cond-mat.mtrl-sciSep 11, 2026
SaveSymmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design
Organizations: Department of Physics, Northeastern University, Boston, 02155, Massachusetts, USA. · Quantum Materials and Sensing Institute, Northeastern University, Burlington, 01803, Massachusetts, USA.
Abstract
The inverse design of crystalline materials ultimately seeks structures with desired physical properties. However, for many functional responses, a favorable numerical value is meaningful only when supported by the symmetry of the underlying crystal. Without the appropriate crystallographic constraints, an apparent response may be ill defined, accidental, or not symmetry protected. Here we introduce SPARC, a symmetry- and property-aware reinforcement learning framework that optimizes physical objectives while preserving the structural conditions required for their realization. We demonstrate SPARC on two complementary tasks. The first targets strong uniaxial dielectric anisotropy, a tensorial response that is well defined only within appropriate crystal classes. The second maximizes the spectroscopic limited maximum efficiency, a scalar device-level objective without a prescribed symmetry class, allowing the framework to identify favorable crystallographic motifs. These results show that symmetry is not merely an additional design constraint, but a physical foundation for generating candidates with meaningful, robust, and realizable functional properties.
Figures & tables
Figure 1: Schematic of the SPARC pipeline. Each reinforcement-learning iteration consists of five stages. (1) The policy samples a space group from the proposal distribution and the number of representative asymmetric-unit sites from the space-group-dependent prior . (2) The diffusion policy denoises the asymmetric-unit representation . (3) The full crystal is reconstructed by applying the symmetry operations of . (4) The reconstructed structures are relaxed, filtered, and evaluated for the target property using a surrogate model. (5) Each resulting structure is assigned a terminal reward . The terminal reward drives two decoupled optimization processes. In the inner loop, the diffusion policy parameters are fine-tuned over multiple steps using a reward-weighted policy gradient together with a KL-divergence penalty that anchors the policy to the pretrained generator . In the outer loop, the space-group proposal is updated using rewards aggregated according to the realized space group, thereby increasing the sampling probability of productive symmetries relative to the dataset prior . Throughout the figure panels, SG abbreviates space group.
Figure 2: Steering generation toward uniaxial dielectric anisotropy. (a) The SPARC loop specialized to the dielectric task. A space group is drawn from the current proposal , and the space-group-conditioned diffusion model generates a crystal. The TSENN surrogate predicts the static dielectric tensor and band gap . The uniaxial reward is defined as , where , , and excludes cubic tensors. The band-gap term favors – eV. The resulting reward updates both the policy and the space-group proposal . Generated structures are filtered to stable, unique, and novel (S.U.N.) candidates before scoring. The worked example is a generated with symmetry whose predicted tensor is uniaxial, with and vanishing off-diagonal entries. (b) Mean reward versus RL step during fine-tuning (raw per-step mean in grey, exponential moving average with span 10 in black), and the uniaxial-anisotropy score climbs from to . (c) The fifteen most frequent realized space groups among the generated structures (inset, full distribution over all 230 groups). Sampling concentrates on the hexagonal, trigonal, and tetragonal groups that admit a uniaxial dielectric tensor, led by and , followed by , , and . Together, these five groups account for approximately 44% of the generated structures.
Figure 3: Ablation of symmetry enforcement and pre-reward relaxation. SymmCD and DiffCSP are fine-tuned using the same uniaxial-anisotropy reward. For each backbone, MLIP relaxation is either enabled or disabled before reward evaluation. (a) Mean reward as a function of RL step. All runs improve during fine-tuning, and the reward trajectories overlap substantially. Similar reward values therefore do not indicate whether the targeted tensor relation is symmetry-enforced. (b) Distribution of the highest rotational axis among generated crystals for the four ablation settings. Without relaxation, DiffCSP generates almost exclusively or structures. Relaxation recovers mainly twofold symmetry, with only rare threefold and fourfold structures and no sixfold structures. SymmCD retains substantial populations with threefold, fourfold, and sixfold axes because symmetry is imposed during generation. (c) Representative DiffCSP-generated structure in space group 1 ( ). (d) SymmCD asymmetric unit for the same composition under the space-group operations. Dashed lines connect representative atoms to their symmetry images. (e) Full SymmCD crystal reconstructed from the asymmetric unit. The resulting structure belongs to space group 176 ( ) and contains a sixfold rotational axis.
Figure 4: RL-guided generation and DFT verification of dielectric-anisotropy candidates. (a) Distribution of generated structures in the plane of surrogate-predicted band gap and uniaxial dielectric reward . Kernel-density contours compare structures generated during the first 10 RL rounds with those generated during the last 10 rounds. The black and green circles indicate the corresponding population medians, and the arrow highlights the shift toward the finite-gap, high-reward region during fine-tuning. The dashed lines indicate the band-gap and reward thresholds defining the target region. The four numbered markers denote the candidates selected for first-principles verification: , , blue phosphorus, and TeAs . (b) DFT-relaxed structures and calculated properties of the four selected candidates. The table reports the realized space group, , PBE band gap , and dielectric tensor . The quantities in panel (a) are surrogate predictions before DFT validation, whereas those in panel (b) are obtained after DFT relaxation and may therefore differ. The three high-reward candidates, ( ), ( ), and blue phosphorus ( ), retain the symmetry-constrained form . Blue phosphorus shows the largest dielectric anisotropy, with . In contrast, orthorhombic TeAs ( ) has three distinct diagonal components and serves as a low-reward biaxial comparison.
Figure 5: Steering crystal generation toward high-efficiency single-junction absorbers. (a) Mean reward as a function of RL step during SPARC fine-tuning. The gray line shows the raw mean reward at each step, while the black line shows the exponential moving average with a span of 10 steps. The shaded regions indicate the prior window, defined by the first 10 RL steps, and the steered window, defined by the last 10 RL steps. (b) Spectroscopic limited maximum efficiency (SLME), , as a function of band gap for structures generated in the prior window (left, 149 structures) and the steered window (right, 279 structures). The points are colored by their kernel-density estimate, and the dashed curve indicates the Shockley–Queisser limit. The light-gray area marks the target region, defined by – eV and . The fraction of structures within this region increases from to after steering. (c) Periodic-table heatmaps showing the percentage of generated structures containing each element. The number of represented elements decreases from 53 in the prior population to 46 in the steered population. Steering redistributes the elemental frequencies and enriches sulfur- and antimony-containing structures. (d) Relative frequencies of the ten most commonly realized space groups in the prior and steered populations. The cubic space group remains the most frequent symmetry after steering. The orthorhombic and triclinic space groups become substantially more prevalent.
Appendix figures & tables16 assets
Supplementary material from the paper’s appendix.
Appendix
| Setting | Shared | Dielectric RL | SLME RL |
| Initial policy | SymmCD MP-20 checkpoint | – | – |
| Reference policy | Frozen initial checkpoint | – | – |
| Optimizer | Adam | – | – |
| Initial learning rate | – | ||
| Learning-rate multiplier per round | – | ||
| Weight decay | – | – |
Supplementary Table 1: Reinforcement-learning optimization settings. Shared and task-specific settings used for SPARC fine-tuning. Both campaigns were run for RL rounds.
Supplementary Figure 1: Symmetry-constrained decomposition of a crystal . On a two-dimensional toy example (wallpaper group , fictitious species X, Y, Z). (a) The asymmetric unit, comprising the representative atoms with their fractional coordinates, species, and site symmetries (X on a site, Y on a mirror line, Z on a general site). (b) The corresponding space-group symmetry elements, the mirror lines and rotation centres. (c) Applying these operations to the asymmetric unit replicates it across the unit cell and recovers the full crystal. SPARC generates only the asymmetric unit in (a) and reconstructs the complete structure through (b) and (c) , so every generated crystal is symmetry-valid by construction.
Supplementary Figure 2: Site-symmetry marginal for space group 225 ( ). Empirical site-symmetry marginal , aggregated over all training materials in and shown as a heatmap over the crystallographic axes and site-symmetry categories. Color encodes probability.
Supplementary Figure 3: Discrete-diffusion transition toward the site-symmetry marginal (space group 225, , axis ). The forward process for the site-symmetry labels along one crystallographic axis, shown through the cumulative transition at diffusion steps . (a) The space-group marginal over the site-symmetry classes, the absorbing prior the forward process relaxes onto; for this axis the mass concentrates on a few classes. (b) The noise-schedule weights and the cumulative over diffusion time, with the five sampled timesteps marked; decreases from to as the clean signal is replaced by the marginal. (c) The self-transition probability , the probability of remaining in the initial class , versus for three representative classes; each decays from toward its marginal weight. (d) The row distribution for source class at the five timesteps, showing how probability initially localized on class spreads toward the marginal as grows.
| Crystal Family | Space Group No. | Lattice Shape | Constraint of Symmetric Bases |
| Triclinic | No Constraint | No Constraint | |
| Monoclinic | |||
| Orthorhombic | |||
| Tetragonal | |||
| Trigonal/Hexagonal | |||
| Cubic |
Supplementary Table 2: Relationship between the lattice shape and the constraint on the symmetric-basis coefficients , where and denote the lengths and angles of the lattice bases, respectively (after DiffCSP++ Jiao et al. (2024) ).
| Component | Value |
| Backbone | CSPNet message-passing graph neural network (fully connected over representatives) |
| Message-passing layers | 8 |
| Hidden / representation dim. | 1024 |
| Timestep embedding dim. | 10 |
| Coordinate Fourier frequencies | 128 |
| Activation | SiLU |
Supplementary Table 3: Denoising-network architecture hyperparameters (inherited from the SymmCD MP-20 model).
| Hyperparameter | Value |
| Maximum spherical harmonic order | 2 |
| Embedding dimension | 128 |
| Number of layers | 4 |
| Channel multiplicity | 16 |
| Batch size | 128 |
Supplementary Table 4: Hyperparameters for the TSENN band-gap surrogate.
Supplementary Figure 4: Predicted versus DFT-computed band gap. Parity plots of the predicted band gap against the DFT reference, shown for the train, validation, and test splits (left to right). The dashed line indicates perfect agreement ( ). On the held-out test set, the model achieves with .
| Hyperparameter | Value |
| Maximum spherical harmonic order | 2 |
| Embedding dimension | 128 |
| Number of layers | 4 |
| Channel multiplicity | 64 |
| Batch size | 24 |
Supplementary Table 5: Hyperparameters for the TSENN SLME surrogate.
Supplementary Figure 5: Predicted versus DFT-computed SLME. Parity plots of the predicted SLME against the value computed from the DFT dielectric spectrum, shown for the train, validation, and test splits (left to right). The dashed line indicates perfect agreement ( ). On the held-out test set, the model achieves with (absolute efficiency percentage points).
| Hyperparameter | Value |
| Maximum spherical harmonic order | 2 |
| Embedding dimension | 64 |
| Number of layers | 2 |
| Channel multiplicity | 32 |
| Batch size | 16 |
Supplementary Table 6: Hyperparameters for the TSENN dielectric-tensor surrogate.
| Component | MAE | |
| 0.993 | 0.087 | |
| 0.992 | 0.083 | |
| 0.991 | 0.088 | |
| 0.986 | 0.005 | |
| 0.976 | 0.006 | |
| 0.956 | 0.009 |
Supplementary Table 7: Per-component performance of the TSENN dielectric-tensor surrogate on the held-out test split. MAE is reported in units of the relative dielectric constant.
Supplementary Figure 6: Predicted versus DFT-computed static-limit dielectric tensors across all three splits. Parity plots of each independent Cartesian component ( ) of the dielectric tensor, evaluated on the train, validation, and test splits. The dashed line indicates perfect agreement ( ). The inset showing the MAE for per-component performance, and all the metrics are also reported in Supplementary Table 7 .
Supplementary Figure 7: Reward surfaces. Reward as a function of the predicted band gap and the task-specific property for (a) the uniaxial dielectric objective and (b) the SLME objective. In (a), the tensor-based property score is , and the final reward is . The vertical dotted line marks the onset of the band-gap contribution at eV. In (b), the final reward is , where and is a symmetric tent centered at eV. The dashed curve shows the Shockley–Queisser detailed-balance limit evaluated using the same AM1.5G spectrum.
Supplementary Figure 8: First-principles validation of generated uniaxial dielectric candidates. (a) Nine generated structures whose dielectric tensors retain the targeted relation with a distinct out-of-plane component. Each entry shows the DFT-relaxed structure, realized space group, calculated dielectric tensor, DFT band gap , and . The viewing direction is indicated below each structure, and denotes an axis pointing out of the page. The hexagonal and tetragonal structures exhibit symmetry-enforced in-plane equality, while the monoclinic and orthorhombic structures exhibit approximate equality that is not required by symmetry. (b) Comparison cases illustrating three rejection mechanisms. retains an approximately uniaxial tensor but exceeds the stability cutoff, has a nearly isotropic response and is suppressed by the out-of-plane contrast term, and has a biaxial response that fails the in-plane isotropy requirement.
Supplementary Figure 9: First-principles optical response of eight generated candidates. For each material the left panel shows the relaxed crystal structure (ball-and-stick, unit cell outlined) viewed down the crystallographic axis marked in the lower-left corner. The denotes a viewing axis pointing out of the page. Each structure panel associated to a spectra panel comparing the imaginary dielectric function from first-principles DFT (QE, black solid) with the TSENN prediction (red dashed) over – eV. Annotations give the PBE indirect band gap and the SLME evaluated following the parameters throughout the work describe in main text.