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Representations of Materials for Machine Learning

James Damewood, Jessica Karaguesian, Jaclyn R. Lunger, Aik Rui Tan, Mingrou Xie, Jiayu Peng, Rafael Gómez-Bombarelli

arXiv:2301.08813v1cond-mat.mtrl-sci

TL;DR

Materials ML depends on representations that turn data into inputs capturing relevant chemistry and physics, yet datasets and prediction tasks vary. This review surveys context-dependent representation strategies and concludes that the relative benefit of structural information varies by property, while important scope limits remain.

  • Problem

    Materials data must be represented numerically for ML, and constructing features that capture relevant physics and chemistry under varying data constraints remains a central challenge.

    Source: §1 · p. 3

  • Method

    The review surveys context-dependent material representations, learned representations, transfer learning, and open research questions.

    Source: §1 · p. 3, §6 · p. 16, §8 · p. 19

  • Results

    Structural approaches generally outperform compositional descriptors, but the improvement depends on the property, with potentially smaller gains for global properties such as band gap.

    Source: §8.3 · p. 20

  • Takeaways & Limitations

    Representation strategies and future research should account for the target property and data constraints, including unresolved challenges beyond idealized crystals.

    Source: §1 · p. 3, §8.4 · p. 20

    Recent generative approaches are promising for small unit cells, but microstructure, dimensionality, and surfaces remain important extensions.

    Source: §8.4 · p. 20

Abstract

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High-throughput data generation methods and machine learning (ML) algorithms have given rise to a new era of computational materials science by learning relationships among composition, structure, and properties and by exploiting such relations for design. However, to build these connections, materials data must be translated into a numerical form, called a representation, that can be processed by a machine learning model. Datasets in materials science vary in format (ranging from images to spectra), size, and fidelity. Predictive models vary in scope and property of interests. Here, we review context-dependent strategies for constructing representations that enable the use of materials as inputs or outputs of machine learning models. Furthermore, we discuss how modern ML techniques can learn representations from data and transfer chemical and physical information between tasks. Finally, we outline high-impact questions that have not been fully resolved and thus, require further investigation.

1 INTRODUCTION

Materials ML can help identify promising materials, but its predictions depend on numerical representations that capture relevant chemistry and physics. This review surveys representation strategies in light of data constraints and open problems.

1 INTRODUCTION

  • Robotized laboratories can speed synthesis and characterization, while virtual screening reduces screening costs at the expense of fidelity.The number of possible design parameters can still make exhaustive analysis impractical.Source: §1 · p. 2
  • Predictive ML can approach or exceed first-principles accuracy at much lower cost, but only when its inputs make the desired physics and chemistry emergent.Model choice matters, but the numerical representation is also a key dependency.Source: §1 · p. 2
  • Useful representations should preserve relevant similarities, apply across the domain of interest, and cost less to calculate than the target property.The review frames these as practical criteria for representation design.Source: §1 · p. 2
  • Representation features must be obtainable across training and deployment domains and cost less than determining the property; otherwise, ML screening may offer no practical benefit.Using DFT-derived inputs to predict a DFT property at the same theory level is one example of this cost problem.Source: §1 · pp. 2–3
  • Structural diversity across length scales, defects, microstructure, and interfaces complicates representation design, while unavailable structures and scarce data can require compositional descriptors or transfer learning.Related-task representations can help address scarcity; compositional descriptors are more readily available for unexplored materials.Source: §1 · p. 3
  • The review evaluates representation strategies against data format, quantity, and quality, emphasizing selected methods and open problems rather than exhaustive literature coverage.It aims to introduce the field’s current state and future research directions.Source: §1 · p. 3

2 STRUCTURAL FEATURES FOR ATOMISTIC GEOMETRIES

Atomistic structure representations encode atom types, positions, and cell parameters as machine-readable features. This section surveys local, global, and topological descriptors for representing structures in machine learning.

2 STRUCTURAL FEATURES FOR ATOMISTIC GEOMETRIES

  • Because scalar properties are unchanged by translations, rotations, or atom-index permutations, structural representations should share those invariances.The input structure comprises atom types, positions, and primitive-cell parameters.Source: §2 · p. 3

2.1 Local Descriptors

  • Local descriptors encode atom neighborhoods through distance-and-angle symmetry functions or SOAP density overlaps, enabling local environment comparisons.With localized symmetry functions, neural networks can predict local property contributions and approximate global properties as their sum.Source: §2.1 · pp. 3–5
  • Voronoi descriptors combine cell geometry and neighbor properties with composition; this representation outperformed partial radial distribution functions for ICSD formation-enthalpy predictions.Features include effective coordination, nearby-cell lengths and volumes, and neighbor atomic properties weighted by shared-face area.Source: §2.1 · p. 5

2.2 Global Descriptors

  • Global descriptors include Coulomb matrices for electrostatic interactions and MBTR tensors for k-body interactions and chemical similarity.MBTR is translationally, rotationally, and permutation invariant, but its accuracy may not extend to larger systems.Source: §2.2 · pp. 6–7
  • Cluster expansions predict configuration properties from symmetrized cluster correlations, but assume a parent lattice and standard implementations are restricted to binary and ternary systems.The framework can be extended to continuous environments through the Atomic Cluster Expansion.Source: §2.2 · p. 7

2.3 Topological Descriptors

  • Persistent homology encodes births, deaths, and persistences of holes across distance-scale filtrations, but is size-dependent and can cost O(N3) per filtration.The cutoff radius or supercell size must encompass significant features; scaling is poor when geometric features exceed unit-cell lengths.Source: §2.3 · pp. 7–8
  • Persistent-homology descriptors outperform traditional structural descriptors for CO2 adsorption under varying conditions in MOFs and methane storage capacity in zeolites.Persistent 2D deaths correspond to the largest sphere that can fit inside a pore.Source: §2.3 · p. 8

3 LEARNING ON PERIODIC CRYSTAL GRAPHS

Periodic crystal graphs turn atomic structures into neighborhood-based features for property prediction, but local graph models can miss periodicity-dependent and long-range behavior. Equivariant features and other extensions address some of these limits, while structural models still depend on accurate atomic positions.

3 LEARNING ON PERIODIC CRYSTAL GRAPHS

  • Crystal graphs encode atoms as nodes and nearby-atom connections as edges, then convolve neighborhood features and pool them into a material-level vector for prediction.Node features may encode atomic identity or other elemental properties, while edge features typically encode interatomic distance.Source: §3 · p. 8, Figure 3 · p. 9
  • CGCNN and MEGNet serve as benchmark models across bulk, surface, disordered, and 2D materials, while ALIGNN adds triplet features and outperforms CGCNN on several regression tasks.Reported tasks include formation energy, bandgap, and shear modulus; other variants use Voronoi neighborhoods or nearest-neighbor geometry and electron configuration.Source: §3 · pp. 8–9
  • Local-neighborhood graph convolutions may miss long-range interactions or extensive properties, and performance can vary substantially by target, especially for periodicity-dependent properties such as lattice parameters [63].Concatenating pooled graph features with human-tuned descriptors reduced errors by 90% for related phonon internal-energy and heat-capacity predictions [63].Source: §3 · p. 9
  • Extensions for long-range information include Gaussian and plane-wave convolution masks, global pooling with added context, reciprocal-space features, and Gaussian-mixture representations of phonon spectra [65, 66, 67, 68].Source: §3 · pp. 9–10
  • Equivariant models can reach state-of-the-art benchmark accuracy and descriptor-comparable performance with about 100 datapoints; many-body interactions can further improve accuracy.They use higher-order tensor features and update functions equivariant to translations, rotations, and reflections.Source: §3 · p. 10
  • Graph-based approaches require atomic positions, which may be costly to obtain; graphs from pristine structures omit relaxed distortions and can substantially reduce accuracy.Training on perturbed structures labeled with the relaxed structure's property improved accuracy on an unrelaxed test set [76]; alternative options include energy-based prototype search or surrogate relaxation.Source: Figure 3 · p. 9, §3 · p. 10

4 CONSTRUCTING REPRESENTATIONS FROM STOICHIOMETRY

Compositional representations use elemental fractions and properties when structural information is unavailable, offering accessible routes to screening and experimental-data modeling. Recent composition-only neural models improve performance with smaller datasets, but they cannot distinguish polymorphs of the same composition and may be less suitable for metastable systems.

4 CONSTRUCTING REPRESENTATIONS FROM STOICHIOMETRY

  • When phase, crystal system, or atomic positions are unavailable, composition-only representations can use readily available elemental properties and standard ML libraries for screening.Their predictions suggest elemental concentrations rather than specific geometries, making them less dependent on assumed synthesis outcomes.Source: §4 · p. 11
  • Compositional features include elemental properties such as radii, electronegativity, and valence states; expanded feature sets have supported formation-energy classification and compositionally driven predictions for fixed prototypes.Source: §4 · p. 11
  • Compressive sensing can select interpretable descriptors from feature-combination sets exceeding 10^9 possibilities, including descriptors for structure formation and an improved perovskite tolerance factor.Source: §4 · p. 11
  • Composition-based embeddings can be learned from Materials Project compositions or scientific literature, and have been used to identify promising compositions by similarity to known materials.Source: §4 · p. 11
  • With more than 3,000 training points, ElemNet, using only fractional stoichiometry, outperformed a MagPie-based model on formation-enthalpy prediction.Source: §4 · p. 12
  • ROOST outperforms MagPie with only hundreds of training examples, while CrabNet achieves similar performance; both reduce ElemNet's approximate O(10^3)-datapoint threshold.CrabNet's log-scale fractional embedding allows low-concentration dopants to influence predictions.Source: §4 · p. 12
  • Compositional models can learn from experimental data and guide materials design, including validated hardness predictions, Heusler-compound discovery, and structure-prototype prediction above 96% accuracy.A model predicting prototype probabilities was used to engineer TiFeP with multiple competing phases.Source: §4 · p. 12
  • A composition model gives one prediction per stoichiometry regardless of polymorph, may have limited metastable-system extrapolation, and is generally outperformed by structural models.It is most applicable when atomistic resolution is unavailable.Source: §4 · p. 12

5 DEFECTS, SURFACES, AND GRAIN BOUNDARIES

Materials properties often depend on environments beyond a small bulk unit cell, so useful representations must reflect the relevant length scales. The review describes approaches for defects, surfaces, and grain boundaries, alongside limits in their scope and generality.

Scope

  • Practical materials design requires representations at relevant length scales because surfaces, defects, and grain boundaries can affect properties beyond the bulk unit cell.The (100) RuO2 facet has an order of magnitude higher OER current than the thermodynamically stable (110) facet.Source: §5 · p. 13

Defects

  • Defect models range from bulk-derived predictors to representations comparing defect and pristine structures, with broader chemistry and defect coverage still a goal.Bulk band-structure descriptors predicted vacancy formation energy using pristine-bulk DFT alone, while a relative-difference representation required no DFT at prediction time.Source: §5 · pp. 13–14, Figure 4 · p. 13

Surfaces

  • Surface binding-energy models combine pristine-bulk electronic and geometric descriptors, but descriptor generality across chemistries remains limited.With more than 10,000 examples, modified graph-convolutional models predicted binding energies across datasets with up to 37 elements; errors remain high for some adsorbates and non-metallic surface elements.Source: Figure 4 · p. 13, §5 · p. 14

Grain boundaries

  • Grain-boundary representations have progressed from geometry and energetics approaches benchmarked on up to three elements toward local-structure methods.Earlier approaches often omitted constituent-atom features; more recent work approximated excess energy across compositions using five orientation and bond-length variables.Source: §5 · pp. 14–15
  • Representations of grain-boundary environments support predictions of energy, temperature-dependent mobility, and shear coupling, while helping interpret structural effects.SOAP-based local-environment classes were predictive of these properties; related methods address thermal conductivity and atomic mobility.Source: §5 · p. 15

6 TRANSFERABLE INFORMATION BETWEEN REPRESENTATIONS

Materials models often face sparse data and limited accuracy across compositions and structures. The review describes transfer learning and related representation strategies for sharing information across tasks, while emphasizing that their effectiveness depends on data availability and task relationships.

Challenges

  • Generalization across materials spaces remains challenging, and collecting more data may be impractical when measurements or calculations are costly or limited.Some deep-learning approaches discussed in the review require at least 10^3 training points; predictive modeling may instead use human-engineered descriptors.Source: §6 · p. 15

Transfer learning

  • Transfer learning typically involves pretraining on a related task, freezing network layers, then fine-tuning remaining layers or using extracted features.The procedure has been used with composition-only, crystal-graph, and equivariant-convolution models, and across experimental and DFT-calculated targets.Source: §6 · p. 16

Dataset size

  • Transfer learning can help models perform when task B has few examples, especially when task A has orders of magnitude more data.A larger task A dataset can yield a more robust learned representation, while sparse task B data can make that prior information useful for interpolation.Source: §6 · p. 16

Task relationship

  • Transfer performance depends on the physical relationship between tasks; empirically testing candidate source tasks can identify better pairings.Transfer from Open Catalyst data performed significantly better for absorption of new species than for less-related small molecules.Source: §6 · p. 16

Layer depth

  • The best feature-extraction depth depends on task relatedness, and combining activations from multiple layers has also been proposed.More distant tasks may benefit from retraining more layers, while early layers can provide more general representations for cross-property transfer.Source: §6 · p. 16

Alternative representations

  • When tasks differ substantially, predicted properties or descriptors can serve as inputs to expand screening beyond cases with directly available structures or labels.Examples include using predicted structural descriptors for Li conductivity and GAN-generated density-of-states features for absorption spectra.Source: §6 · pp. 16–17

7 GENERATIVE MODELS FOR INVERSE DESIGN

Crystal generation must handle diverse atomic species, periodic structures, and multiple equivalent descriptions of the same material. The review contrasts voxel, symmetry-aware, and topology-constrained representations, each with different scope and trade-offs.

Challenges

  • Crystal generative models face large search spaces because structures require species, atomic positions, and lattice parameters, with equivalent structures under rotations, translations, and unit-cell scaling.Some relevant catalytic materials have unit cells with more than 100 atoms.Source: §7 · p. 17

Voxel representations

  • Voxel models represent unit cells as discretized volumes, but early approaches were not general across crystal space and some were unstable or confined to narrow material classes.iMatgen rediscovered stable structures but was limited to vanadium oxides; other voxel approaches generated materials within cubic systems or a fixed perovskite prototype.Source: §7 · pp. 17–18, Figure 5 · p. 17

Continuous coordinates

  • Continuous-coordinate representations proposed lower-energy V-O structures than iMatgen and generated materials with negative formation energy, but retained symmetry and sampling limitations.These approaches lacked translational, rotational, and scale invariance and were restricted to particular material classes.Source: §7 · p. 18

Symmetry-aware models

  • Symmetry-aware periodic graph models use invariant or equivariant representations and were reported as more flexible than voxel models across crystal systems.Crystal Diffusion VAEs also outperformed point-cloud representations at identifying crystals with low formation energy.Source: Figure 5 · p. 17, §7 · p. 18

Large-cell materials

  • For large unit cells, physical constraints can be built into generation through post-processing or representations that encode feasible topologies.RFcode representations correspond to structurally possible MOFs and supported interpolation and property optimization; the review frames stable generation versus discovery of new motifs as a trade-off.Source: Figure 5 · p. 17, §7 · p. 18

8 DISCUSSION

The review surveys representation strategies for materials ML and argues that representation choice is pivotal to materials-design research. It identifies trade-offs and open questions across structural, compositional, and generative approaches.

8 DISCUSSION

  • The review also covers learned atomic-scale representations, non-ideal crystals, and limited-data strategies using compositional descriptors or transfer learning.It frames representation choice and novel approaches as pivotal to materials-design research.Source: §8 · p. 19

8.1 Trade-offs of Local and Global Structural Descriptors

  • Local descriptors such as SOAP predict well for short-range properties and scale to large systems, but may miss long-range physics; global descriptors trade away generality.Combining local and long-range features may help address these trade-offs.Source: §8.1 · p. 19

8.2 Prediction from Unrelaxed Crystal Prototypes

  • Requiring relaxed structures restricts screening to known optimized geometries; ML force-field relaxations [162] could broaden availability, with relaxed inputs improving energy predictions.The review identifies training-set scale, material-class generalizability, and tasks beyond stability as open questions.Source: §8.2 · p. 19

8.3 Applicability of Compositional Descriptors

  • Composition-only descriptors are readily available but underperform the best structural approaches, and structural gains vary by property.Structural information may help less for phonons and band gaps; sparse structural data motivates transfer learning or combining composition and structure models.Source: §8.3 · p. 20

8.4 Extensions of Generative Models

  • Symmetry considerations and diffusion-based architectures improved generative models over voxel approaches, but broader generation should address microstructure, dimensionality, and surfaces.Stoichiometry-only sampling limits full computational characterization but may identify promising regions for experimental analysis.Source: §8.4 · p. 20

DISCLOSURE STATEMENT

The authors report no affiliations, memberships, funding, or financial holdings that might affect the review’s objectivity.

DISCLOSURE STATEMENT

  • The authors report no affiliations, memberships, funding, or financial holdings that might affect the review’s objectivity.Source: DISCLOSURE STATEMENT · p. 20
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