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An invertible crystallographic representation for general inverse design of inorganic crystals with targeted properties
Zekun Ren, Siyu Isaac Parker Tian, Juhwan Noh, Felipe Oviedo, Guangzong Xing, Jiali Li, Qiaohao Liang, Ruiming Zhu, Armin G. Aberle, Shijing Sun, Xiaonan Wang, Yi Liu, Qianxiao Li, Senthilnath Jayavelu, Kedar Hippalgaonkar, Yousung Jung, Tonio Buonassisi
TL;DR
Existing materials-design approaches are limited in targeting user-specified property combinations across arbitrary compositions and crystal structures. FTCP addresses this with an invertible real- and reciprocal-space representation and a property-structured VAE latent space, generating crystals with targeted properties and achieving 7.1%–38.9% first-principles success rates, while experimental synthesizability remains challenging.
Problem
Materials-property databases and generative models remain limited for discovering materials with user-specified property combinations across general chemical compositions and crystal structures.
Method
FTCP combines an invertible crystallographic representation with CIF-like real-space and reciprocal-space features and a VAE-based property-structured latent space for sampling and decoding crystals.
Results
FTCP generated unique crystals with targeted properties across three design cases, achieving success rates from 7.1% to 38.9% and improvements over random ranging from 38.8% to 560%.
Takeaways & Limitations
FTCP enables composition- and structure-varying property-driven inverse design and can incorporate an ICSD-based synthesizability metric alongside target properties.
Takeaways & Limitations
Imperfect reconstruction and latent-space interpolation introduce validity-rate errors, while negative formation energy and low energy above hull do not guarantee experimental synthesizability.
Abstract
from arXiv · showhide
Realizing general inverse design could greatly accelerate the discovery of new materials with user-defined properties. However, state-of-the-art generative models tend to be limited to a specific composition or crystal structure. Herein, we present a framework capable of general inverse design (not limited to a given set of elements or crystal structures), featuring a generalized invertible representation that encodes crystals in both real and reciprocal space, and a property-structured latent space from a variational autoencoder (VAE). In three design cases, the framework generates 142 new crystals with user-defined formation energies, bandgap, thermoelectric (TE) power factor, and combinations thereof. These generated crystals, absent in the training database, are validated by first-principles calculations. The success rates (number of first-principles-validated target-satisfying crystals/number of designed crystals) ranges between 7.1% and 38.9%. These results represent a significant step toward property-driven general inverse design using generative models, although practical challenges remain when coupled with experimental synthesis.
Introduction
FTCP addresses the lack of a general generative inverse-design framework for inorganic crystals by varying both composition and structure. It combines an invertible real- and reciprocal-space representation with a property-structured VAE and demonstrates targeted designs across multiple material 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.
- 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.
- Framework: 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.
- Framework: 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.
- 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.
- 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%.
Results and Discussion
FTCP combines an invertible crystal representation with a property-structured VAE to generate composition- and structure-varying candidates, then relaxes and validates them using first-principles calculations. Across design cases, it achieved target-satisfying crystals while exposing modeling, synthesizability, and experimental-validation challenges.
- 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.
- Framework and workflow: 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.
- 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.
- 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.
- Design outcomes: 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.
Conclusion and future work
FTCP provides general, property-driven inverse design across varying crystal chemistries and structures, producing targeted materials validated with first-principles calculations. The framework also addresses synthesizability considerations and reports substantial improvement over random selection.
- FTCP performs inverse design across varying chemistries and crystal structures for targeted Ef, Eg, and thermoelectric power factor.
- 38.8% to 560% improvement over random was achieved for finding materials with user-specified target properties.
- Designed crystals were structurally relaxed with DFT and their properties were confirmed using DFT and BoltzTraP calculations.
- FTCP can simultaneously consider an ICSD-entry synthesizability metric alongside user-specified target properties.
Experimental procedures
The experimental procedures construct FTCP from CIF-like real-space features and reciprocal-space elemental-property projections, then use a VAE and target-learning branch to sample candidate crystals. Candidates undergo postprocessing and validation, while datasets and implementation details are documented.
- The power-factor data came from a reference, while the remaining dataset was queried from Materials Project in November 2019.
- The FTCP representation concatenates CIF-like real-space features with reciprocal-space features derived from elemental-property projections.
- The VAE encodes FTCP into a 256-dimensional probabilistic latent space, while target learning organizes latent points by continuous property variation.
- The model uses reconstruction, KL-divergence, and property-mapping losses, with a separate power-factor loss for semi-supervised learning in case 3.
S1. Prior Art on Invertible Crystallographic Representations
Prior invertible crystallographic representations used in generative inverse design are characterized by their encoded features, invariances, and algorithmic categories. The cited comparison distinguishes representations associated with GANs, VAEs, and other optimization approaches.
- Table S1 compares invertible crystallographic representations used in generative-model-based inverse design algorithms.
- The comparison identifies Bayesian optimization and generative-model approaches, including GAN- and VAE-based algorithms.
- Representation-level invariances exclude invariances achieved through algorithmic implementations such as rotational augmentation with 3D convolutional networks.
- The table notes that one referenced method appeared later than the authors’ arXiv preprint.
- Full material invariances include translational, rotational, permutational, and supercell invariances.
S2. The FTCP Framework
The FTCP framework uses a property-structured latent space with multiple sampling strategies and evaluates reciprocal-space formulations and representation ablations. Results show improved reconstruction from reciprocal-space features, while sampling exposes an exploration–exploitation tradeoff and reconstruction introduces validity errors.
- The FTCP Framework: The 256-dimensional latent space shows densely packed encoded training data and continuous property gradients.
- 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.
- S2.3 Ablation Study of the FTCP Representation: Reciprocal-space features improve reconstruction overall by adding correlations, despite diverting some capacity from real-space reconstruction.
- S2.5 Comparison of Sampling Strategies: Three sampling strategies are explored: local perturbation, spherical linear interpolation, and global perturbation.
- S2.5 Comparison of Sampling Strategies: Local perturbation mainly produces elemental substitutions, whereas spherical interpolation and global perturbation induce more structural change but higher rediscovery errors.
- 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.
S3. FTCP-Designed Crystals, and Their First-Principles Calculations
This section lists valid FTCP-designed compositions and describes the first-principles methods and chromium correction procedure used to evaluate them.
- Only structurally relaxed crystals are listed in Table S6, with compositions provided and CIFs available from the FTCP repository.
- Spin-polarized PBE and PBE+U calculations used PAW pseudopotentials in VASP, with chromium oxide calculations adopting U = 3.7.
- GGA+U energies for Cr-O compounds were corrected using the chromium correction term and an O2 correction of -1.4046 eV/atom.
- 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.
- 520 eV plane-wave cutoffs and specified Brillouin-zone spacings were used for structure relaxations and total-energy calculations, with BoltzTraP calculating transport coefficients.
S4. Design Cases
The design cases produced valid crystals spanning diverse structures and chemistries, while DFT formation-energy evaluations and dissimilarity analyses assessed target satisfaction and structural uniqueness.
- 14 of 18 FTCP-designed crystals were valid and unique relative to the Materials Project database; four atom-overlapping CeFeGe4 variants were excluded.
- Eight crystal structures and more than 30 elements occur among the 14 valid crystals, demonstrating access to varied structures and chemistries.
- Figure S5 presents DFT-calculated formation-energy boxplots for the three targets, and Figure S6 marks target-satisfying crystals with red dots.
- DFT formation energies were evaluated for targets of -0.3, -0.6, and -0.7 eV/atom, with success rates reported in Table 1.
- 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.
- 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.
S5. Invariance Study of the FTCP Representation
The invariance study quantifies how translation, rotation, site-order permutation, and supercell changes affect FTCP property prediction and mapping performance.
- 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.
- FTCP has no built-in invariances, so Table S8 evaluates performance degradation from translation, rotation, permutation, and different supercells.