Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design

• Crystal properties emerge from atomic species, positions, and lattice geometry, while symmetry can determine whether functional responses are allowed.The introduction identifies inverse design as finding structures that meet a physical objective while remaining chemically and crystallographically plausible.Source: §1 · p. 1
• Latent and free-coordinate steering can produce apparent property success while losing required crystallographic symmetry, often collapsing to P1 when symmetry is unenforced.Source: §1 · pp. 1–2
• SPARC couples property objectives with crystallographic constraints during generation to search for structures optimized for target functionality and consistent with meaningful symmetry.The framework is presented as a way to target functional properties that cannot be specified by scalar objectives alone.Source: §1 · p. 2
• The evaluation pairs symmetry-sensitive uniaxial dielectric anisotropy with scalar SLME, which has no prescribed symmetry class and can reveal favorable crystallographic motifs.The dielectric task tests access to and preservation of relevant symmetry classes; the SLME task tests discovery of motifs or space-group distributions for a device-level objective.Source: §1 · p. 2

2.1 SPARC pipeline

• Each SPARC iteration samples a space group and asymmetric unit, denoises and reconstructs the crystal, then relaxes, filters, scores, and assigns a terminal reward.Reward-weighted policy updates tune the diffusion model, while aggregated rewards separately update the space-group proposal.Source: §2.1 · p. 2, Figure 1 · p. 3

2.2 Inverse design of uniaxial dielectric responses

• The mean uniaxial-anisotropy reward rises from 0.17 to 0.56 over 120 RL steps, while five realized space groups account for approximately 44% of generated structures.Reward feedback enriches trigonal, tetragonal, and hexagonal groups that enforce the target tensor form, though these classes are not prescribed in advance.Source: §2.2 · pp. 3–4, Figure 2 · p. 4
• With relaxation enabled, 50.1% of SymmCD structures versus 1.4% of DiffCSP structures have three-, four-, or sixfold axes, yielding a 35× higher symmetry yield.Comparable reward curves do not establish symmetry enforcement; DiffCSP relaxation rarely recovers the higher rotational axes retained by SymmCD.Source: §2.2 · p. 5, Figure 3 · p. 6
• After DFT relaxation, the three high-reward candidates retain the symmetry-constrained uniaxial tensor form, with blue phosphorus showing ε∥/ε⊥≈3.46.Orthorhombic TeAs instead has three distinct diagonal components and serves as a low-reward biaxial comparison.Source: §2.2 · p. 6, Figure 4 · p. 7

2.3 Inverse design of high-efficiency photovoltaic absorbers

• The share of candidates in the photovoltaic target region (Eg ≈0.8–1.9 eV, η ≥25%) rises from 11% (149 structures) to 32% (279 structures) after steering.The steered population retains multiple crystallographic routes to high efficiency.Source: §2.3 · p. 8, Figure 5 · p. 9
• Photovoltaic steering reduces represented elements from 53 to 46 while enriching sulfur- and antimony-containing structures rather than collapsing sampling to one chemical family.The space-group distribution also shifts while retaining multiple crystallographic families, with Fm¯3m remaining most frequent.Source: §2.3 · p. 8, Figure 5 · p. 9

3 Discussion

• Symmetry and target properties must be optimized together: uniaxial dielectric anisotropy calls for specific rotational symmetries, whereas SLME permits multiple favorable crystal families.For dielectric anisotropy, the required in-plane equivalence is associated with threefold, fourfold, or sixfold rotational symmetry; SLME depends on the band gap and full optical spectrum but does not prescribe a unique crystal class.Source: §3 · p. 9
• Similar dielectric rewards from SymmCD and DiffCSP can conceal different crystallographic distributions; low-symmetry tensor equality may be accidental and lost under structural distortions.Local relaxation cannot reliably recover missing global symmetry.Source: §3 · pp. 9–10
• When a target’s symmetry class is unknown, SPARC can keep symmetry channels available and use reward-guided search to identify favorable crystallographic families.For known symmetry requirements, the space-group proposal can instead be restricted to classes that permit the desired response.Source: §3 · p. 10
• A gapped trigonal P31 blue-phosphorus candidate with a 1.75 eV band gap was recovered beyond the MP-20 training set without a phosphorus-specific target or template.The allotrope had been proposed independently, and is not the most stable phosphorus allotrope.Source: §3 · p. 10
• The dielectric objective concentrates generation primarily in symmetry space, while SLME concentrates chemical-space sampling and retains broad crystal-symmetry diversity.The discussion distinguishes this selective concentration from uncontrolled mode collapse.Source: §3 · p. 10
• The symmetry-aware design principle extends to magnetocrystalline anisotropy, elastic and piezoelectric response, thermal expansion, and other symmetry-constrained properties.SPARC can couple to surrogates predicting tensorial properties, transport quantities, or electronic Hamiltonians.Source: §3 · p. 10
• Reward accuracy limits the broader potential: the SLME surrogate uses PBE-level data, whose semiconductor band gaps are expected to be systematically underestimated.The authors propose higher-level calculations and multi-fidelity learning as possible ways to improve reward predictions.Source: §3 · p. 10

4.1 Symmetry-constrained crystal representation

• A crystal is generated from representative sites in an asymmetric unit, with atom types, coordinates, site symmetries, and symmetry-constrained lattice coefficients conditioned on space group and site count.Applying the space-group operations reconstructs the full crystal.Source: §4.1 · p. 11

4.2 Symmetry-aware generation

• SymmCD jointly denoises asymmetric-unit species, coordinates, site symmetries, and lattice coefficients, fixing symmetry-constrained lattice entries while denoising only the free ones.The denoiser uses coordinate score matching, masked lattice noise prediction, and categorical losses for species and site symmetries.Source: §4.2 · pp. 11–12

4.3 Reward definition

• Separate task rewards combine target properties with band-gap terms: excluding near-metallic dielectric candidates and steering SLME candidates toward a favored band-gap region.The dielectric reward requires in-plane equality and finite out-of-plane contrast; the SLME band-gap localization score is centered at 1.3 eV and is not a hard constraint.Source: §4.3 · pp. 12–13

4.4 Reward-guided fine-tuning on the asymmetric unit

• Reward-weighted policy gradients fine-tune the asymmetric-unit diffusion process, while a reward-dependent KL penalty keeps low-reward trajectories closer to the pretrained generator.Rewards are evaluated on reconstructed crystals; the update increases the probability of asymmetric units that generate high-reward structures.Source: §4.4 · pp. 13–14

4.5 Adaptive space-group sampling

• After each RL round, SPARC updates its space-group proposal using running reward estimates, while mixing with the base proposal and applying a probability floor to retain exploration.The task-specific base proposals differ: dielectric sampling is nearly uniform, whereas SLME retains the empirical MP-20 marginal.Source: §4.5 · pp. 14–15

4.6 Training and fine-tuning details

• Both tasks fine-tune the pretrained SymmCD checkpoint for 120 RL rounds; the dielectric learning rate stays constant, while the SLME rate decays by 0.98 each round.A frozen copy of the checkpoint serves as the KL reference policy.Source: §4.6 · p. 15

Declarations

• T.-W.H. and Q.Y. acknowledge U.S. Department of Energy support under Award DE-SC0023664; T.-W.H. and A.B. acknowledge NSF ExpandQISE support under Award OMA-2329067.Source: Acknowledgements · p. 15

1 Reinforcement-learning experimental setup

• Both campaigns ran for 120 RL rounds; dielectric generated 96 candidates per round and SLME 128, with frozen property surrogates and replay of 10 eligible stored structures each round.Source: §1 · pp. 19–20
• Each round samples a space group and asymmetric-unit size, generates candidates, reconstructs full crystals, relaxes them, filters them by validity and S.U.N. criteria, and scores up to 30 survivors.Relaxation uses MatterSim with full variable-cell optimization; stability requires Ehull ≤ 0.10 eV/atom.Source: §1 · p. 19
• The terminal reward updates the asymmetric unit that generated each relaxed crystal, while an outer loop groups rewards by realized space group and updates its sampling proposal.The base-distribution anchor and probability floor retain sampling probability across the original support.Source: §1 · p. 20

2 Symmetry-constrained generation

• Generating the asymmetric unit together with site-symmetry assignments lets space-group operations reconstruct a crystal that respects the selected symmetry by construction.The toy example shows representative atoms, space-group symmetry elements, and the reconstructed unit cell.Source: §2 · p. 21, Figure 1 · p. 21
• For each space group, training-material site-symmetry assignments define a marginal over 15 crystallographic axes and 13 categories; discrete diffusion moves assignments toward this group-specific prior.The forward process redistributes probability only among classes supported by the marginal, and reverse denoising maps noisy assignments toward configurations compatible with the target group.Source: §2 · pp. 22–23

2.1 Lattice parameterization and space-group constraints

• The polar decomposition L = Q exp(S) separates rigid rotation from a rotation-invariant symmetric lattice description, whose symmetry-constrained coefficients are held fixed during generation.Only the free coefficients are denoised, ensuring the generated lattice is consistent with the chosen space group.Source: §2.1 · pp. 23–24

2.2 Denoising losses

• The denoiser jointly reconstructs coordinates, lattice coefficients, atom types, and site symmetries using score matching, masked noise prediction, and categorical cross-entropy losses.Source: §2.2 · p. 24

2.3 Denoising-network architecture

• A fully connected graph over asymmetric-unit sites processes atom, coordinate, symmetry, space-group, and timestep features; its output heads predict coordinate, atom-type, site-symmetry, and lattice denoising terms.Fourier features of relative fractional coordinates encode periodic translation invariance.Source: §2.3 · pp. 24–25

3 Reinforcement-learning fine-tuning objective

• The reverse denoising trajectory is treated as a Markov decision process, with per-step transitions as actions and a terminal reward assigned to the decoded crystal.The policy acts on the asymmetric unit, which is replicated deterministically into the full crystal before reward evaluation.Source: §3 · p. 25
• The score-function identity gives a policy gradient equivalent to weighting each per-step denoising-loss gradient by the terminal reward.The trajectory likelihood is intractable, motivating differentiation through the trajectory likelihood with the score-function identity.Source: §3 · p. 26
• A trajectory-level KL bound regularizes the fine-tuned policy toward the pretrained generator because the marginal divergence is intractable.The data-processing inequality bounds the marginal divergence by the divergence of the full trajectories.Source: §3 · p. 26
• The reward-weighted KL penalty decreases as reward rises, allowing high-reward trajectories more deviation from the pretrained policy while constraining low-reward trajectories more strongly.The objective uses λ above the maximum reward so that the weighting factor remains positive.Source: §3 · p. 26

4 Property surrogate models

• The band-gap surrogate uses a scalar output head, while all three surrogates share a TSENN backbone retrained for each task with task-matched output irreps.This common backbone and evaluation protocol allow target properties to be swapped by changing the output head.Source: §4 · p. 26, §4.1 · p. 26

Dataset.

• Band-gap labels come from Materials Project VASP calculations using PBE+U settings, so predictions remain on the semilocal PBE scale and underestimate experimental gaps.The band-gap dataset is restricted to entries with valid band-gap targets.Source: Dataset · p. 26

Train/validation/test splits.

• The band-gap data are split 8:1:1 into 121,065 training, 15,138 validation, and 15,127 test entries.Source: Train/validation/test splits · p. 27

Training.

• R2 = 0.881 with MAE = 0.252 eV on the held-out test set, with predictions closely following the parity plot's y = x reference line.Validation performance is R2 = 0.869 with MAE = 0.265 eV; the held-out estimate is described as suitable for candidates within the training data's chemical regime.Source: Training · p. 27, Figure 4 · p. 27

4.2 Spectroscopic limited maximum efficiency

• The surrogate predicts the dielectric spectrum on a fixed 0–20 eV grid, derives the real part through Kramers–Kronig relations, and evaluates SLME at fixed film thickness, temperature, and solar spectrum.The grid spacing is 0.1 eV; evaluation uses Labs = 0.3 µm, Tcell = 300 K, and AM1.5G.Source: §4.2 · p. 27

Dataset.

• The training data comprise 21,064 DFT dielectric spectra from OptiMate, sampled from the Alexandria database.Source: Dataset · p. 28

Train/validation/test splits.

• The data are split into 16,848 training, 2,106 validation, and 2,110 test materials, stratified by crystal system to preserve symmetry distributions.Source: Train/validation/test splits · p. 28

Training.

• R2 = 0.925 with MAE = 1.35% on the held-out test set, where predicted SLME is compared with DFT-computed SLME.The parity plots show train, validation, and test splits against the perfect-agreement line.Source: Figure 5 · p. 28, Training · p. 28

4.3 Static-limit dielectric tensor

• The output head combines a scalar and a rank-2 traceless tensor, then maps their six coefficients to the Cartesian components of the symmetric dielectric tensor.Separating isotropic trace from anisotropic response prevents larger diagonal entries from dominating the loss.Source: §4.3 · pp. 28–29

Dataset.

• The training set comprises 6,706 Materials Project materials with dielectric tensors computed by density-functional perturbation theory in VASP.Source: Dataset · p. 29

Train/validation/test splits.

• The 5,363 training, 671 validation, and 672 test materials are split by crystal system to preserve symmetry distributions.Source: Train/validation/test splits · p. 29

Training.

• The surrogate maintains high R2 for diagonal and off-diagonal components, while off-diagonal absolute errors are smaller, consistent with their smaller DFT magnitudes.Per-component test metrics are reported in Supplementary Table 7, and the parity plots show all components across the three splits.Source: Training · p. 29, Figure 6 · p. 30

Dielectric task.

• The uniaxial score combines out-of-plane contrast with in-plane isotropy, suppressing near-isotropic and biaxial tensors through separate factors.The additional selection criterion qiso ≥ 0.95 addresses candidates beyond the score’s saturation at quni = 0.20.Source: Dielectric task · pp. 30–31
• The dielectric reward takes the smaller of the uniaxial-score and band-gap terms, so neither a large score nor a large gap can compensate for a weak counterpart.It is nonzero only for quni > 0 and Eg > 0.3 eV, and reaches one at quni ≥ 0.20 and Eg ≥ 0.8 eV.Source: Dielectric task · p. 31

SLME property term.

• SLME is evaluated from the absorption spectrum and band gap under AM1.5G illumination, with Labs = 0.3 µm, Tcell = 300 K, and fr = 1.The efficiency depends on both Eg and the full absorption spectrum α(E).Source: SLME property term · p. 31

SLME reward.

• The SLME reward weights normalized efficiency at 0.8 and adds a 0.2 band-gap term centered at E∗ = 1.3 eV.The tent-shaped gap term is not a hard gate: outside 0.3–2.3 eV, a candidate can still earn reward through predicted efficiency.Source: SLME reward · p. 32
• The Shockley–Queisser reference reaches 33.7% at Eg = 1.34 eV under the same AM1.5G spectrum, but is not included in the reward.Supplementary Figure 7 shows the dielectric minimum-based reward and the weighted SLME reward surface alongside the reference curve.Source: SLME reward · p. 32, Figure 7 · p. 32

6.1 Uniaxial dielectric responses

• Nine generated dielectric candidates retained the targeted in-plane equality and distinct out-of-plane response after first-principles validation.Their crystal systems were hexagonal, tetragonal, orthorhombic, and monoclinic; DFT gaps ranged from 0.72 to 5.47 eV and Ehull from 0 to 67 meV/atom.Source: §6.1 · p. 33, Figure 8 · p. 34
• Hexagonal and tetragonal candidates have symmetry-enforced in-plane tensor equality, while InPS4 has εxx = 5.77, εyy = 5.78, and εzz = 3.57.Its reported tensor has the largest contrast among the listed candidates.Source: §6.1 · p. 33
• Lower-symmetry monoclinic and orthorhombic candidates also show approximate uniaxiality, but their space groups do not require equal in-plane components.The reported responses may be more sensitive to strain, defects, temperature, or other structural perturbations than symmetry-enforced cases.Source: §6.1 · p. 33
• The rejection cases show that stability, out-of-plane contrast, and in-plane equality each constrain dielectric candidate selection.Ti2TeO6 exceeded the stability cutoff, SrTiO3 was nearly isotropic, and TeI2 failed the in-plane isotropy requirement.Source: §6.1 · p. 33

6.2 SLME

• 241 of 576 SLME-filtered candidates (42%) relaxed to gapped semiconductors with well-defined absorption spectra; the remainder became metallic or nearly metallic.The selected pool required predicted band gaps of 0.8–1.9 eV and SLME above 25%.Source: §6.2 · p. 35
• The DFT-confirmed SLME candidates span six crystal systems, with PBE indirect gaps of 1.15–2.12 eV and SLME values of 18–32%.The surrogate closely reproduced DFT ε2(ω) spectra across represented crystal systems; layered orthorhombic PbS had SC = 0.97.Source: §6.2 · p. 35
• The reported band gaps and optical spectra are PBE independent-particle screening quantities, not quantitative quasiparticle or optical excitation energies.PBE generally underestimates band gaps, and the independent-particle approximation neglects electron–hole interactions and microscopic local-field effects.Source: §6.2 · p. 35