An invertible crystallographic representation for general inverse design of inorganic crystals with targeted properties

Motivation

• Existing generative inverse-design demonstrations were often limited to fixed element subsets or crystal structures because general invertible representations are difficult to create.General inverse design requires predicting both a material’s chemistry and structure from user-specified properties.Source: Introduction · pp. 2–3

Framework

• FTCP varies both composition and structure, enabling general and property-driven inverse design of inorganic crystals.Its real-space features are CIF-like and its reciprocal-space features provide an additional crystal-property featurizer inspired by structure-factor calculations.Source: Introduction · p. 3
• The framework combines real-space CIF-like features, reciprocal-space Fourier-transformed features, and an invertible crystallographic representation.The real-space component guarantees conversion back to crystal information, while reciprocal-space features capture periodicity and related crystal characteristics.Source: Introduction · p. 3
• A VAE with a target-learning branch organizes a continuous latent space so sampled latent vectors can decode into crystals with user-specified properties.The encoder maps training crystals into a probabilistic latent space, and the decoder maps sampled latent points back to crystal representations.Source: Introduction · p. 4

Design cases

• Three design cases target formation energy, bandgap with formation-energy constraints, and thermoelectric power factor under bandgap and formation-energy constraints.The cases range from single-property targets to simultaneous constraints, including Ef from −0.3 to −0.7 eV/atom and Eg = 1.5 eV.Source: Introduction · p. 4

Results

• FTCP achieves success rates from 14.3% to 38.9% in case 1, 36.8% in case 2, and 7.1% in case 3, with first-principles validation.Case 3 produces two unique crystals with peak power factors comparable to GeTe, while overall random-baseline improvements range from 38.8% to 560%.Source: Introduction · p. 4

Framework and workflow

• The VAE maps crystals into a 256-dimensional property-structured latent space for sampling candidates associated with user-specified properties.The encoder compresses FTCP representations, while the decoder reconstructs crystals and the target-learning branch organizes latent points by property gradients.Source: Inverse Design Framework, FTCP · p. 8
• Candidates undergo sampling, postprocessing, DFT structural relaxation, filtering of invalid or repeated structures, and first-principles property verification.Success rate counts candidates exiting verification relative to those entering design, whereas validity rate measures successful structural relaxation.Source: Workflow · pp. 9–10

Error sources and implications

• Increasing the number of design targets lowers random success, validity, and success rates because more property-mapping losses must be optimized.The authors report that FTCP’s improvement over random can nevertheless grow as inverse-design problems become more complex.Source: Workflow · p. 12

Design outcomes

• Fourteen of 19 crystals (73.7%) satisfied the formation-energy criterion in case 2, while seven (36.8%) also met Eg = 1.5 ± 0.3 eV.Only 5.5% of the case-2 dataset satisfied both target regions, indicating that random sampling had a low likelihood of meeting the combined criteria.Source: Applying FTCP: Three design cases… · p. 14
• Two of 28 designed crystals (7.1%) achieved power factors comparable with state-of-the-art cubic GeTe while satisfying bandgap and negative-formation-energy criteria.The two candidates were compositionally unique, with structural dissimilarity values of 0.80 and 0.53.Source: Applying FTCP: Three design cases… · pp. 16–17

• FTCP performs inverse design across varying chemistries and crystal structures for targeted Ef, Eg, and thermoelectric power factor.Source: Conclusion and future work · p. 20
• 38.8% to 560% improvement over random was achieved for finding materials with user-specified target properties.Source: Conclusion and future work · p. 21
• Designed crystals were structurally relaxed with DFT and their properties were confirmed using DFT and BoltzTraP calculations.Source: Conclusion and future work · p. 21
• FTCP can simultaneously consider an ICSD-entry synthesizability metric alongside user-specified target properties.Source: Conclusion and future work · p. 21

• The power-factor data came from a reference, while the remaining dataset was queried from Materials Project in November 2019.Source: Resource availability · p. 21
• The FTCP representation concatenates CIF-like real-space features with reciprocal-space features derived from elemental-property projections.Source: Real-Space Features · p. 21, Reciprocal-Space Features · p. 22
• The VAE encodes FTCP into a 256-dimensional probabilistic latent space, while target learning organizes latent points by continuous property variation.Source: Generative model · pp. 22–23
• The model uses reconstruction, KL-divergence, and property-mapping losses, with a separate power-factor loss for semi-supervised learning in case 3.Source: Generative model · p. 23

• Table S1 compares invertible crystallographic representations used in generative-model-based inverse design algorithms.Source: Table S1 · p. 28
• The comparison identifies Bayesian optimization and generative-model approaches, including GAN- and VAE-based algorithms.Source: S1. Prior Art on Invertible… · pp. 28–29
• Representation-level invariances exclude invariances achieved through algorithmic implementations such as rotational augmentation with 3D convolutional networks.Source: S1. Prior Art on Invertible… · p. 29
• The table notes that one referenced method appeared later than the authors’ arXiv preprint.Source: S1. Prior Art on Invertible… · p. 29
• Full material invariances include translational, rotational, permutational, and supercell invariances.Source: S1. Prior Art on Invertible… · p. 29

The FTCP Framework

• The 256-dimensional latent space shows densely packed encoded training data and continuous property gradients.Source: S2. The FTCP Framework · p. 29

S2.3 Ablation Study of the FTCP Representation

• FTCP matches real-space-only performance in property mapping while improving reconstruction of constituent elements, lattice parameters, and site fractional coordinates.Source: S2.3 Ablation Study of the FTCP… · p. 31
• Reciprocal-space features improve reconstruction overall by adding correlations, despite diverting some capacity from real-space reconstruction.Source: S2.3 Ablation Study of the FTCP… · p. 31

S2.5 Comparison of Sampling Strategies

• Three sampling strategies are explored: local perturbation, spherical linear interpolation, and global perturbation.Source: S2.5 Comparison of Sampling Strategies · p. 34
• Local perturbation mainly produces elemental substitutions, whereas spherical interpolation and global perturbation induce more structural change but higher rediscovery errors.Source: S2.5 Comparison of Sampling Strategies · p. 35

S2.6 Sources of Error for Validity Rate, and the Need of DFT Structural Relaxation

• FTCP-designed crystals incur decompression and interpolation errors because the latent representation is reduced-dimensional and sampling shifts atoms and positions.Source: S2.6 Sources of Error for Validity… · p. 36

• Only structurally relaxed crystals are listed in Table S6, with compositions provided and CIFs available from the FTCP repository.Source: Table S6 · p. 36
• Spin-polarized PBE and PBE+U calculations used PAW pseudopotentials in VASP, with chromium oxide calculations adopting U = 3.7.Source: S3.2 First-Principles Calculations in… · p. 37
• GGA+U energies for Cr-O compounds were corrected using the chromium correction term and an O2 correction of -1.4046 eV/atom.Source: S3.2 First-Principles Calculations in… · pp. 37–38
• Chromium correction energies were derived from three convex-hull Cr-O reference structures by fitting formation energy against chromium fraction.The references were Cr2O3, Cr5O12, and Cr6O11, and the adjusted formation energies are listed in Table S7.Source: S3.2 First-Principles Calculations in… · p. 38
• 520 eV plane-wave cutoffs and specified Brillouin-zone spacings were used for structure relaxations and total-energy calculations, with BoltzTraP calculating transport coefficients.Source: S3.3 First-Principles Calculations in… · p. 39

• 14 of 18 FTCP-designed crystals were valid and unique relative to the Materials Project database; four atom-overlapping CeFeGe4 variants were excluded.Source: Figure S4 · p. 39
• Eight crystal structures and more than 30 elements occur among the 14 valid crystals, demonstrating access to varied structures and chemistries.Source: S4. Design Cases · p. 39
• Figure S5 presents DFT-calculated formation-energy boxplots for the three targets, and Figure S6 marks target-satisfying crystals with red dots.Source: Figure S5 · p. 40, Figure S6 · p. 40
• DFT formation energies were evaluated for targets of -0.3, -0.6, and -0.7 eV/atom, with success rates reported in Table 1.Source: S4.2 DFT-Calculated Ef Values of… · p. 40
• Structural dissimilarity is the vector distance between crystals based on local coordination information, with zero indicating identical structures and values above 1 indicating huge dissimilarity.Source: S4.2 DFT-Calculated Ef Values of… · p. 40
• Case 2 had median dissimilarity 0.57 with three values above 0.75, while Case 3 had median 0.67 with five values above 0.75.Source: S4.2 DFT-Calculated Ef Values of… · p. 41

• Table S8 reports mean absolute error and percentage performance drop using five-fold cross-validation on an updated Materials Project dataset.The updated dataset was accessed on 14 September 2021, after the design-case database access dates.Source: Table S8 · p. 41
• FTCP has no built-in invariances, so Table S8 evaluates performance degradation from translation, rotation, permutation, and different supercells.Source: S5. Invariance Study of the FTCP… · p. 41