properties include band gap, bulk modulus, and magnetic density.
  • Contribution: More than twice as many generated MatterGen structures are stable, unique, and novel compared with the previous state-of-the-art generative model.The comparison uses the percentage of generated stable, unique, and novel (S.U.N.) materials.
  • Contribution: More than 15 times closer to their ground-truth structures at the DFT local energy minimum, MatterGen’s generated structures improve over the previous state of the art.This result is reported alongside the S.U.N. comparison.
  • Method: The model reverses a corruption process by iteratively denoising atom types, coordinates, and the periodic lattice from an initially random structure.The forward process independently corrupts these crystal components toward a physically motivated random-material distribution.
  • Method: Adapter modules fine-tune the score network with labeled data to steer generation toward desired chemistry, symmetry, and scalar property constraints.The approach is intended to support inverse design tasks involving rare or conflicting target properties.
  • Contribution: MatterGen demonstrates multi-property design by proposing materials with both high magnetic density and chemical compositions having low supply-chain risk.The paper presents this as a demonstration of the model’s breadth of property-constrained generation.

2 Results

MatterGen combines a crystal-specific diffusion process with adapter-based fine-tuning to generate stable, diverse inorganic materials and steer them toward chemical, symmetry, and property constraints. Across these tasks, it produces novel candidates efficiently, with generated structures close to DFT-relaxed minima.

  • Diffusion process: MatterGen diffuses atom types, coordinates, and periodic lattices using component-specific processes suited to crystalline geometry.Coordinates use a wrapped Normal distribution, lattices approach a density-informed cubic distribution, and atom types are corrupted into a masked state.
  • Fine-tuning: Adapter modules fine-tune the base model on labeled data to steer generation toward chemical composition, symmetry, and scalar property constraints.The two-stage strategy first learns general crystal generation, then adapts it for downstream inverse-design tasks.
  • Stable, diverse materials: 100% of structures are unique at 1,000 samples, while uniqueness remains 86% and novelty about 68% at one million samples.Novelty is measured with respect to Alex-MP-ICSD, indicating limited saturation at large generation scales.
  • Stable, diverse materials: MatterGen-MP achieves a 1.8 times increase in S.U.N. structures and a 3.1 times decrease in average RMSD versus CDVAE, while full MatterGen adds 1.6 times and 5.5 times improvements respectively.The comparison attributes the further gain to scaling from the smaller MP-20 training set.
  • Target chemistry: MatterGen discovers five novel structures on the V-Sr-O combined hull using 10,240 samples, versus four for substitution and two for RSS.For quinary systems, MatterGen achieves strong performance with about 10,240 samples, compared with about 70,000 for substitution and 600,000 for RSS.
  • Target properties: MatterGen shifts generated property distributions toward target values, including targets in the tails of the labeled data distribution.This behavior holds even when substantially fewer DFT labels are available than unlabeled training structures.
  • Multiple constraints: MatterGen jointly generates materials with high magnetic density and chemical compositions shifted toward low supply-chain risk.Joint conditioning shifts the HHI distribution toward its target while retaining high magnetic density.

3 Discussion

MatterGen combines joint diffusion and adapter modules to generate stable, novel materials under diverse single- and multi-property constraints. The authors identify symmetry bias and the need for experimental validation as remaining limitations.

  • MatterGen jointly diffuses atom types, coordinates, and lattice, then uses adapter modules to target chemical, symmetry, and scalar property constraints.The approach is paired with the Alex-MP-20 training dataset and supports constraints across mechanical, electronic, and magnetic properties.
  • MatterGen generates S.U.N. structures satisfying target constraints across a wide range of properties and supports joint design of high magnetic density with low supply-chain risk.Figure 6 presents the low-supply-chain-risk magnet design task as a multi-property application.
  • MatterGen disproportionately generates P1 structures relative to the training data, especially for larger crystals.The authors hypothesize that improved denoising, architecture, and training data could reduce this tendency.
  • Experimental validation and characterization remain the ultimate test because the evaluations cover only some criteria required for real-world applications.The authors also discuss broader evaluation challenges for generative crystalline materials.
  • The authors position MatterGen as a step toward a universal generative model and suggest extending it to catalyst surfaces and metal-organic frameworks.These proposed extensions are presented as future directions rather than demonstrated capabilities in this study.

Declarations

The supplementary material defines crystalline-material representations and diffusion processes used by MatterGen. It specifies invariances, coordinate and lattice conventions, and the independent corruption of atom types, coordinates, and lattice.

  • A crystal material is represented by atom species A, fractional coordinates X, and lattice L within a repeating unit cell.The lattice contains the unit-cell vectors, while fractional coordinates locate atoms relative to those vectors.
  • The lattice volume is Vol(L) = |det L|, so physically sensible cells require a non-singular lattice matrix.The three lattice vectors form the columns of the 3 × 3 lattice matrix.
  • Fractional coordinates are periodic and convert to Cartesian coordinates through the lattice vectors, with x ∼ x + k for k ∈ Z3.For example, x = (0.2, 0.3, 0.5)⊤ maps to 0.2l1 + 0.3l2 + 0.5l3.
  • Material energy is invariant to atom permutation, translation, rotation, periodic cell choice, and supercell choice, while forces and stresses have corresponding equivariance properties.These transformation properties constrain how crystalline structures and physical quantities are represented.
  • Diffusion models learn to reverse a Markov corruption chain, using score models to approximate noise-distribution gradients and sampling from a simple terminal prior.MatterGen uses variance-exploding diffusion for fractional coordinates and variance-preserving diffusion for the lattice.
  • MatterGen diffuses atom types, coordinates, and lattice independently, with atom types using masked categorical diffusion and the limit lattice distribution tending toward cubic cells.The lattice prior has mean proportional to the identity and lattice-vector angles concentrated mostly between 60° and 120°.

A.8 Architecture of the score network

MatterGen uses an SE(3)-equivariant score network to predict lattice, coordinate, and atom-type updates during denoising, with lattice-specific inputs resolving equivalent-cell ambiguities. Adapter modules then incorporate property labels for conditional generation.

  • Score network: The score network predicts lattice, atom-position, and atom-type scores during the denoising process.It uses an adapted GemNet-dT architecture with four message-passing layers and a 7 Å neighbor cutoff.
  • Score network: Cartesian coordinate scores are rotation- and permutation-equivariant and translation-invariant before conversion to fractional scores.
  • Lattice prediction: Lattice scores are built from edge representations that predict how edge lengths should change, then averaged and transformed into lattice updates.The chain-rule construction is scale-invariant and rotation-equivariant after normalization.
  • Lattice prediction: Because chain-rule lattice scores lacked expressiveness, the model adds lattice-angle information so it can distinguish equivalent unit-cell choices.Training structures are converted to Niggli-reduced cells to address the resulting loss of cell-choice equivariance.
  • Property conditioning: Adapter modules inject property embeddings into message-passing layers, enabling conditional and unconditional score prediction for classifier-free guidance.Zero-initialized mix-in layers preserve the unconditional score at initialization, supporting fine-tuning from a stable-materials model.

C.1 Data sources

The datasets combine crystal structures from MP, Alexandria, and ICSD-derived sources, followed by validation, deduplication, corrections, and phase-diagram construction. Alex-MP-ICSD provides the reference set, while Alex-MP-20 is a filtered training subset.

  • Data sources: Crystal structures were collected from MP, Alexandria, and ICSD-derived sources.The sources include DFT-relaxed structures from experimentally known and hypothetical crystals.
  • Data processing: The processing pipeline validates calculations, deduplicates equivalent structures, applies PBE corrections, and constructs convex-hull phase diagrams.Deduplication groups the same crystal structure when it appears in multiple data sources.
  • Reference dataset: The reference Alex-MP-ICSD dataset contains 1,081,850 unique structures with DFT-calculated energy-above-hull values.
  • Training dataset: Alex-MP-20 retains structures with at most 20 atoms and energy above hull below 0.1 eV/atom.Well-explored chemical systems are excluded, and structures present only in ICSD are reserved for testing.
  • Computational settings: DFT calculations use VASP with MP-consistent PBE and Hubbard U settings for energies, band gaps, and elastic tensors.

D.1.1 Hyperparameters for training base model

The base model is trained with Adam on eight A100 GPUs, while property fine-tuning uses a smaller learning rate, gradient clipping, and validation-based stopping. Generation uses 1,000 reverse-diffusion steps with predictor-corrector sampling.

  • Base-model training: The unconditional base model was trained for 1.74 million steps with batch size 64 per GPU across eight A100 GPUs.Adam optimization used an initial learning rate of 0.0001 with ReduceLROnPlateau scheduling.
  • Property fine-tuning: Fine-tuning used global batch size 128, Adam, gradient clipping at 0.5, and an initial learning rate of 6 × 10^-5.Training stopped after validation loss failed to improve for 100 epochs.
  • Sampling: Unconditional and conditional generation discretized reverse diffusion over [0, 1] into T = 1000 steps.Each ancestral predictor step was followed by a Langevin corrector step.
  • Evaluation model: The machine-learning force field used for evaluation was trained on 1.08 million crystalline structures with the M3GNet architecture.It contained 890,000 parameters and used MP-compatible energy corrections.

D.2 Qualitative analysis of generated structures

The qualitative analysis evaluates generated crystals with computational metrics and human-assisted inspection, while emphasizing that synthesizability remains only partially captured by theory and computation. The evaluation is further constrained by ordered, three-dimensional periodic material representations.

  • Evaluation scope: Generated-crystal quality is assessed computationally, with additional human review used to identify failure modes that metrics may miss.The human analysis examines generated structures in multiple appendices.
  • Synthesizability: Energy above the convex hull at 0 K and 0 GPa is used as a signal of possible ambient-condition stability, not a conclusive test of synthesizability.Phonon calculations and other analyses could improve the approximation, but robust synthesizability assessment remains open.
  • Synthesizability: Energy above hull alone is insufficient because metastable off-hull materials can be synthesized and acceptable thresholds vary by chemical system.Carbides and nitrides can tolerate higher energies above the convex hull than some other material classes.
  • Human assessment: Empirical synthesizability priors depend on expert chemical intuition and may be biased toward previously synthesized materials.The distribution of possible crystal structures is not yet known even under constraints such as composition or size.
  • Representation limits: MatterGen assumes ordered, three-dimensional periodic crystals, excluding or mishandling amorphous, non-3D-periodic, and disordered materials.Disordered materials may be represented by multiple ordered approximations, complicating novelty assessment.
  • Metrics: RMSD measures distance to the DFT-relaxed structure, while S.U.N. fraction measures global stability and partly captures diversity; novelty uses StructureMatcher comparisons.The novelty definition may classify ordered approximations of disordered structures as novel.

D.3.2 Additional qualitative analysis of structures

The supplementary analysis reports 43 unique, on-hull structures from 1024 unconditional generations and details the evaluation design for chemical-system exploration. It also describes comparison workflows against RSS and substitution baselines.

  • 43 unique, on-hull crystal structures were identified among 1024 unconditional generations: 11 binaries, 22 ternaries, and 10 quaternaries.Three had P1 symmetry, and three contained molecules or were molecular crystals.
  • Chemical-system exploration covered 27 ternary, quaternary, and quinary systems grouped as well explored, partially explored, or not explored.The groups tested recovery of unseen stable structures, expansion of known convex hulls, and exploration of systems without near-hull data.
  • Near-the-convex-hull structures were defined as lying between 0.0 and 0.1 eV/atom above the convex hull.Nine systems were randomly selected for each element count within each exploration group.
  • MatterGen was fine-tuned jointly on chemical system and energy above hull, conditioning generation on 0.0 eV/atom and the target chemical system.Both labels were available for all base-model training structures.
  • Baseline comparisons used MLFF relaxation, uniqueness filtering, selection of 100 lowest predicted-energy structures, and subsequent DFT calculations.RSS generated 600,000 MLFF-relaxed structures across two rounds, while substitution used 5,143 ICSD prototypes.

D.4.2 Additional qualitative analysis of structures

The supplementary symmetry analysis describes target-space-group conditioning and reports chemically diverse generated structures with high novelty and generally robust target symmetry. A V-Sr-O case study produced four new on-hull structures with plausible local environments.

  • Four new on-hull crystal structures were generated for the V-Sr-O chemical system: SrV2O6, SrVO3, Sr3V2O8, and SrV2O4.The system is well studied, and SrVO3 is a known perovskite.
  • All generated vanadate structures had plausible atomic environments containing ideal or distorted VO4 units and oxygen-coordinated Sr atoms.The coordination environments varied consistently with vanadium oxidation state.
  • One SrV2O4 structure had P¯1 symmetry, edge-sharing VO4 tetrahedral layers, and Sr-separated one-dimensional void channels.The Sr atoms formed triangular-prismatic bonding environments.
  • MatterGen represented target space groups with one-hot embeddings and evaluated conditioned generation across sampled lattice systems.The first task sampled two space groups from each of seven lattice systems, restricted to groups with at least 1000 training structures.
  • Highlighted symmetry-conditioned examples combined uncommon elements such as DyScNiPd and CeAsRh and showed few matches to known prototypes.Half could be assigned formal valences, and most remained robust under a higher symmetry-finding tolerance.

D.6.1 Additional experimental details

The experimental details describe property-conditioned generation for magnetic density, band gap, and bulk modulus, followed by relaxation, stability, uniqueness, novelty, and property filtering. The resulting numbers of S.U.N. structures differed substantially across properties.

  • MatterGen was fine-tuned on 605,000 magnetic-density labels, 42,000 band-gap labels, and 5,000 bulk-modulus labels.Scalar properties were represented with sinusoidal encodings.
  • 512 samples per property were conditioned on 0.2 Å^-3 magnetic density, 3.0 eV band gap, or 400 GPa bulk modulus.Generated structures underwent MLFF relaxation before DFT-based filtering and property calculation.
  • 251 magnetic-density, 142 band-gap, and 22 bulk-modulus structures remained after stability, uniqueness, novelty, DFT, and outlier filtering.These were the final S.U.N. structures obtained for the three property targets.
  • For a larger magnetic-density experiment, 15,360 samples produced 5,365 MLFF-filtered candidates, of which 540 of 600 DFT-tested structures were stable.The workflow then sampled structures according to the available DFT property-calculation budget.
  • The bulk-modulus predictor was trained on 7,108 structures and achieved a 9.5 GPa mean absolute error.The data split was 80% training, 10% validation, and 10% testing.

D.6.2 Additional qualitative analysis of structures

Qualitative review found property-conditioned structures with recognizable chemical trends and plausible local environments, but also exposed important limits of novelty and ordered-cell representations. These issues were especially evident in magnetic-density generation.

  • Eight of ten randomly reviewed high-magnetic-density structures were ordered approximations of Fe-Co alloys.The generated structures resembled α-Fe with 10%–40% Co substitution, a known soft-magnetic system.
  • The high-magnetic-density task showed strong compositional bias toward Gd-containing materials and Fe-Co-like structures.Gd was favored because Gd3+ and elemental Gd have large magnetic moments.
  • Some apparent novelty may reflect specific ordered unit cells absent from reference databases rather than genuinely novel alloy phases.For alloys, energy-above-hull estimates based on a single ordered approximation can be misleading.
  • Band-gap generations covered a wide range of elements and could nominally charge balance without the strong compositional dependence seen in other single-property tasks.VBiO4 had local bonding similar to known bismuth vanadate but a different generated space group.
  • High-bulk-modulus structures were compositionally biased toward refractory elements Re, W, Mo, and Ir, often alongside B and C.This trend was consistent with compositions reported for superhard materials.

D.7.1 Additional experimental details

The experiment fine-tunes MatterGen to jointly target magnetic density and HHI score, then filters generated structures for stability, uniqueness, and novelty.

  • 512 samples were generated while conditioning on magnetic density 0.2 ˚A−3 and HHI score 1200.
  • 130 samples remained after stability and uniqueness filtering following DFT relaxation.
  • 112 structures passed the novelty check against the reference dataset and were reported in Fig. 6(a).

D.7.2 Additional qualitative analysis of structures

Jointly targeting low HHI and high magnetic density steers MatterGen away from cobalt-containing structures, while the qualitative outputs include known systems and expose limitations in magnetic-state realism and alloy handling.

  • Targeting low HHI alongside high magnetic density steers MatterGen away from Co, which is associated with poor HHI scores.
  • Fex Mn1–x O rocksalt alloys showed high magnetization density only in a hypothetical ferromagnetic state, not their actual antiferromagnetic ground state.
  • Defected FeO with vacancies and body-centered-cubic Fe8Au were among the example outputs and are well-known experimentally.
  • More reasonable candidates could result from penalizing expensive elements and favoring metallic systems more likely to be ferromagnetic.
  • Better treatment of alloy systems may be needed to improve high magnetic density generation and ensure generated structures are truly novel.
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