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Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design
Ting-Wei Hsu, Arun Bansil, Qimin Yan
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
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Summary
The paper introduces SPARC, a reinforcement learning framework for inverse materials design that generates crystalline structures while preserving crystallographic symmetry. SPARC combines a symmetry-constrained diffusion generator (SymmCD) with property-aware reward optimization. It is demonstrated on two tasks: maximizing uniaxial dielectric anisotropy (a tensorial property requiring specific symmetry classes) and maximizing spectroscopic limited maximum efficiency (a scalar device-level objective). Results show that SPARC successfully enriches generated structures in symmetry classes that support the target properties, outperforming symmetry-agnostic baselines like DiffCSP in generating physically meaningful and robust candidates.
Entities (10)
Relation Signals (8)
SymmCD → enforces → crystallographic symmetry
confidence 95% · SymmCD treats GG as a fixed conditioning input... generating under explicit symmetry constraints
SPARC → optimizes → uniaxial dielectric anisotropy
confidence 95% · The first targets strong uniaxial dielectric anisotropy... We demonstrate SPARC on two complementary tasks.
SPARC → optimizes → spectroscopic limited maximum efficiency
confidence 95% · The second maximizes the spectroscopic limited maximum efficiency... We demonstrate SPARC on two complementary tasks.
SPARC → uses → SymmCD
confidence 95% · SPARC couples a symmetry-constrained SymmCD Levy et al. () generator with reward optimization
TSENN → predicts → dielectric tensor
confidence 92% · The TSENN surrogate predicts the static dielectric tensor
DiffCSP → doesnotenforce → crystallographic symmetry
confidence 90% · DiffCSP... diffuses atoms freely in continuous coordinate space without enforcing crystallographic symmetry.
Cd2SnBr6 → hasspacegroup → P-31m
confidence 90% · It belongs to the tetragonal space group P3bar1m
MatterSim → relaxes → crystal structures
confidence 90% · Each decoded structure is first relaxed using MatterSim
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Abstract
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.
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- Source: https://arxiv.org/abs/2609.13468v1
- Canonical: https://arxiv.org/abs/2609.13468v1
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Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design Ting-Wei Hsu Email: hsu.ting@northeastern.edu Affiliation: Department of Physics, Northeastern University, Boston, 02155, Massachusetts, USA Affiliation: Quantum Materials and Sensing Institute, Northeastern University, Burlington, 01803, Massachusetts, USA Arun Bansil Affiliation: Department of Physics, Northeastern University, Boston, 02155, Massachusetts, USA Affiliation: Quantum Materials and Sensing Institute, Northeastern University, Burlington, 01803, Massachusetts, USA Qimin Yan Email: q.yan@northeastern.edu Affiliation: Department of Physics, Northeastern University, Boston, 02155, Massachusetts, USA Affiliation: 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. keywordsMaterials design, Generative model, Optical properties, Solar cell efficiency, Reinforcement Learning 1 Introduction The physical properties of functional materials are deeply constrained by the symmetry of their underlying crystal structures. Many technologically relevant responses are not determined by composition alone, but emerge from the coupled effects of atomic species, atomic positions, and lattice geometry. In the inverse design of functional materials, the central goal is therefore to identify crystal structures that satisfy a desired physical objective while remaining chemically and crystallographically plausible. Symmetry plays a particularly important role in this process. From tensorial responses such as dielectric anisotropy, to nonlinear optical effects such as second-harmonic generation, and symmetry-protected electronic phenomena all depend on the allowed crystal symmetry. In some cases, the target response is permitted only within specific symmetry classes. For example, electric-dipole second-order nonlinear optical responses require the absence of inversion symmetry, while chiral charge-density-wave states can depend on layer-resolved charge-order patterns and their interlayer phase or stacking relationships. Thus, symmetry is not merely a descriptor of a generated structure, but a fundamental constraint that determines which physical properties can occur. Recent advances in generative modeling have opened new opportunities for crystal structure design. Diffusion-based crystal generators can sample periodic structures across large chemical and configurational spaces Xie et al. (2022); Jiao et al. (2023); Zeni et al. (2025), while steering strategies such as classifier-free guidance Ho and Salimans (2022); Guo et al. (2025) and reinforcement learning (RL) Chen et al. (2025); Park and Walsh (2026); Govindarajan et al. (2025); Cao and Wang (2026) allow the generation process to be biased toward user-specified objectives. Most demonstrations focus on scalar targets such as formation energy, band gap, or magnetic density, whereas many functional materials are defined by tensorial, spectral, or device-level responses. A key distinction among these approaches lies in the structural representation on which the steering is applied. Recent work, for example, applies group-relative policy optimization to a latent diffusion model and combines novelty, stability, and diversity rewards to direct sampling toward underexplored regions of chemical space Park and Walsh (2026). This strategy controls where the generator samples in property and chemical space, but the latent and free-coordinate representations being optimized do not preserve crystallographic symmetry by construction. As a result, a generated crystal may appear successful in property space while losing the crystallographic symmetry required for the intended functionality, often collapsing toward the P1P1 space group when symmetry is not explicitly enforced during generation. This limitation reflects a deeper mismatch between local, continuous structure updates and global crystallographic constraints. Closing this gap requires generative frameworks that optimize functional objectives while preserving the symmetry conditions under which those objectives are physically admissible. Here we introduce SPARC (Symmetry- and Property-Aware Reinforcement Learning for Crystal Generation), a framework for symmetry-aware, property-driven inverse design of crystalline materials. SPARC couples complex property objectives with crystallographic constraints during generation, enabling the search for structures that are simultaneously optimized for target functionality and consistent with physically meaningful symmetry. By integrating symmetry into the design loop, SPARC provides a route to generate candidates for functional materials whose desired properties cannot be specified by scalar objectives alone. We evaluate SPARC on two representative inverse-design tasks that test complementary aspects of symmetry-aware property optimization. First, we consider a symmetry-sensitive optical target, the strong uniaxial dielectric anisotropy. This response is defined by a large contrast between the in-plane and out-of-plane components of the static-limit dielectric tensor (hereafter the dielectric tensor), as commonly found in layered van der Waals crystals Liu et al. (2016). Because the special form of the dielectric tensor is constrained by crystal symmetry, this task tests whether SPARC can access and preserve the symmetry classes required for directional optical response. Second, we consider a device-level optical-efficiency target by maximizing the spectroscopic limited maximum efficiency (SLME) η. Unlike the anisotropy objective, SLME is a scalar figure of merit and does not prescribe a known symmetry class. It depends on the full frequency-resolved dielectric response and implicitly rewards an optimal band gap near the Shockley–Queisser maximum Yu and Zunger (2012). This task therefore provides a complementary test of whether symmetry-aware generation can discover preferential crystallographic motifs or space-group distributions for a complex device-level objective. Together, these case studies show that incorporating symmetry into complex-property-guided crystal generation is essential for producing realistic candidates and suggest a broader route toward symmetry-aware generative modeling for inverse materials design. 2 Results 2.1 SPARC pipeline SPARC couples a symmetry-constrained SymmCD Levy et al. () generator with reward optimization for property-directed crystal generation. Figure 1 traces this pipeline as a left-to-right sequence of five stages, from sampling a crystal structure through denoising, reconstruction, relaxation and scoring, to the terminal reward, and the reward then closes the two feedback loops to update the diffusion model and the space-group proposal. At generation time the policy first samples a space group G from the proposal π(j)π^(j), then draws the number of representative asymmetric-unit sites M from a per-space-group prior p(M∣G)p(M G) estimated from Materials Project (MP-20 dataset) Jain et al. (2013); Xie et al. (2022). Conditioned on (G,M)(G,M), it denoises the asymmetric unit ℳ′=(′,′,′,)M =( A , F , , k) over T reverse steps, jointly generating the atom types ′ A , fractional coordinates ′ F , site-symmetry labels ′ , and symmetric-matrix lattice basis of its representative sites k. Applying the group operations of G to these representatives then regenerates the full crystal. The reward is therefore optimized over symmetry-constrained crystals rather than arbitrary coordinate representations. Each decoded structure is first relaxed using MatterSim, a machine-learning interatomic potential (MLIP) Yang et al. (2024). The relaxed structure is then evaluated according to the stable, unique, and novel (S.U.N.) criteria following MatterGen Zeni et al. (2025), and its target property is evaluated by the surrogate model to assign the terminal reward. Each generated batch drives two decoupled updates, both derived in full details in Methods. In simple words, the inner loop fine-tunes the diffusion policy pθ(′,′,′,)p_θ( A , F , , k) to adjust the asymmetric-unit generation. Treating the reverse trajectory as a Markov decision process, it applies a reward-weighted policy gradient anchored to the pretrained generator pprep_pre by an explicit Kullback–Leibler (KL) term to discourage reward hacking and preserves the broad chemical and structural coverage learned during pretraining Chen et al. (2025). The experience replay and a diversity filter to improve sample efficiency and downweight near-duplicate candidates. The outer loop updates the space-group proposal itself. Because SymmCD treats G as a fixed conditioning input that the policy gradient cannot reshape, we accumulate reward feedback across the realized space groups and tilt the proposal π(j)π^(j) toward desirable space groups. At every round, we mix the proposal back with the initial base distribution π(0)π^(0) so that multiple symmetry classes remain active and the optimization does not collapse onto a few low-symmetry structures. Figure 1: Schematic of the SPARC pipeline. Each reinforcement-learning iteration consists of five stages. (1) The policy samples a space group G from the proposal distribution π(j)π^(j) and the number of representative asymmetric-unit sites M from the space-group-dependent prior p(M∣G)p(M G). (2) The diffusion policy pθp_θ denoises the asymmetric-unit representation ℳ′=(′,′,′,)M =( A , F , , k). (3) The full crystal ℳ=(,,)M=( A, F, L) is reconstructed by applying the symmetry operations of G. (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 r(ℳ)r(M). The terminal reward drives two decoupled optimization processes. In the inner loop, the diffusion policy parameters θ 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 pprep_pre. In the outer loop, the space-group proposal π(j)(G)π^(j)(G) is updated using rewards aggregated according to the realized space group, thereby increasing the sampling probability of productive symmetries relative to the dataset prior π(0)π^(0). Throughout the figure panels, SG abbreviates space group. 2.2 Inverse design of uniaxial dielectric responses We first examine whether SPARC can generate crystals whose symmetry is dictated by a target physical response. We consider a uniaxial dielectric tensor, =(ε∥000ε∥000ε⟂), = pmatrix _ &0&0\\ 0& _ &0\\ 0&0& _ pmatrix, (1) where ε∥ _ is the response within the basal plane and ε⟂ _ is the response along the unique axis. The target requires both in-plane isotropy, εxx=εyy _x= _y, and a finite out-of-plane contrast, εzz≠εxx _z≠ _x. It therefore defines one distinguished dielectric axis without selecting a preferred direction within the perpendicular plane. Crucially, this target is not merely a numerical relation among tensor components. Under Neumann’s principle, the dielectric tensor must reflect the point-group symmetry of the crystal. Three-, four-, and six-fold rotational symmetries enforce εxx=εyy _x= _y, making the uniaxial form symmetry-compatible with trigonal, tetragonal, and hexagonal crystals. Cubic symmetry instead removes the desired anisotropy by requiring εxx=εyy=εzz _x= _y= _z. Lower-symmetry crystals do not generally enforce the in-plane degeneracy, so any observed εxx≈εyy _x≈ _y is accidental rather than symmetry-protected. We therefore seek structures in which the target dielectric response follows naturally from the underlying crystal symmetry. A large contrast |ε⟂−ε∥|| _ - _ | is useful when a device requires different in-plane and out-of-plane responses without sensitivity to the in-plane orientation of the crystal. This tensor form is relevant to layered dielectric environments, gate dielectrics, and van der Waals heterostructures Dean and others (2010); Laturia et al. (2018); Illarionov and others (2020). At finite frequencies, uniaxial anisotropy also supports birefringence and polarization control. It can produce hyperbolic light propagation when the in-plane and out-of-plane permittivities have opposite signs Niu and others (2018); Ermolaev and others (2021). Although the present task uses the static-limit dielectric tensor, it provides a direct test of whether SPARC can generate crystal symmetries that support a prescribed tensorial response. Black phosphorus (BP) provides a useful example of the technological value of dielectric anisotropy. Its strong in-plane optical anisotropy has enabled polarization-sensitive photodetectors and birefringent optical devices Yuan et al. (2015); Qiao et al. (2014); Xia et al. (2014). BP is orthorhombic and therefore exhibits a biaxial rather than uniaxial dielectric response. Its εxx _x and εyy _y components are not symmetry-equivalent. Exfoliated BP can also undergo oxidation and progressive degradation under ambient conditions Island et al. (2015). These characteristics motivate the search for complementary anisotropic materials with uniaxial symmetry, thermodynamic accessibility, and synthetic feasibility. SPARC provides a route to explore this broader multi-constraint materials space. To express the target response as a scalar objective, we evaluate the dielectric tensor of each generated structure using a modified Tensorial Spectra Equivariant Neural Network (TSENN) surrogate Hsu et al. (2026). The tensor is evaluated in the standardized conventional cell. For trigonal, tetragonal, and hexagonal structures, this convention aligns the unique axis with z. We then assign a uniaxial anisotropy reward runi∈[0,1]r_uni∈[0,1]. The reward combines an in-plane equality term with an out-of-plane contrast gate gzg_z. The first favors εxx=εyy _x= _y. The second requires a finite difference between εzz _z and the mean in-plane response, thereby excluding the fully isotropic cubic limit. Because the target materials are insulating or semiconducting, the reward additionally includes a band-gap term that suppresses metallic structures. The full reward definition is provided in Methods, and the surrogate benchmark is reported in the Supplementary Information (SI). Figure 2(a) shows the SPARC loop specialized to this task. The initial space-group proposal π(0)π^(0) is obtained by temperature reweighting the MP-20 distribution Kirkpatrick et al. (1983), yielding a nearly uniform distribution that reduces the strong space-group frequency bias of the training data. Starting from this broad prior, SPARC iteratively updates the proposal using reward feedback, allowing the optimization to identify which symmetry classes best support the target dielectric response. During RL fine-tuning, the mean uniaxial-anisotropy reward increases from 0.170.17 to 0.560.56 over 120 RL steps (Figure 2(b)). At each step, 30 structures that pass the S.U.N. filter are scored, giving a total of 3,600 evaluated candidates. Over the same optimization trajectory, the realized space-group distribution becomes increasingly enriched in trigonal, tetragonal, and hexagonal groups that symmetry-enforce the target tensor form (Figure 2(c)). Importantly, these symmetry classes are not prescribed in advance, but emerge through the reward-guided redistribution of the space-group proposal. Among the 3,600 scored structures, P3¯1mP 31m, P42/mnmP4_2/mnm, R3¯R 3, P3¯P 3 and P31P3_1 together account for approximately 44%44\% of the generated population. The generated Cd2SnBr6Cd X_ [t]2SnBr X_ [t]6 structure in Figure 2(a) provides a representative symmetry-enforced result. It belongs to the tetragonal space group P3¯1mP 31m and has the TSENN-predicted dielectric tensor =(6.20006.20004.7), = pmatrix6.2&0&0\\ 0&6.2&0\\ 0&0&4.7 pmatrix, (2) Its equality εxx=εyy _x= _y and vanishing off-diagonal components follow directly from its tetragonal symmetry. The finite difference between the in-plane and out-of-plane components provides the desired uniaxial contrast. The structure retains this response after MLIP relaxation and is selected for subsequent density function theory (DFT) validation. Figure 2: Steering generation toward uniaxial dielectric anisotropy. (a) The SPARC loop specialized to the dielectric task. A space group G is drawn from the current proposal π(j)π^(j), and the space-group-conditioned diffusion model pθ(ℳ′|G,M)p_θ(M |G,M) generates a crystal. The TSENN surrogate predicts the static dielectric tensor and band gap EgE_g. The uniaxial reward is defined as runi=min[Φ(quni),Φ(Eg)]r_uni= [ (q_uni), (E_g)], where quni=gz(1−m)q_uni=g_z(1-m), m=|εxx−εyy|/(|εxx|+|εyy|)m=| _x- _y|/(| _x|+| _y|), and gzg_z excludes cubic tensors. The band-gap term favors Eg=0.3E_g=0.3–0.80.8 eV. The resulting reward updates both the policy pθp_θ and the space-group proposal π(j)π^(j). Generated structures are filtered to stable, unique, and novel (S.U.N.) candidates before scoring. The worked example is a generated Cd2SnBr6Cd X_ [t]2SnBr X_ [t]6 with P3¯1mP 31m symmetry whose predicted tensor is uniaxial, =diag(6.2,6.2,4.7) =diag(6.2,6.2,4.7) with εxx=εyy≠εzz _x= _y≠ _z 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 runir_uni climbs from 0.170.17 to 0.560.56. (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 P3¯1mP 31m and P42/mnmP4_2/mnm, followed by R3¯R 3, P3¯P 3, and P31P3_1. Together, these five groups account for approximately 44% of the generated structures. Lower-symmetry groups also remain present in Figure 2(c), including the orthorhombic PbcaPbca, and PbamPbam group. These structures may approximately satisfy the component-level reward, but the in-plane equality is not symmetry-enforced. This motivates the backbone ablation below. To test the necessity of a symmetry-aware diffusion backbone, we performed a controlled ablation by replacing SymmCD with DiffCSP Jiao et al. (2023), which diffuses atoms freely in continuous coordinate space without enforcing crystallographic symmetry. Both backbones were fine-tuned under the same uniaxial-anisotropy objective. For each backbone, we performed separate runs with MLIP relaxation either enabled or disabled before reward evaluation. Figure 3(a) shows that the four runs reach comparable mean rewards during fine-tuning. Their symmetry distributions, however, differ sharply (Figure 3(b)). Without relaxation, approximately 90% of DiffCSP structures have no nontrivial rotational axis and belong to P1P1 or P1¯P 1. MLIP relaxation recovers mainly twofold symmetry, with only a small fraction acquiring threefold or fourfold axes and almost no sixfold symmetry. SymmCD shows the opposite behavior. Because each structure is generated under explicit symmetry constraints, substantial populations with threefold, fourfold, and sixfold axes remain present both with and without relaxation. The quantitative contrast is even more pronounced. With relaxation enabled, 50.1%50.1\% of SymmCD structures contain a threefold, fourfold, or sixfold axis, compared with only 1.4%1.4\% of DiffCSP structures. This corresponds to one uniaxially symmetric candidate every 2.02.0 generations for SymmCD, versus every 70.670.6 generations for DiffCSP, giving a 35×35× higher symmetry yield. Sixfold symmetry appears in 546546 SymmCD structures but only once among 3,6003,600 DiffCSP samples. Because the two backbones required comparable computational wall-clock time for the same number of fine-tuning loops, this yield advantage reduced the computational cost per uniaxially symmetric candidate by approximately 34×34× in our runs (4343 s versus 14651465 s per candidate). The structural examples in Figure 3(c)–(e) illustrate the origin of this contrast. DiffCSP generates CrTe3WCrTe X_ [t]3W in space group 1 (P1P1), with no imposed rotational symmetry. Because the generated atoms lie far from a high-symmetry arrangement, local relaxation cannot reliably recover the missing rotational symmetry afterward. For the same composition, SymmCD generates an asymmetric unit under P63/mP6_3/m operations. Applying these operations reconstructs the full crystal in space group 176 (P63/mP6_3/m) and produces the required sixfold axis. These results show that reward optimization can satisfy the targeted dielectric relation, but a symmetry-aware backbone is needed when that relation must also be enforced by the crystal symmetry. We therefore adopt SymmCD as a central component of SPARC. 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 P1P1 or P1¯P 1 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 CrTe3WCrTe X_ [t]3W structure in space group 1 (P1P1). (d) SymmCD asymmetric unit for the same composition under the P63/mP6_3/m 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 (P63/mP6_3/m) and contains a sixfold rotational axis. To validate the reward-guided predictions with first-principles calculations, we performed DFT relaxations and computed dielectric tensors using density functional perturbation theory (DFPT) Petousis et al. (2016) for a broader set of generated candidates. Figure 4 highlights four representative examples and summarizes the evolution of the generated population during fine-tuning. As fine-tuning progresses, the reward function shifts the generated population toward the finite-gap, high-reward region. Figure 4(a) compares the distributions obtained during the first and last 10 RL steps. The displacement between their population medians illustrates the corresponding shift in the sampled property space. The four highlighted candidates probe different regions of this distribution. RbGePO4RbGePO X_ [t]4, ZrTi2O6ZrTi X_ [t]2O X_ [t]6, and blue phosphorus occupy the predicted high-reward region, whereas TeAs provides a low-reward biaxial comparison. The positions of the candidates in Figure 4(a) are determined by the surrogate-predicted band gaps and rewards before first-principles validation, whereas the values reported in panel (b) are obtained after DFT relaxation. Differences between the two panels are therefore expected and directly reveal the remaining surrogate error. Figure 4(b) presents the DFT-relaxed structures together with their realized space groups, energies above the convex hull EhullE_hull, Perdew–Burke–Ernzerhof (PBE) band gaps EgE_g, and dielectric tensors 0 _0. The DFT band gaps range from 0.490.49 to 4.224.22 eV, while EhullE_hull ranges from 0.00.0 to 60.860.8 meV/atom. The three candidates selected from the high-reward region retain the symmetry-constrained uniaxial form εxx=εyy≠εzz _x= _y≠ _z after DFT relaxation. Among these candidates, blue phosphorus exhibits the strongest dielectric anisotropy, with εxx=8.80 _x=8.80, εyy=8.79 _y=8.79, and εzz=2.54 _z=2.54. This corresponds to an anisotropy ratio ε∥/ε⟂≈3.46 _ / _ ≈ 3.46. ZrTi2O6ZrTi X_ [t]2O X_ [t]6 and RbGePO4RbGePO X_ [t]4 retain weaker but symmetry-consistent uniaxial responses, with ratios of approximately 1.181.18 and 1.131.13, respectively. By contrast, orthorhombic TeAs has three distinct diagonal components, εxx=15.98 _x=15.98, εyy=9.76 _y=9.76, and εzz=13.33 _z=13.33, consistent with its low uniaxial reward. 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 EgE_g and uniaxial dielectric reward runir_uni. 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: RbGePO4RbGePO X_ [t]4, ZrTi2O6ZrTi X_ [t]2O X_ [t]6, blue phosphorus, and TeAs. (b) DFT-relaxed structures and calculated properties of the four selected candidates. The table reports the realized space group, EhullE_hull, PBE band gap EgE_g, and dielectric tensor 0 _0. 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, RbGePO4RbGePO X_ [t]4 (P63P6_3), ZrTi2O6ZrTi X_ [t]2O X_ [t]6 (P3¯1mP 31m), and blue phosphorus (P31P3_1), retain the symmetry-constrained form εxx=εyy≠εzz _x= _y≠ _z. Blue phosphorus shows the largest dielectric anisotropy, with ε∥/ε⟂≈3.46 _ / _ ≈ 3.46. In contrast, orthorhombic TeAs (PbcaPbca) has three distinct diagonal components and serves as a low-reward biaxial comparison. 2.3 Inverse design of high-efficiency photovoltaic absorbers We next consider a second optical inverse-design task. Our objective is the SLME η of a single-junction photovoltaic absorber Choudhary et al. (2019); Yu and Zunger (2012). Although SLME is reported as a single percentage, it depends on the material’s full optical response. The complex dielectric function ε(ω) (ω) determines the absorption coefficient α(ω)α(ω). This absorption spectrum controls the short-circuit current under the AM1.5G solar spectrum and the radiative dark current associated with black-body emission. Optimizing η therefore requires control over the absorption spectrum above the band edge, rather than the response at a single frequency. This objective complements the uniaxial dielectric task considered above. The uniaxial reward depends only on the dielectric tensor in the ω→0ω→ 0 limit. It is also restricted to crystal classes that permit the target tensor form. In contrast, SLME depends on the full frequency-dependent optical response and does not prescribe a particular crystal class. This task therefore tests whether SPARC can discover favorable chemical and structural distributions when the target property does not impose a preferred symmetry. For each generated crystal, we evaluate η under fixed device conditions. We use an absorber thickness of Labs=0.3 µmL_abs=$0.3\, m$, a cell temperature of Tcell=300 KT_cell=$300\,K$, and a radiative recombination fraction of fr=1f_r=1. The modified TSENN surrogate predicts the imaginary dielectric response ε2(ω) _2(ω). We recover the real part ε1(ω) _1(ω) using the Kramers–Kronig relation. The resulting complex dielectric function is then converted to the absorption coefficient through α(ω)=2ωk(ω)c,α(ω)= 2ω k(ω)c, (3) where k(ω)k(ω) is the extinction coefficient Fox (2010); Gajdoš et al. (2006). The architecture and performance of the modified surrogate are benchmarked in the SI. SLME depends jointly on the band gap EgE_g and the spectral response above the gap. The band-gap term in the reward is therefore aligned with the photovoltaic objective rather than acting as a separate competing target. It steers the population away from the near-metallic regime, where the predicted efficiency is low and the optical surrogate is less reliable. We center the search near Eg≈1.3E_g≈ 1.3 eV, close to the optimal single-junction band gap under standard assumptions. We define a broader target region of Eg≈0.8E_g≈ 0.8–1.91.9 eV and η≥25%η≥ 25\%. Reaching this region requires more than placing the band edge at an appropriate energy. The absorption above the edge must also be strong enough to convert a large fraction of the incident photon flux into photocurrent. Figure 5(a) shows that fine-tuning produces a clear increase in the reward. The mean reward rises rapidly during the early RL steps and remains elevated throughout the optimization process. Its exponential moving average increases from approximately 0.280.28 in the prior window to approximately 0.460.46 in the steered window. Figure 5(b) shows the corresponding redistribution in property space. Before steering, the generated structures are concentrated mainly in the low-gap, low-efficiency region. Fine-tuning moves the population toward intermediate band gaps and higher SLME values. Many of the steered candidates approach the Shockley–Queisser limit within the target band-gap range. As a result, the fraction of structures satisfying both Eg≈0.8E_g≈ 0.8–1.91.9 eV and η≥25%η≥ 25\% increases from 11%11\% in the prior population (149 structures) to 32%32\% in the steered population (279 structures). The improved photovoltaic response is accompanied by changes in the sampled chemistry. Figure 5(c) shows the number of represented elements decreases only modestly, from 53 in the prior population to 46 after steering. The optimization therefore does not collapse the search onto a single chemical family. Instead, it redistributes the elemental frequencies while preserving substantial compositional diversity. Sulfur becomes particularly prominent after steering, and antimony is also enriched in the high-SLME population. This trend is consistent with the broader relevance of sulfide semiconductors to photovoltaics, such as kesterite Cu2ZnSnS4Cu X_ [t]2ZnSnS X_ [t]4 and Sb2S3Sb X_ [t]2S X_ [t]3, which have been extensively investigated as thin-film solar absorbers Yan et al. (2018); Yang et al. (2019). The steered population also remains crystallographically diverse. Figure 5(d) compares the ten most frequently realized space groups before and after steering. Since SLME does not favor a specific crystal class, fine-tuning shifts the symmetry distribution without collapsing it to a single family. The cubic Fm3¯mFm 3m group remains the most common after steering. The orthorhombic PnmaPnma and triclinic P1¯P 1 groups also become more prevalent. This shows that high SLME can be reached through several distinct crystallographic families. This behavior differs from the dielectric task, where the reward favors a much narrower set of rotational symmetries. Taken together, these results establish SLME optimization as a multi-constraint inverse-design task. A successful candidate cannot be identified by optimizing the band gap or a single spectral feature alone. It must combine a suitable band gap with strong absorption across the relevant solar spectrum while remaining structurally valid and crystallographically plausible. SPARC addresses these coupled requirements by concentrating the generated population in favorable regions of property, compositional, and crystallographic space, while still preserving multiple structural routes to high efficiency. The increase in the target-region population from 11%11\% to 32%32\% therefore reflects more than an improvement in a single predicted quantity. It demonstrates a coordinated redistribution of the generated population toward candidates that simultaneously satisfy electronic, optical, and structural constraints. Additional candidates sampled during the RL runs, together with their first-principles validation, are reported in the SI. Overall, this task shows that SPARC can navigate a complex photovoltaic design space governed by the full optical response rather than by a single target descriptor. 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), η, as a function of band gap EgE_g 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 Eg≈0.8E_g≈ 0.8–1.91.9 eV and η≥25%η≥ 25\%. The fraction of structures within this region increases from 11%11\% to 32%32\% 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 Fm3¯mFm 3m space group remains the most frequent symmetry after steering. The orthorhombic PnmaPnma and triclinic P1¯P 1 space groups become substantially more prevalent. 3 Discussion The central result of this work is that symmetry and functional properties must be optimized together in crystal inverse design. They are not independent design objectives. Crystal symmetry constrains the form of many physical responses and determines whether a desired behavior is allowed, enforced, or forbidden, while the targeted property in turn determines which symmetry classes are relevant to the design problem. A lower-symmetry structure may still achieve the targeted dielectric response and therefore receive a high reward, but the required tensor relation is not enforced by symmetry and may occur only for a particular atomic configuration. Reward optimization alone therefore cannot guarantee that the desired response is realized in a symmetry-protected form. It must operate within a structural representation that can access the appropriate symmetry while simultaneously improving the target property. The two optical tasks illustrate complementary forms of this coupling. For the uniaxial dielectric objective, the target tensor directly restricts the admissible crystal classes because the required in-plane equivalence is associated with threefold, fourfold, or sixfold rotational symmetry. For SLME, the property depends on the band gap and the full optical spectrum but does not prescribe a unique crystal class, allowing multiple chemical and structural families to achieve favorable photovoltaic performance. Together, these results show that symmetry and property cannot generally be separated into independent stages of crystal design and should instead be treated within the same optimization problem. The backbone ablation makes the physical role of symmetry explicit. SymmCD and DiffCSP reach similar reward values under the same dielectric objective, yet produce markedly different crystallographic distributions. Both can satisfy the targeted tensor relation numerically, but in a low-symmetry structure the equality εxx≈εyy _x≈ _y is accidental rather than symmetry-enforced. This relation can be lost under small structural distortions, while local relaxation cannot reliably recover the missing global symmetry. The same reasoning extends to properties with known symmetry requirements. For example, inversion symmetry forbids piezoelectricity and electric-dipole second-harmonic generation, so the space-group proposal can be restricted to classes that permit the desired response. When the relevant symmetry class is not known in advance, SPARC can instead keep all symmetry channels available and use the reward-guided outer loop to identify which crystallographic families best support the target property, as demonstrated in the SLME task. Blue phosphorus provides a useful benchmark for whether symmetry-aware steering can recover physically meaningful candidates. SPARC generates a layered phosphorus structure with trigonal P31P3_1 symmetry that remains gapped after first-principles relaxation and exhibits strong uniaxial dielectric anisotropy, with a band gap of 1.751.75 eV. The structure is absent from the MP-20 training set, although a blue-phosphorus allotrope was proposed independently in prior work Zhu and Tománek (2014). No phosphorus-specific target or structural template was supplied during optimization. Its recovery therefore provides a positive benchmark. It shows that SPARC can reach a known and physically plausible motif beyond the training distribution, even though blue phosphorus is not the most stable phosphorus allotrope. The two tasks also show that property steering can concentrate upon different dimensions of materials space. The dielectric objective acts primarily in symmetry space because only specific crystal classes enforce the target tensor. SLME leaves several crystallographic routes available and instead produces a stronger redistribution in property and chemical space. This selective concentration is an important output of inverse design. The goal is not to preserve the full diversity of the pretrained model, but to identify a smaller region that is more likely to contain the desired response. Such behavior should be distinguished from uncontrolled mode collapse because diversity remains along directions that are less strongly constrained. In the SLME task, the generated population becomes concentrated in a smaller region of chemical space while retaining a broad distribution of crystal symmetries. The same principle extends beyond optical properties. Symmetry-aware generation can benefit the design of magnetocrystalline anisotropy, elastic and piezoelectric response, thermal expansion, and other symmetry-constrained material properties (2021); Yan et al. (2024). More broadly, SPARC is not tied to a specific surrogate and can be coupled to models that predict tensorial properties, transport quantities, or the electronic Hamiltonian directly Dong et al. (2025); Zeng et al. (2026); Li et al. (2022); Gong et al. (2023); Zhong et al. (2026). Electronic Hamiltonians are particularly attractive because they provide a common starting point for evaluating band structures, Berry-phase and Berry-curvature quantities, and topological properties through standard electronic-structure post-processing Gong et al. (2023); Zhong et al. (2026); Wang et al. (2019). The reward may change across applications, but the generated structure must still possess the symmetry needed to support the intended physics. Realizing this broader potential will also require improving the accuracy of the reward signal. Learned surrogates inherit both their own prediction errors and the approximations of the reference calculations used for training. This is particularly relevant for the SLME task, where the efficiency depends sensitively on the band gap and absorption edge. The present surrogate is trained on PBE-level data, for which semiconductor band gaps are expected to be systematically underestimated Sham and Schlüter (1983); Borlido et al. (2019). Promising candidates could instead be evaluated using higher-level methods such as hybrid functionals Heyd et al. (2006) or GWGW Govoni and Galli (2015); Deslippe et al. (2012). Because such calculations are substantially more expensive, the resulting high-fidelity dataset will naturally be much smaller than the available PBE dataset. Multi-fidelity learning provides a natural way to address this imbalance by combining abundant low-fidelity data with a smaller number of high-fidelity calculations to improve predictions at the desired level of accuracy Chen et al. (2021). This strategy could be incorporated into an active-learning extension of SPARC, where candidates with high reward or high predictive uncertainty are prioritized for higher-level electronic-structure calculations. The most promising candidates could then be validated experimentally, providing an additional level of fidelity. These theoretical and experimental data could be fed back into the surrogate to improve the reward signal for subsequent rounds of generation and fine-tuning. In this way, increasingly accurate theory and experiment become part of the optimization loop rather than serving only as final validation steps. Taken together, these results establish SPARC as a framework for crystal inverse design in which symmetry and functional properties are optimized together rather than treated as separate stages. By keeping crystallographic symmetry explicit during reward-guided generation, SPARC can enforce known symmetry requirements or identify favorable symmetry classes when they are not known in advance. More broadly, this provides a route toward inverse design in which the search is guided not only toward high-performing materials, but toward structures that can physically support the targeted response. 4 Methods 4.1 Symmetry-constrained crystal representation A crystal ℳM is the periodic arrangement of atoms specified by the triplet ℳ=(,,)M=( A, F, L), where =[1,…,N]∈ℝh×N A=[ a_1,…, a_N] ^h× N collects the one-hot atom types of the N atoms in the cell over the h=94h=94 supported chemical species, =[1,2,3]∈ℝ3×3 L=[ l_1, l_2, l_3] ^3× 3 is the lattice whose columns span one period of the crystal, and =[1,…,N]∈[0,1)3×N F=[ f_1,…, f_N]∈[0,1)^3× N collects the fractional coordinates, related to the Cartesian coordinates by = X= L F, from which physical quantities such as interatomic distances are computed. The space group G is the set of Euclidean operations g=(,)g=( O, t), with ∈O(3) O∈ O(3) and ∈ℝ3 t ^3, that leave the periodic crystal invariant. In three dimensions, space groups are classified into 230230 types. The operations that leave a given site fixed form its site-symmetry group. Sites related by operations of G belong to the same crystallographic orbit, and these orbits partition the atomic sites into symmetry-equivalent classes. Points whose site-symmetry groups are conjugate within G belong to the same Wyckoff position. SPARC adopts the symmetry-constrained asymmetric-unit representation used in SymmCD Levy et al. (). For a fixed space group G, a crystal can be reconstructed from a symmetry-compatible lattice and one representative atom from each occupied orbit. Applying the operations of G to these representatives generates the remaining symmetry-equivalent sites. The representatives form the asymmetric unit, and their number is denoted by M. Conditioned on the sampled pair (G,M)(G,M), the asymmetric-unit representation is encoded as ℳ′=(′,′,′,),M =( A , F , , k), (4) where ′ A and ′ F contain the atom types and fractional coordinates of the M representative sites. The labels ′ specify their site symmetries within G, and ∈ℝ6 k ^6 parameterizes the lattice subject to the constraints imposed by G. The space group G and the number of representatives M are supplied as conditioning variables and are therefore not included among the generated components of ℳ′M . The model generates ℳ′M conditioned on (G,M)(G,M), after which the full crystal ℳM is reconstructed by applying the space-group operations. Following DiffCSP++ Jiao et al. (2024), the lattice is parameterized in a rotation-invariant way rather than through the raw matrix L. The polar decomposition =exp() L= Q ( S) separates a rigid rotation ∈SO(3) Q∈ SO(3) from a symmetric matrix S, and S expands uniquely in a fixed six-dimensional symmetric basis ii=16\ B_i\_i=1^6 as =∑i=16kii S= _i=1^6k_i B_i, yielding the coordinate vector =(k1,…,k6) k=(k_1,…,k_6). For each crystal family a subset of the kik_i is fixed by symmetry while the remainder are free, so a space group is enforced simply by holding the constrained entries at their symmetry-determined values. The explicit basis i\ B_i\ and the per-family constraints on k are reproduced in the SI. 4.2 Symmetry-aware generation We use the symmetry-constrained diffusion model SymmCD Levy et al. (), which builds on DiffCSP++ Jiao et al. (2024). Given a space group G and the number of representative sites M, the model jointly generates the four components of the asymmetric-unit representation, pθ(ℳ′∣G,M)=pθ(′,′,′,∣G,M).p_θ(M G,M)=p_θ( A , F , , k G,M). (5) For simplicity, we do not write G and M in every equation in this work. But they are still used as inputs in all generation and denoising steps. Before RL fine-tuning, G is sampled from the empirical MP-20 distribution pmarg(G)p_marg(G), and M is sampled from pmarg(M∣G)p_marg(M G). These distributions provide the initial sampling rule used for property steering. The reverse diffusion process jointly denoises the atomic species ′ A , fractional coordinates ′ F , site-symmetry descriptors ′ , and lattice coefficients k. Crystal symmetry restricts the allowed values of k. A binary mask (G)∈0,16 m(G)∈\0,1\^6 selects the coefficients that remain free under the chosen space group. Only these coefficients are denoised, while the symmetry-constrained coefficients remain fixed. The reconstructed lattice is therefore consistent with G by construction. The site-symmetry descriptors are also defined relative to G. For each representative site, ′ is encoded as a 15×1315× 13 binary array. The 15 rows correspond to crystallographic axes such as [001][001], [010][010], [100][100], [110][110], and [111][111]. The 13 columns represent the possible site-symmetry operations associated with these axes, including rotations, reflections, and rotoinversions. Each entry indicates whether the corresponding operation is present, giving a 195-dimensional descriptor for each representative site. During the forward discrete diffusion process, these descriptors approach a marginal distribution defined for the selected space group. The complete encoding and transition process are provided in the SI. The denoiser is trained to reconstruct all four components from their noisy representations. We use a score-matching loss for the fractional coordinates, a masked noise-prediction loss for the lattice coefficients, and categorical cross-entropy losses for the atomic species and site-symmetry descriptors. The total objective is ℒ=λℒ+λ′ℒ′+λ′ℒ′+λ′ℒ′,L= _ k\,L_ k+ _ F \,L_ F + _ A \,L_ A + _ \,L_ , (6) where the loss weights are λ=5 _ k=5, λ′=1 _ F =1, λ′=0.1 _ A =0.1, and λ′=10 _ =10. The individual component losses and the transition matrices used for the discrete variables are given in the SI. 4.3 Reward definition We define a separate reward for each inverse-design task. Both rewards depend on the target property and the predicted band gap, but the band gap plays a different role in each case. For the dielectric task, it penalizes near-metallic candidates for which the static dielectric response is not physically meaningful. For the SLME task, it guides the generated population toward the band-gap region favored by the Shockley–Queisser limit. We use the clamped linear function Φ(x,a,b)=clip(x−ab−a,0,1) (x;a,b)=clip ( x-ab-a,0,1 ) (7) to map selected quantities to the interval [0,1][0,1]. Uniaxial dielectric anisotropy. For the uniaxial dielectric task, we evaluate the dielectric tensor of each generated structure using the TSENN surrogate. Before evaluation, we standardize the refined cell so that the symmetry-determined unique axis is aligned with z. We define the out-of-plane response as ε⟂≡εzz _ ≡ _z and the mean in-plane response as ε∥≡12(εxx+εyy) _ ≡ 12 ( _x+ _y ). We then introduce an out-of-plane contrast factor gz=clip(|ε⟂−ε∥|δz,0,1),g_z=clip ( | _ - _ | _z,0,1 ), (8) where δz=0.01 _z=0.01 sets the contrast required to receive full credit. The factor gzg_z is zero for a fully isotropic tensor. It increases linearly with the dielectric contrast and reaches one when the contrast exceeds δz _z. The residual mismatch between the two in-plane components is m=|εxx−εyy||εxx|+|εyy|,m= | _x- _y | | _x |+ | _y |, (9) The uniaxial-anisotropy score is then quni=gz(1−m)∈[0,1].q_uni=g_z(1-m)∈[0,1]. (10) The factor (1−m)(1-m) penalizes differences between εxx _x and εyy _y, while gzg_z penalizes insufficient contrast between the in-plane and out-of-plane responses. A high score therefore requires both approximate in-plane equality and finite out-of-plane anisotropy. We further penalize structures with vanishing band gaps by defining runi=min[Φ(quni,auni,buni),Φ(Eg,ag,bg)].r_uni= [ (q_uni;a_uni,b_uni ), (E_g;a_g,b_g ) ]. (11) The band-gap score limits the final reward when EgE_g approaches the metallic regime. A high anisotropy score therefore cannot compensate for a vanishing band gap. This keeps the generated population within the finite-gap regime where the static dielectric response is well defined and can be verified using DFPT. Spectroscopic limited maximum efficiency. For the photovoltaic task, the target property is the spectroscopic limited maximum efficiency η. It is evaluated under the AM1.5G solar spectrum using an absorber thickness Labs=0.3 µmL_abs=$0.3\, m$, a cell temperature Tcell=300 KT_cell=$300\,K$, and a radiative recombination fraction fr=1f_r=1 Yu and Zunger (2012); Hung et al. (2024). The full calculation of η from the predicted absorption spectrum is provided in the SI. High SLME alone does not explicitly localize the generated population near the band-gap region favored for single-junction solar cells. We therefore combine an efficiency score with a band-gap localization score, rSLME=wηrη+wgrg,SLME,r_SLME=w_ηr_η+w_gr_g,SLME, (12) where rη=Φ(η,0,0.35),r_η= (η;0,0.35), (13) and rg,SLME(Eg)=clip(1−|Eg−E∗|Δ,0,1).r_g,SLME(E_g)=clip (1- |E_g-E^* | ,0,1 ). (14) We use wη=0.8w_η=0.8, wg=0.2w_g=0.2, E∗=1.3E^*=1.3 eV, and Δ=1.0 =1.0 eV. The band-gap score forms a symmetric tent centered at E∗=1.3E^*=1.3 eV and becomes zero outside Eg∈[0.3,2.3]E_g∈[0.3,2.3] eV. It therefore encourages the search to remain near the theoretically favorable single-junction region without acting as a hard constraint. The larger value of wηw_η keeps SLME as the dominant optimization objective. The smaller band-gap term discourages both near-metallic candidates and overly wide-gap materials with weak solar absorption. The reward therefore favors structures that combine high predicted photovoltaic efficiency with a band gap near the single-junction optimum. The ramp intervals and the resulting reward surfaces for both tasks are provided in the SI. 4.4 Reward-guided fine-tuning on the asymmetric unit Reinforcement-learning fine-tuning shifts the reverse diffusion process toward high-reward crystals by weighting the likelihood of each denoising transition according to the terminal reward. Recent reinforcement-learning approaches to crystal generation optimize either the full crystal representation, including atom types, atomic coordinates, and lattice parameters Chen et al. (2025), or a learned continuous latent representation Park and Walsh (2026). SPARC instead fine-tunes the diffusion process directly in the symmetry-constrained asymmetric-unit representation ℳ′=(′,′,′,)M =( A , F , , k). The generated variables are constrained by the selected space group, and the full crystal is reconstructed from the final asymmetric unit through the space-group operation ℛGR_G. Because the reverse trajectory, policy, and KL anchor are all defined on ℳ′M , symmetry is built into the optimization rather than imposed as a post hoc filter. We retain the reward-weighted objective with an explicit KL anchor introduced for crystal generation in MatInvent Chen et al. (2025). The choice of gradient estimator is independent of the representation, allowing other policy-gradient methods to be used without changing the symmetry constraints imposed during generation. For a sampled space group G and number of representative sites M, the reverse process generates a trajectory ℳT′,…,ℳ0′M _T,…,M _0. As above, we omit the explicit conditioning on (G,M)(G,M) for notational simplicity. At reverse step t, the current noisy asymmetric unit ℳt′M _t is the policy state, and the transition to ℳt−1′M _t-1 is sampled from pθ(ℳt−1′∣ℳt′).p_θ(M _t-1 _t). (15) The terminal sample ℳ0′M _0 is expanded into a full crystal ℳM by applying the operations ℛGR_G. The reward is evaluated only on this reconstructed crystal. Intermediate noisy states are not assigned separate rewards. The reinforcement-learning objective maximizes the expected terminal reward, RL(θ)=pθ(ℳ0′)[r(ℳ)].J_RL(θ)=E_p_θ(M _0) [r(M) ]. (16) Using the reverse trajectory as a policy rollout, the corresponding loss gradient can be written as ∇θℒRL=pθ(ℳ′0:T)[−r(ℳ)∑t=1T∇θlogpθ(ℳt−1′∣ℳt′)]. _θL_RL=E_p_θ(M _0:T) [-r(M) _t=1^T _θ p_θ(M _t-1 _t) ]. (17) The same terminal reward weights every transition along the trajectory that produced the crystal. This is the standard policy-gradient formulation for reinforcement-learning fine-tuning of diffusion models Black et al. (2023); Fan et al. (2023) and follows its application to crystal generation in MatInvent Chen et al. (2025). Optimizing the terminal reward alone can move the policy too far from the pretrained crystal distribution. The model may then generate implausible structures or exploit errors in the property surrogates. We therefore regularize the fine-tuned reverse process against the frozen pretrained process pprep_pre. At each reverse step, we evaluate the KL divergence between the fine-tuned and pretrained transition distributions. The KL contribution for each trajectory is weighted by λ−r(ℳ)λ-r(M), where λ is chosen above the maximum attainable reward. This coefficient remains positive and decreases as the reward increases. Low-reward trajectories are therefore kept close to the pretrained process, while high-reward trajectories are allowed to deviate more strongly. Combining the reward and KL contributions gives the gradient estimator used for fine-tuning, ∇θℒ(θ)=pθ(ℳ′0:T)[ _θL(θ)=E_p_θ(M _0:T) [ −αr(ℳ)∑t=1T∇θlogpθ(ℳ′t−1∣ℳ′t) -α\,r(M) _t=1^T _θ p_θ(M _t-1 _t) (18) +β(λ−r(ℳ))∑t=1T∇θKL(pθ(ℳ′t−1∣ℳ′t)∥ppre(ℳ′t−1∣ℳ′t))], +β (λ-r(M) ) _t=1^T _θ\,KL (p_θ(M _t-1 _t)\,\|\,p_pre(M _t-1 _t) ) ], where α and β control the reward and KL contributions, respectively. The sampled reward is treated as a fixed weight during each parameter update. The derivation of the policy-gradient term and the data-processing-inequality bound that replaces the intractable terminal-distribution KL with a sum of per-step conditional KL terms are provided in the SI. Because the full crystal ℳM is reconstructed deterministically from ℳ0′M _0 and ℛGR_G, its terminal reward provides a learning signal for the asymmetric-unit trajectory that generated it. The policy update therefore increases the probability of asymmetric units whose reconstructed crystals achieve high rewards, while the KL term limits unsupported deviations from the pretrained generator. The space-group proposal is updated separately across reinforcement-learning rounds using the outer-loop procedure described next. 4.5 Adaptive space-group sampling The reward-weighted policy gradient updates the conditional generator pθ(′,′,′,∣G,M)p_θ( A , F , , k G,M) but does not directly update the proposal distribution over G. At the beginning of each trajectory, a space group G is sampled from the current proposal and the number of representative sites M is sampled from pmarg(M∣G)p_marg(M G). Both variables remain fixed throughout the T reverse-diffusion steps. Although G is provided to the denoising model as a conditioning variable, it is sampled outside the diffusion policy and is not parameterized by θ. The policy-gradient update therefore changes the generated distribution conditioned on (G,M)(G,M) but not the probability of selecting each space group. We address this limitation using a discrete outer-loop update of the space-group proposal after each RL round. Let π(0)π^(0) denote the base proposal over the supported space groups. It is obtained from the empirical MP-20 marginal using an optional temperature reweighting Kirkpatrick et al. (1983), π(0)(G)=pmarg(G)1/Tsg∑G′∈pmarg(G′)1/Tsg,π^(0)(G)= p_marg(G)^1/T_sg _G p_marg(G )^1/T_sg, (19) where =G:pmarg(G)>0S=\G:p_marg(G)>0\ is the support of the training distribution. The first RL round samples G from π(0)π^(0), while subsequent rounds use the updated proposal defined below. For each generated crystal, we determine the realized space group from the reconstructed full structure using SpacegroupAnalyzer with symprec=0.01 symprec=0.01 Å. The detected label can differ from the conditioning label because of accidental higher symmetry or small numerical deviations in the reconstructed structure. We associate the terminal reward with this realized space-group label. Let r¯(j)(G) r^(j)(G) denote the mean terminal reward of structures assigned to space group G during RL round j. We update a running reward estimate for each space group using μ(j)(G)=γμ(j−1)(G)+(1−γ)r¯(j)(G),Gobserved in round j,γμ(j−1)(G),Gunobserved in round j,μ^(j)(G)= casesγ\,μ^(j-1)(G)+(1-γ)\, r^(j)(G),&G\ observed in round j,\\ γ\,μ^(j-1)(G),&G\ unobserved in round j, cases (20) where γ∈(0,1)γ∈(0,1) controls the decay of previous reward estimates. The second branch gradually removes stale reward information for space groups that are not observed in the current round. The running reward estimates are used to reweight the base proposal, π~(j)(G)=π(0)(G)exp(κμ(j)(G))∑H∈π(0)(H)exp(κμ(j)(H)), π^(j)(G)= π^(0)(G) (κμ^(j)(G) ) _H π^(0)(H) (κμ^(j)(H) ), (21) where κ controls the sensitivity of the proposal to the accumulated reward. We then mix the reward-reweighted distribution with the base proposal, π(j)(G)=ρπ(0)(G)+(1−ρ)π~(j)(G),π^(j)(G)=ρ\,π^(0)(G)+(1-ρ)\, π^(j)(G), (22) where ρ∈[0,1]ρ∈[0,1] controls the strength of the base-distribution anchor. A minimum-probability floor is subsequently applied to every G∈G , followed by renormalization. Both terms in the mixture retain the base proposal π(0)π^(0). The adaptive component favors space groups with larger running rewards, while the anchored component continually restores probability mass across the original support. The probability floor provides an additional safeguard against eliminating rarely sampled space groups. Together, these terms limit premature concentration of the proposal and preserve exploration throughout fine-tuning. We use different base proposals for the two design tasks, π(0)(G)∝pmarg(G)1/Tsg.π^(0)(G) p_marg(G)^1/T_sg. (23) For the uniaxial dielectric objective, we use Tsg=100T_sg=100, γ=0.9γ=0.9, κ=5.0κ=5.0, and ρ=0.15ρ=0.15. The large temperature produces a nearly uniform base proposal over S, reducing the influence of the MP-20 space-group frequencies for this strongly symmetry-dependent objective. For the SLME objective, we use Tsg=1T_sg=1, γ=0.9γ=0.9, κ=3.0κ=3.0, and ρ=0.1ρ=0.1. This choice retains the empirical MP-20 marginal as the base proposal because SLME is not restricted to a narrow set of crystal symmetries. 4.6 Training and fine-tuning details SPARC uses the SymmCD generator Levy et al. () as its diffusion backbone. The generator was pretrained on the MP-20 dataset. We initialize the policy pθp_θ from the released checkpoint and retain a frozen copy as the KL reference policy pprep_pre. Only the reward-guided fine-tuning stage is performed in this work. The original diffusion process is retained, with T=1000T=1000 reverse-diffusion steps. Both objectives are fine-tuned for 120 RL rounds using an initial learning rate of 3×10−53× 10^-5. The learning rate remains constant for the dielectric-anisotropy objective and is multiplied by 0.980.98 after each round for the SLME objective. The policy-gradient coefficient is set to α=1α=1, and the reward-anchor threshold is set to λ=1.1λ=1.1. The KL coefficient is initialized as β0=0.025 _0=0.025 and decays with the fine-tuning update index ν according to βν=β01+ν/50. _ν= _0 1+ν/50. (24) Complete optimization, diffusion, and sampling settings are provided in SI. Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request. The datasets and trained model weights will be deposited in a publicly accessible repository upon acceptance of the manuscript. Code Availability The code that supports the findings of this study is available from the corresponding author upon reasonable request. It will be released in a publicly accessible repository upon acceptance of the manuscript. Declarations Acknowledgements T.-W.H. and Q.Y. acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0023664. T.-W.H. and A.B. acknowledge support from the U.S. National Science Foundation through the Expanding Capacity in Quantum Information Science and Engineering (ExpandQISE) program under Award No. OMA-2329067. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231, through NERSC Award No. BES-ERCAP0029544. This work also benefited from Northeastern University’s Quantum Materials and Sensing Institute (QMSI). Author Contribution T.-W.H. and Q.Y. conceived the study. T.-W.H. developed the methodology, performed the calculations, trained the models, analyzed the results, and prepared the figures. Q.Y. and A.B. supervised the project. 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External Links: Document, Link Cited by: §3. Supplementary Information for Symmetry- and Property-Aware Crystal Generation with Reinforcement Learning for Inverse Materials Design 1 Reinforcement-learning experimental setup This section provides the complete experimental workflow and optimization settings used for reinforcement-learning (RL) fine-tuning. SPARC uses the publicly released SymmCD model Levy et al. (), pretrained on MP-20, as the initial generative model and RL policy. A frozen copy of the same checkpoint is retained as the reference policy pprep_pre for Kullback–Leibler (KL) regularization. Both design campaigns are run for 120120 RL rounds. Each round consists of the following steps. 1. For each generation trajectory, a space group G is sampled from the current proposal distribution π(j)(G)π^(j)(G), and the number of representative asymmetric-unit sites M is sampled from pmarg(M∣G)p_marg(M G). The initial proposal π(0)(G)π^(0)(G) is obtained from the MP-20 space-group marginal using the task-specific temperature transformation described in the main text. The proposal over G is updated between RL rounds, while pmarg(M∣G)p_marg(M G) remains fixed. 2. Conditioned on the sampled pair (G,M)(G,M), the diffusion model generates asymmetric-unit candidates ℳ′M through the reverse denoising process. The uniaxial dielectric task generates 9696 candidates per round, while the spectroscopic limited maximum efficiency (SLME) task generates 128128 candidates per round. Each asymmetric-unit candidate is expanded using the space-group reconstruction operator ℛGR_G to obtain the corresponding full crystal, ℳ=ℛG(ℳ′).M=R_G(M ). (1) Generation and policy updates are performed on the compact asymmetric-unit representation. Relaxation, filtering, property prediction, and reward evaluation are performed on the reconstructed full crystal. 3. The reconstructed structures are relaxed using the MatterSim-v1.0.0-5M machine-learning interatomic potential (MLIP) Yang et al. (2024). Relaxation is performed with the FIRE optimizer under full variable-cell relaxation, with the atomic positions and lattice optimized jointly using an exponential cell filter. We use a force-convergence threshold of fmax=0.05 eV Å−1f_ =$0.05\,eV\, ^-1$ and a maximum of 50005000 optimization steps. 4. The relaxed structures are first checked for structural validity and then evaluated using the stable, unique, and novel (S.U.N.) criteria following MatterGen Zeni et al. (2025). A structure is considered stable when its MLIP-estimated energy above the full Materials Project convex hull satisfies Ehull≤0.10 eV/atomE_hull≤$0.10\,eV$/atom Jain et al. (2013). Uniqueness is evaluated by comparing the generated structures with one another using the pymatgen StructureMatcher. Novelty is evaluated by comparing the generated structures with the Materials Project reference set using the same matcher. Only structurally valid candidates satisfying all three S.U.N. criteria proceed to property evaluation. 5. Up to 3030 surviving structures are evaluated by the fixed property surrogates in each RL round. When more than 3030 structures pass the filters, 3030 are selected for surrogate evaluation. Otherwise, all surviving structures are evaluated. Each scored structure is assigned the task-specific terminal reward defined in the main Methods. These are runir_uni for the uniaxial dielectric task and rSLMEr_SLME for the photovoltaic task. 6. The terminal reward r(ℛG(ℳ0′))r(R_G(M _0)) is assigned to the asymmetric unit that generated the relaxed full crystal. For each policy update, a diffusion time t is sampled and the clean asymmetric unit ℳ0′M _0 is forward-noised to ℳt′M _t. The policy is then updated using the reward-weighted denoising loss together with the KL penalty relative to pprep_pre. The noising and denoising operations act on the asymmetric-unit variables ′ A , ′ F , ′ , and k. The full crystal is recovered through ℛGR_G, allowing the policy to improve the target reward while preserving compatibility with the conditioning space group. 7. After reward evaluation and policy fine-tuning, the realized space group of each reconstructed structure is determined using SpacegroupAnalyzer with symprec=0.01 symprec=0.01 Å. The rewards are grouped by the realized space-group labels and used to update π(j)(G)π^(j)(G) according to the outer-loop rule defined in the main text. Space groups that repeatedly yield high-reward structures receive increased proposal probability. The base-distribution anchor and minimum-probability floor retain sampling probability across the original support. The outer-loop parameters are the proposal temperature TsgT_sg, the reward-decay factor γ, the tilt strength κ, and the base-proposal mixture ρ, set to Tsg=100T_sg=100, γ=0.9γ=0.9, κ=5.0κ=5.0, ρ=0.15ρ=0.15 for the dielectric task and Tsg=1T_sg=1, γ=0.9γ=0.9, κ=3.0κ=3.0, ρ=0.1ρ=0.1 for the SLME task. The updated proposal is used in the next RL round. The filtering stage can substantially reduce the number of structures available for property evaluation. Candidates may be removed because of invalid geometries, insufficient stability, duplication within the generated population, or matches to known Materials Project structures. We therefore generate more structures than are used in each policy update. The dielectric task generates 9696 candidates per round, while the SLME task generates 128128. At most 3030 surviving structures are evaluated and used for fine-tuning in each round. Over 120120 rounds, the SLME campaign generated 15,36015,360 structures. Of these, 4,8254,825 passed the validity checks and 3,5223,522 entered the capped per-round evaluation pool and were scored by the property surrogates. To improve sample efficiency, high-reward structures from previous rounds are reused through an experience-replay buffer. The buffer stores up to 100100 structures, and 1010 stored structures are replayed in each RL round. Only structures with reward r≥0.1r≥ 0.1 are eligible to enter or be sampled from the buffer. A diversity filter is also applied to reduce the influence of near-duplicate candidates. The filter uses the implementation parameters of tolerance 33 and buffer size 66. Fine-tuning retains the diffusion settings of the pretrained SymmCD model. The diffusion process contains T=1000T=1000 steps. A cosine variance schedule with offset s=0.008s=0.008 is used for the lattice coefficients k and the discrete atom-type and site-symmetry variables. A geometric noise-scale schedule from σmin=0.005 _ =0.005 to σmax=0.5 _ =0.5 is used for the fractional coordinates. The initial learning rate is 3×10−53× 10^-5 for both tasks. It remains constant during dielectric fine-tuning and is multiplied by 0.980.98 after each RL round for the SLME task. The KL coefficient is initialized at β0=0.025 _0=0.025 and decays with the fine-tuning update index ν according to βν=β01+ν/50. _ν= _0 1+ν/50. (2) The policy-gradient coefficient is α=1α=1, and the reward-anchor threshold is λ=1.1λ=1.1. Each RL round contains three inner fine-tuning epochs. The training mini-batch size is 3030, and gradients are accumulated over 5050 steps before each optimizer update. The property surrogates remain frozen throughout RL fine-tuning. The uniaxial dielectric reward uses the band-gap and dielectric-tensor surrogates, while the SLME reward uses the band-gap and spectral surrogates. Their architectures, training datasets, and predictive benchmarks are described separately in Section 4. Supplementary Table 1: Reinforcement-learning optimization settings. Shared and task-specific settings used for SPARC fine-tuning. Both campaigns were run for 120120 RL rounds. Setting Shared Dielectric RL SLME RL Initial policy SymmCD MP-20 checkpoint – – Reference policy Frozen initial checkpoint – – Optimizer Adam – – Initial learning rate – 3×10−53× 10^-5 3×10−53× 10^-5 Learning-rate multiplier per round – 1.01.0 0.980.98 Weight decay 00 – – Diffusion steps T 10001000 – – Cosine-schedule offset s 0.0080.008 – – Coordinate noise range σmin=0.005 _ =0.005, σmax=0.5 _ =0.5 – – Denoising loss weights λ=5 _ k=5, λ′=1 _ F =1, λ′=0.1 _ A =0.1, λ′=10 _ =10 – – Training mini-batch size 3030 – – Gradient accumulation steps 5050 – – Inner fine-tuning epochs per round 33 – – RL rounds 120120 – – Generated candidates per round – 9696 128128 Total generated candidates – 11,52011,520 15,36015,360 Scored candidates per round 3030 – – Policy-gradient coefficient α 11 – – Initial KL coefficient β0 _0 0.0250.025 – – KL schedule βν=β0/1+ν/50 _ν= _0/ 1+ν/50 – – Reward-anchor threshold λ 1.11.1 – – Replay-buffer capacity 100100 – – Replayed structures per round 1010 – – Replay reward threshold 0.10.1 – – 2 Symmetry-constrained generation SPARC inherits its symmetry-constrained generation from SymmCD Levy et al. (). The central idea is to diffuse not only over atom types, fractional coordinates, and the lattice, but also over the site symmetries of the representative atoms, conditioned on the space group G. Because the asymmetric unit is generated together with its site-symmetry assignments, applying the operations of G regenerates a crystal that respects G by construction, rather than one whose symmetry must be recovered after generation. This decomposition is illustrated on a two-dimensional toy example in Supplementary Figure 1. Supplementary Figure 1: Symmetry-constrained decomposition of a crystal. On a two-dimensional toy example (wallpaper group p4mmp4m, fictitious species X, Y, Z). (a) The asymmetric unit, comprising the representative atoms with their fractional coordinates, species, and site symmetries (X on a 4mm4m 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. The symmetry enters the generator through a data-driven prior. For a given space group G we aggregate the site-symmetry assignments of every training material in G to form the site-symmetry marginal pmarg(′∣G)p_marg( G). For example, G=225G=225 (Fm3¯mFm 3m), the marginal pmarg(′∣G=225)p_marg( G=225) is naturally displayed as a 15×1315× 13 heatmap over the 1515 crystallographic axes and 1313 site-symmetry categories, with color encoding probability (Supplementary Figure 2). Supplementary Figure 2: Site-symmetry marginal for space group 225 (Fm3¯mFm 3m). Empirical site-symmetry marginal pmarg(′∣G=225)p_marg( G=225), aggregated over all training materials in Fm3¯mFm 3m and shown as a 15×1315× 13 heatmap over the 1515 crystallographic axes and 1313 site-symmetry categories. Color encodes probability. The site-symmetry representation ′ is treated as a collection of categorical variables. For each representative atom i, the site symmetry is represented by 15 one-hot vectors i,u′∈0,113 _i,u∈\0,1\^13, where u∈1,…,15u∈\1,…,15\ indexes the crystallographic symmetry axis and the 13 categories correspond to the possible site-symmetry operations along that axis. We apply discrete diffusion independently to each axis. Given the initial site-symmetry representation 0,i,u′ _0,i,u and space group G, the noising process is defined as q(t,i,u′∣0,i,u′,G)=Cat(t,i,u′;=0,i,u′¯t,u,G⊤),q( _t,i,u _0,i,u,G)=Cat ( _t,i,u; p= _0,i,u Q_t,u,G ), with the cumulative transition matrix ¯t,u,G=∏τ=1tτ,u,G,t,u,G=αt+βtΣ,u,G⊤. Q_t,u,G= _τ=1^t Q_τ,u,G, Q_t,u,G= _t I+ _t1 m_ ,u,G . Here, Σ,u,G m_ ,u,G denotes the empirical marginal distribution of site-symmetry operations for axis u within space group G, so the forward process drives each axis from its clean assignment toward this space-group-specific marginal (Supplementary Figure 2). The denoising network predicts the clean site-symmetry distribution ^0′=ϕΣ(ℳt′,t,G) _0= _ (M _t,t,G) and is trained using the averaged cross-entropy loss ℒΣ′=t∼(1,T)t′∼q(t′∣0′,G)[115M∑i=1M∑u=115CrossEntropy(0,i,u′,^0,i,u′)].L_ =E_t (1,T)E_ _t q( _t _0,G) [ 115M _i=1^M _u=1^15CrossEntropy ( _0,i,u, _0,i,u ) ]. where M is the number of representative atoms in the asymmetric unit. Supplementary Figure 3 makes this forward process concrete for a single crystallographic axis of space group 225. Panel (a) showing the initial prior is the space-group marginal Σ,u,G m_ ,u,G at axis u=1u=1, which for this axis places its mass on only site-symmetry classes number 7, 8, and 12. Panel (b) showing the noise schedule drives the cumulative weight α¯t α_t from 11 to 00 over the trajectory, interpolating from the clean assignment at t=0t=0 toward this marginal at t=Tt=T. Panels (c) and (d) trace the resulting relaxation. Panel (c) shows that the self-transition probability ¯t[i,i] Q_t[i,i] decays from 1 as t increases, so an initially one-hot class progressively transfers probability to other allowed classes. Panel (d) shows the full row distribution i⊤¯t e_i Q_t for a representative source class, i=12i=12. As time progresses, this distribution evolves from a sharp peak at the initial class toward the space-group marginal, with probability redistributed only among the symmetry classes supported by the marginal. Because this absorbing distribution is space-group-specific, the forward noise process is itself symmetry-aware. During sampling, the reverse denoising model learns to invert this process, mapping noisy site-symmetry assignments back toward configurations compatible with the target space group G. Thus, the model samples within a symmetry-conditioned space rather than an unconstrained categorical space. Results in the generated structures preserve local site-symmetry assignments consistent with the prescribed global symmetry. Supplementary Figure 3: Discrete-diffusion transition toward the site-symmetry marginal (space group 225, Fm3¯mFm 3m, axis u=1u=1). The forward process for the site-symmetry labels along one crystallographic axis, shown through the cumulative transition ¯t,u,G Q_t,u,G at diffusion steps t=0,250,500,750,1000t=0,250,500,750,1000. (a) The space-group marginal Σ,u,G m_ ,u,G over the 1313 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 αt _t and the cumulative α¯t α_t over diffusion time, with the five sampled timesteps marked; α¯t α_t decreases from 11 to 00 as the clean signal is replaced by the marginal. (c) The self-transition probability ¯t[i,i] Q_t[i,i], the probability of remaining in the initial class i, versus t for three representative classes; each decays from 11 toward its marginal weight. (d) The row distribution i⊤¯t e_i Q_t for source class i=12i=12 at the five timesteps, showing how probability initially localized on class 1212 spreads toward the marginal as t grows. 2.1 Lattice parameterization and space-group constraints SPARC inherits its lattice parameterization from DiffCSP++ Jiao et al. (2024). Rather than denoising the raw 3×33× 3 lattice matrix L, we use the polar decomposition =exp() L= Q ( S), where ∈SO(3) Q∈ SO(3) is orthogonal, =⊤ S= S is symmetric, and exp()=∑n=0∞n/n! ( S)= _n=0^∞ S^n/n! is the matrix exponential. Any rigid rotation of L is absorbed entirely by Q and leaves S unchanged, so S furnishes a rotation-invariant description of the lattice. Every symmetric S expands uniquely in a fixed six-dimensional symmetric basis, =∑i=16kii,=(k1,…,k6)∈ℝ6, S= _i=1^6k_i B_i, k=(k_1,…,k_6) ^6, (3) with basis matrices 1=(010100000), B_1= ( array[]l0&1&0\\ 1&0&0\\ 0&0&0 array ), 2=(001000100),3=(000001010), B_2= ( array[]l0&0&1\\ 0&0&0\\ 1&0&0 array ), B_3= ( array[]l0&0&0\\ 0&0&1\\ 0&1&0 array ), (4) 4=(1000−10000), B_4= ( array[]c1&0&0\\ 0&-1&0\\ 0&0&0 array ), 5=(10001000−2),6=(100010001). B_5= ( array[]c1&0&0\\ 0&1&0\\ 0&0&-2 array ), B_6= ( array[]l1&0&0\\ 0&1&0\\ 0&0&1 array ). For a chosen space group G, the crystal family fixes a subset of the kik_i while the rest remain free, as listed in Supplementary Table 2. During generation we hold the constrained coefficients at their required values through a mask ∈0,16 m∈\0,1\^6 and denoise only the free entries, guaranteeing that every generated lattice is consistent with G. Supplementary Table 2: Relationship between the lattice shape and the constraint on the symmetric-basis coefficients k, where a,b,ca,b,c and α,β,γα,β,γ denote the lengths and angles of the lattice bases, respectively (after DiffCSP++ Jiao et al. (2024)). Crystal Family Space Group No. Lattice Shape Constraint of Symmetric Bases Triclinic 1∼21 2 No Constraint No Constraint Monoclinic 3∼153 15 α=γ=90∘α=γ=90 k1=k3=0k_1=k_3=0 Orthorhombic 16∼7416 74 α=β=γ=90∘α=β=γ=90 k1=k2=k3=0k_1=k_2=k_3=0 Tetragonal 75∼14275 142 α=β=γ=90∘α=β=γ=90 a=ba=b k1=k2=k3=0k_1=k_2=k_3=0 k4=0k_4=0 Trigonal/Hexagonal 143∼194143 194 α=β=90∘,γ=120∘α=β=90 ,\ γ=120 a=ba=b k2=k3=0,k1=−log(3)/4k_2=k_3=0,\ k_1=- (3)/4 k4=0k_4=0 Cubic 195∼230195 230 α=β=γ=90∘α=β=γ=90 a=b=ca=b=c k1=k2=k3=0k_1=k_2=k_3=0 k4=k5=0k_4=k_5=0 2.2 Denoising losses Following SymmCD Levy et al. (), the denoiser reconstructs the four asymmetric-unit components jointly, minimizing ℒ=λℒ+λ′ℒ′+λ′ℒ′+λ′ℒ′L= _ kL_ k+ _ F L_ F + _ A L_ A + _ L_ , with the weights listed in Supplementary Table 1. The fractional coordinates use a denoising score-matching loss, ℒ′=t′∼q′(t′∣0′),t∼(1,T)[λt‖∇t′logq′(t′∣0′)−ϵ^′(ℳt′,t)‖22],L_ F =E_ F_t q ( F_t F_0 ),t (1,T) [ _t \| _ F_t q ( F_t F_0 )- ε_ F (M_t ,t ) \|_2^2 ], (5) the lattice coefficients a masked ϵ ε-prediction loss, in which the mask m zeroes the symmetry-constrained entries, ℒ=ϵ∼(0,),t∼(1,T)[‖⊙ϵ−ϵ^(ℳt′,t)‖22],L_ k=E_ ε_ k (0, I),t (1,T) [ \| m ε_ k- ε_ k (M_t ,t ) \|_2^2 ], (6) and the atom types a categorical cross-entropy under the discrete-diffusion transition ¯t Q_t, ℒ′=t∼Cat(0⊤¯t),t∼(1,T)∑i=1MCrossEntropy(i′,^i′).L_ A =E_ a_t ( a_0 Q_t ),t (1,T) _i=1^MCrossEntropy ( a_i , a_i ). (7) The site-symmetry loss ℒ′L_ has the same cross-entropy form and is defined earlier in this section. 2.3 Denoising-network architecture SPARC retains the denoising architecture of the released SymmCD model Levy et al. (), which is adapted from the graph neural network used in DiffCSP Jiao et al. (2023). The network operates on a fully connected graph whose M nodes correspond to the representative sites in the asymmetric unit. Following SymmCD, the denoiser is not explicitly rotation-equivariant because the conventional unit-cell axes define a canonical reference frame. Periodic translation invariance is incorporated through Fourier features of the relative fractional coordinates. For representative site i, the initial node feature i h_i is constructed from its atom type i a_i, fractional coordinate i x_i, site-symmetry descriptor i′ _i, the space group G, and the diffusion timestep t. The timestep is encoded using a sinusoidal embedding ψt(t) _t(t), while the space group is embedded using a multilayer perceptron (MLP) ϕG _G. The site-symmetry descriptor is embedded separately along each of the 15 crystallographic axes using a shared MLP ϕU _U, and the resulting axis-wise embeddings are combined using a second MLP ϕS _S, i←ϕh(i,i,ϕS(⨁u=115ϕU(i,u′)),ϕG(G),ψt(t)). h_i← _h ( a_i, x_i, _S ( _u=1^15 _U ( _i,u ) ), _G(G), _t(t) ). (8) At each message-passing layer, the message from representative j to representative i is computed as ij=ϕm(i,j,,ψx(i−j)), m_ij= _m ( h_i, h_j, k, _x ( x_i- x_j ) ), (9) where ψx _x is a Fourier embedding of the relative fractional coordinate. The node features are then updated by aggregating the incoming messages, i←i+ϕn(i,∑j=1Mij). h_i← h_i+ _n ( h_i, _j=1^M m_ij ). (10) The edge and node networks ϕm _m and ϕn _n are MLPs with SiLU activations, and layer normalization is applied at each message-passing layer. After the final layer, node-level output heads predict the fractional-coordinate denoising term, atom-type logits, and site-symmetry logits. The site-symmetry output contains one categorical distribution over 13 possible operations for each of the 15 crystallographic axes. The lattice denoising term is predicted from the aggregated graph representation, graph=∑i=1Mi. h_graph= _i=1^M h_i. (11) The resulting outputs are denoted by ϵ^′,^′,^′,ϵ ε_ F , A , , ε_ k. The architecture and diffusion hyperparameters are summarized in Supplementary Table 3. Supplementary Table 3: Denoising-network architecture hyperparameters (inherited from the SymmCD MP-20 model). 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 Normalization LayerNorm (each layer) Lattice parameterization ∈ℝ6 k ^6 Site-symmetry dimension 15×13=19515× 13=195 Atom-type 94 Prediction heads (linear) k (6), ′ F (3), ′ A (94), ′ (195) Parameters 61,010,350 3 Reinforcement-learning fine-tuning objective This section reproduces the reward-weighted KL objective of MatInvent Chen et al. (2025), see also the reinforcement-learning treatments of diffusion samplers in Black et al. (2023); Fan et al. (2023). In SPARC the diffusion policy acts on the asymmetric unit rather than the full crystal, so we write the objective directly in the asymmetric-unit representation ℳ′=(′,′,′,)M =( A , F , , k). Given a space group G, a generated asymmetric unit is decoded into a full crystal by the deterministic space-group replication map ℛGR_G, ℳ=ℛG(ℳ′)M=R_G(M ), and the reward is evaluated on this decoded crystal. Because ℛGR_G is fixed and non-learned, a reward on ℳM acts as a reward on the ℳ′M that generated it, and the entire trajectory, policy, and regularizer below live in ℳ′M -space (all conditioned on G, which we suppress in the notation for brevity). We treat the length-T reverse denoising trajectory ℳ′0:TM _0:T as a Markov decision process whose actions are the per-step transitions pθ(ℳt−1′∣ℳt′)p_θ(M _t-1 _t), and assign the terminal reward r(ℛG(ℳ0′))r(R_G(M _0)) to the decoded crystal. The fine-tuning objective is minθpθ(ℳ0′)[−r(ℛG(ℳ0′))]. _θE_p_θ(M _0) [-r(R_G(M _0)) ]. (12) Since logpθ(ℳ0′) p_θ(M _0) is intractable, we differentiate through the trajectory likelihood using the score-function identity ∇θpθ=pθ∇θlogpθ _θp_θ=p_θ\, _θ p_θ, giving the policy gradient ∇θpθ(ℳ0′)[−r(ℛG(ℳ0′))]=pθ(ℳ′0:T)[r(ℛG(ℳ0′))∑t=1T(−∇θlogpθ(ℳt−1′∣ℳt′))], _θ\,E_p_θ(M _0) [-r(R_G(M _0)) ]=E_p_θ(M _0:T) [r(R_G(M _0)) _t=1^T (- _θ p_θ(M _t-1 _t) ) ], (13) i.e. the per-step denoising-loss gradient weighted by the terminal reward. To keep the fine-tuned policy near the pretrained generator we regularize the marginal divergence KL(pθ(ℳ0′)∥ppre(ℳ0′))KL(p_θ(M _0)\,\|\,p_pre(M _0)). This marginal is intractable, but the data-processing inequality bounds it by the divergence of the full trajectories, KL(pθ(ℳ0′)∥ppre(ℳ0′))≤KL(pθ(ℳ0:T′)∥ppre(ℳ0:T′)),KL (p_θ(M _0)\,\|\,p_pre(M _0) ) (p_θ(M _0:T)\,\|\,p_pre(M _0:T) ), (14) and the Markov factorization of the reverse process reduces the right-hand side to a sum of per-step KL divergences, KL(pθ(ℳ0:T′)∥ppre(ℳ0:T′))=∑t=1Tpθ(ℳt′)KL(pθ(ℳt−1′∣ℳt′)∥ppre(ℳt−1′∣ℳt′)),KL (p_θ(M _0:T)\,\|\,p_pre(M _0:T) )= _t=1^TE_p_θ(M _t)\,KL (p_θ(M _t-1 _t)\,\|\,p_pre(M _t-1 _t) ), (15) where the shared terminal prior pθ(ℳT′)=ppre(ℳT′)p_θ(M _T)=p_pre(M _T) eliminates the boundary term. Weighting each trajectory’s per-step KL by (λ−r(ℛG(ℳ0′)))(λ-r(R_G(M _0))) yields the reward-weighted regularizer pθ(ℳ′0:T)[(λ−r(ℛG(ℳ0′)))∑t=1TKL(pθ(ℳt−1′∣ℳt′)∥ppre(ℳt−1′∣ℳt′))],E_p_θ(M _0:T) [ (λ-r(R_G(M _0)) ) _t=1^TKL (p_θ(M _t-1 _t)\,\|\,p_pre(M _t-1 _t) ) ], (16) where λ is chosen above the maximum reward so that λ−r(ℛG(ℳ0′))λ-r(R_G(M _0)) remains positive. The KL penalty decreases as the reward increases. High-reward trajectories are therefore allowed to deviate more from pprep_pre, whereas low-reward trajectories remain more strongly constrained. Combining this regularization term with the policy-gradient contribution using weights α,β>0α,β>0 gives the SPARC training gradient reported in the main text. 4 Property surrogate models This section documents the three property surrogates used as reward models in the inverse-design loop, a band-gap EgE_g predictor, an SLME η predictor, and a static-limit dielectric-tensor predictor (hereafter the dielectric-tensor surrogate). All three use the same Tensorial Spectra Equivariant Neural Network (TSENN) Hsu et al. (2026) backbone but differ in the output irreducible representations (irreps), training dataset, and task-specific hyperparameters. The output irreps are chosen to match the symmetry of each target property. Although pretrained weights are publicly available for some related tasks, we retrain a TSENN model from scratch on every task so that all three surrogates share a common e3n-based backbone and a consistent evaluation protocol, allowing us to swap target properties by changing only the output head, without maintaining a separate model definition per objective. The trained surrogate weights are released alongside the code repository. 4.1 Band gap For the band gap, the output head is a single scalar irrep (1x0e), so the network predicts the density functional theory (DFT) band gap EgE_g as one real number per material. Dataset. The training data are drawn from the Materials Project dataset Jain et al. (2013), which contains 151,330151,330 entries. The full dataset is not uniformly labeled for every target property, particularly, band-gap labels are available only for a subset of the structures. We therefore restrict the band-gap surrogate to entries with valid band-gap targets. The band-gap labels were obtained from VASP calculations using Materials Project-consistent Perdew–Burke–Ernzerhof (PBE) +U+U settings. Full details of the underlying DFT protocols are provided in the original dataset references. As these are semilocal PBE(+U) gaps, they underestimate experimental band gaps as expected at this level of theory, and the surrogate predicts on the same PBE scale. Train/validation/test splits. We stratified the dataset into an 8:1:1 ratio, resulting in 121,065121,065 for training, 15,13815,138 for validation, and 15,12715,127 for testing. Training. Learning rate 1.5×10−31.5× 10^-3, weight decay 5×10−25× 10^-2; remaining hyperparameters in Supplementary Table 4. Supplementary Table 4: Hyperparameters for the TSENN band-gap surrogate. Hyperparameter Value Maximum spherical harmonic order ℓmax _ 2 Embedding dimension 128 Number of layers 4 Channel multiplicity 16 Batch size 128 The surrogate attains R2=0.902R^2=0.902 with mean absolute error MAE=0.223eVMAE=0.223\,eV on the training set, R2=0.869R^2=0.869 with MAE=0.265eVMAE=0.265\,eV on the validation set, and R2=0.881R^2=0.881 with MAE=0.252eVMAE=0.252\,eV on the held-out test set. The corresponding parity plots are shown in Supplementary Figure 4, where the train, validation, and test predictions closely follow the y=xy=x reference line. The close agreement between validation and test metrics, together with the high R2R^2 across this chemically diverse dataset, indicates that the surrogate provides a reliable held-out estimate and is suitable for scoring generated candidates within the chemical regime represented by the training data. Supplementary Figure 4: Predicted versus DFT-computed band gap. Parity plots of the predicted band gap EgE_g against the DFT reference, shown for the train, validation, and test splits (left to right). The dashed line indicates perfect agreement (y=xy=x). On the held-out test set, the model achieves R2=0.881R^2=0.881 with MAE=0.252eVMAE=0.252\,eV. 4.2 Spectroscopic limited maximum efficiency The output head is a sequence of scalar irreps: 201×0e201× 0e, predicting the frequency-dependent trace of the imaginary part of the dielectric tensor, ε2(ω)=13Tr2,ij(ω) _2(ω)= 13Tr _2,ij(ω), on a fixed frequency grid from 00 to 20 eV20\,eV with spacing ΔE=0.1 eV E=$0.1\,eV$. The corresponding real part ε1(ω) _1(ω) is recovered from the predicted ε2(ω) _2(ω) via the Kramers–Kronig (K) relations, and the absorption coefficient is obtained from the standard expression α(ω)=2ωc[ε12(ω)+ε22(ω)−ε1(ω)]1/2.α(ω)= 2\,ωc [ _1^2(ω)+ _2^2(ω)- _1(ω) ]^1/2. (17) Given α(E)α(E), the SLME η is evaluated at a fixed film thickness Labs=0.3 µmL_abs=$0.3\, m$, device temperature Tcell=300 KT_cell=$300\,K$, and reference solar spectrum (AM1.5G). Dataset. The training data are drawn from OptiMate Grunert et al. (2024), a large-scale DFT dataset of dielectric spectra comprising 21,06421,064 materials sampled from the Alexandria database Schmidt et al. (2023). The underlying ground-state calculations were performed with Quantum ESPRESSO Giannozzi et al. (2009), and the frequency-dependent dielectric spectra were obtained with Yambo Sangalli et al. (2019). Train/validation/test splits. Following the protocol of the original TSENN study Hsu et al. (2026), we partition the data into 16,84816,848 training, 2,1062,106 validation, and 2,1102,110 test materials — an 80/10/1080/10/10 split stratified by crystal system, so that the symmetry distribution is preserved across the three splits. Training. Learning rate 1×10−21× 10^-2, weight decay 5×10−25× 10^-2; remaining hyperparameters in Supplementary Table 5. Supplementary Table 5: Hyperparameters for the TSENN SLME surrogate. Hyperparameter Value Maximum spherical harmonic order ℓmax _ 2 Embedding dimension 128 Number of layers 4 Channel multiplicity 64 Batch size 24 The parity plots in Supplementary Figure 5 compare the predicted SLME values with those computed for the train, validation, and test splits. The surrogate attains R2=0.943R^2=0.943, MAE=1.15%MAE=1.15\% on the training sets, R2=0.921R^2=0.921, MAE=1.37%MAE=1.37\% on the validation sets, and R2=0.925R^2=0.925, MAE=1.35%MAE=1.35\% on the test sets. The high R2R^2 on the held-out test set indicates that the predictor remains accurate on held-out materials, which is the most relevant benchmark for inverse design because generated candidates are not part of the training set. Supplementary Figure 5: Predicted versus DFT-computed SLME. Parity plots of the predicted SLME η 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 (y=xy=x). On the held-out test set, the model achieves R2=0.925R^2=0.925 with MAE=1.35%MAE=1.35\% (absolute efficiency percentage points). 4.3 Static-limit dielectric tensor The output head emits two irreps of O(3)O(3): a scalar (1x0e) and a symmetric rank-2 traceless tensor (1x2e), giving 1+5=61+5=6 independent coefficients. These coefficients are mapped back to the Cartesian symmetric dielectric tensor εij _ij via the spherical-to-Cartesian transformation, recovering the six independent entries εxx,εyy,εzz,εxy,εxz,εyz\ _x, _y, _z, _xy, _xz, _yz\. This decomposition separates the isotropic trace from the anisotropic traceless response. This is important because the diagonal entries are typically much larger than the off-diagonal entries; separating the two scales prevents the loss from being dominated by the diagonal components Hsu et al. (2026). Dataset. The training data comprise 6,7066,706 materials drawn from the Materials Project database Jain et al. (2013), with dielectric tensors computed by density-functional perturbation theory (DFPT) in VASP Petousis et al. (2016). Train/validation/test splits. We partition the data into 5,3635,363 training, 671671 validation, and 672672 test materials, an ≈80/10/10≈ 80/10/10 split stratified by crystal system, so that the symmetry distribution is preserved across the three splits. Training. Learning rate 1×10−21× 10^-2, weight decay 5×10−25× 10^-2; remaining hyperparameters in Table 6. Supplementary Table 6: Hyperparameters for the TSENN dielectric-tensor surrogate. Hyperparameter Value Maximum spherical harmonic order ℓmax _ 2 Embedding dimension 64 Number of layers 2 Channel multiplicity 32 Batch size 16 Per-component test-set metrics are reported in Supplementary Table 7, and the corresponding parity plots across all splits are shown in Supplementary Figure 6. The off-diagonal components have substantially smaller absolute errors than the diagonal components, consistent with their smaller magnitudes in the DFT data. Despite this scale separation, the model maintains high R2R^2 values for both diagonal and off-diagonal components. These results indicate that the scalar and rank-2 irrep decomposition enables the surrogate to learn the full symmetric dielectric tensor without letting the large diagonal components dominate the prediction. The resulting model is therefore suitable as a tensorial scorer for inverse-design tasks, including those targeting materials with finite off-diagonal dielectric responses. 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. Component R2R^2 MAE εxx _x 0.993 0.087 εyy _y 0.992 0.083 εzz _z 0.991 0.088 εxy _xy 0.986 0.005 εxz _xz 0.976 0.006 εyz _yz 0.956 0.009 Supplementary Figure 6: Predicted versus DFT-computed static-limit dielectric tensors across all three splits. Parity plots of each independent Cartesian component εij _ij (i,j∈x,y,zi,j∈\x,y,z\) of the dielectric tensor, evaluated on the train, validation, and test splits. The dashed line indicates perfect agreement (y=xy=x). The inset showing the MAE for per-component performance, and all the metrics are also reported in Supplementary Table 7. 5 Reward shaping For both inverse-design tasks, the reward depends on a task-specific property and the predicted band gap. Selected quantities are mapped to [0,1][0,1] using the clamped ascending linear ramp Φ(x,a,b)=clip(x−ab−a, 0, 1), (x;a,b)=clip ( x-ab-a,\,0,\,1 ), (18) which is zero for x≤ax≤ a, increases linearly between a and b, and saturates at one for x≥bx≥ b. The dielectric and SLME tasks combine their property and band-gap terms differently. The dielectric reward uses the smaller of the two terms, while the SLME reward uses a weighted sum that favors band gaps near the single-junction optimum. Dielectric task. The tensor-based uniaxial dielectric score is quni=gz(1−m)∈[0,1],q_uni=g_z(1-m)∈[0,1], (19) where gzg_z measures the contrast between the out-of-plane and mean in-plane dielectric responses, and m measures the residual mismatch between εxx _x and εyy _y. These quantities are defined in the main Methods. It is convenient to name the in-plane isotropy factor separately, qiso≡1−m∈[0,1],quni=gzqiso,q_iso≡ 1-m∈[0,1], q_uni=g_z\,q_iso, (20) since the two factors suppress different failure modes, where gzg_z rejects near-isotropic tensors, while qisoq_iso rejects biaxial ones. The uniaxial score is mapped to [0,1][0,1] using rq=Φ(quni,0,0.20),r_q= (q_uni;0,0.20), (21) while the band-gap contribution is rg,uni(Eg)=Φ(Eg,0.3,0.8).r_g,uni(E_g)= (E_g;0.3,0.8). (22) The final dielectric reward is runi=min[rq,rg,uni(Eg)]=min[Φ(quni,0,0.20),Φ(Eg,0.3,0.8)].r_uni= [r_q,\,r_g,uni(E_g) ]= [ (q_uni;0,0.20),\, (E_g;0.3,0.8) ]. (23) The minimum operator makes the smaller term the limiting contribution. A large uniaxial score cannot compensate for a small predicted band gap, and a large band gap cannot compensate for a poor uniaxial response. The reward is therefore nonzero only when quni>0q_uni>0 and Eg>0.3E_g>0.3 eV. It reaches one when quni≥0.20q_uni≥ 0.20 and Eg≥0.8E_g≥ 0.8 eV. Because the ramp saturates at quni=0.20q_uni=0.20, the reward does not discriminate among candidates with quni>0.20q_uni>0.20. We therefore apply an additional in-plane isotropy criterion, qiso≥0.95q_iso≥ 0.95, when selecting candidates for first-principles validation. This construction prevents near-metallic candidates from receiving a high reward solely because of their predicted dielectric response. It also keeps the search within the finite-gap regime represented by the dielectric surrogate and used for subsequent DFPT validation. The resulting reward surface is shown in Supplementary Figure 7(a). SLME property term. For the photovoltaic task, the property term is the SLME η. We evaluate η under the AM1.5G illumination spectrum using a film thickness Labs=0.3 µmL_abs=$0.3\, m$, a cell temperature Tcell=300 KT_cell=$300\,K$, and a radiative fraction fr=1f_r=1 Yu and Zunger (2012); Hung et al. (2024). The efficiency is defined as η=PmaxPsolar,η= P_ P_solar, (24) where the incident solar power is Psolar=∫0∞Isolar(E)E.P_solar= _0^∞I_solar(E)\,dE. (25) Here, Isolar(E)I_solar(E) is the AM1.5G energy irradiance. The maximum electrical output power is obtained from the diode J–VJ--V curve, Pmax=maxVJ(V)V,J(V)=Jsc−J0(eqV/(kBTcell)−1),P_ = _V\J(V)V\, J(V)=J_sc-J_0 (e^qV/(k_BT_cell)-1 ), (26) where q is the elementary charge and kBk_B is the Boltzmann constant. The material absorptance is calculated from the predicted absorption coefficient as A(E)=1−e−2α(E)Labs,A(E)=1-e^-2α(E)L_abs, (27) where E=ℏωE= ω. The short-circuit current density is Jsc=q∫Eg∞A(E)ϕsolar(E)E,J_sc=q _E_g^∞A(E)\, _solar(E)\,dE, (28) where EgE_g defines the absorption edge and ϕsolar(E) _solar(E) is the incident solar photon flux. When the photon energy E is expressed in electronvolts, the photon flux is obtained from the energy irradiance as ϕsolar(E)=Isolar(E)qE, _solar(E)= I_solar(E)qE, (29) where qEqE converts the photon energy from electronvolts to joules. The dark saturation current is modeled as radiative-recombination limited, J0=J0radfr=qπfr∫Eg∞A(E)ϕb(E,Tcell)E,J_0= J_0^radf_r= qπf_r _E_g^∞A(E)\, _b(E,T_cell)\,dE, (30) where ϕb(E,Tcell) _b(E,T_cell) is the blackbody photon radiance per unit solid angle, ϕb(E,Tcell)=2E2h3c21exp(E/kBTcell)−1. _b(E,T_cell)= 2E^2h^3c^2 1 (E/k_BT_cell )-1. (31) Here, h is Planck’s constant and c is the speed of light. The factor π performs the hemispherical angular integration that converts the radiance into an emitted photon flux. Because η depends on both the band gap and the full absorption spectrum α(E)α(E), it provides the task-specific photovoltaic property term. SLME reward. Although η already depends on the band gap, we include an additional band-gap term to guide the generated population toward the favorable region for single-junction solar cells. The final SLME reward is rSLME=wηrη+wgrg,SLME,r_SLME=w_ηr_η+w_gr_g,SLME, (32) where the normalized efficiency term is rη=Φ(η,0,0.35),r_η= (η;0,0.35), (33) and the band-gap localization term is rg,SLME(Eg)=clip(1−|Eg−E∗|Δ,0,1).r_g,SLME(E_g)=clip (1- |E_g-E^* | ,0,1 ). (34) We use wη=0.8w_η=0.8, wg=0.2w_g=0.2, E∗=1.3E^*=1.3 eV, and Δ=1.0 =1.0 eV. Because both terms lie in [0,1][0,1] and the weights sum to one, the final reward also satisfies rSLME∈[0,1]r_SLME∈[0,1]. The band-gap term forms a symmetric tent centered at E∗=1.3E^*=1.3 eV. It reaches one at the center and decreases linearly to zero at Eg=0.3E_g=0.3 and 2.32.3 eV. Outside this interval, the band-gap contribution remains zero. This term does not act as a hard gate. A candidate outside the preferred interval can still receive reward through its predicted efficiency, but it does not receive the additional contribution from band-gap localization. The larger weight wη=0.8w_η=0.8 keeps SLME as the dominant optimization objective. The smaller band-gap term discourages the search from drifting toward very small-gap or overly wide-gap regions. As shown in Supplementary Figure 7(b), the reward increases with SLME and contains an additional ridge centered near Eg=1.3E_g=1.3 eV. For reference, the dashed curve in Supplementary Figure 7(b) shows the Shockley–Queisser detailed-balance limit ηSQ(Eg) _SQ(E_g) evaluated using the same AM1.5G spectrum. In our implementation, the curve reaches a maximum efficiency of 33.7%33.7\% at Eg=1.34E_g=1.34 eV. This curve provides a theoretical reference and is not included in the reward. Supplementary Figure 7: Reward surfaces. Reward as a function of the predicted band gap EgE_g and the task-specific property for (a) the uniaxial dielectric objective and (b) the SLME objective. In (a), the tensor-based property score is quni=gz(1−m)q_uni=g_z(1-m), and the final reward is runi=min[Φ(quni,0,0.20),Φ(Eg,0.3,0.8)]r_uni= [ (q_uni;0,0.20), (E_g;0.3,0.8)]. The vertical dotted line marks the onset of the band-gap contribution at Eg=0.3E_g=0.3 eV. In (b), the final reward is rSLME=0.8rη+0.2rg,SLMEr_SLME=0.8r_η+0.2r_g,SLME, where rη=Φ(η,0,0.35)r_η= (η;0,0.35) and rg,SLMEr_g,SLME is a symmetric tent centered at E∗=1.3E^*=1.3 eV. The dashed curve shows the Shockley–Queisser detailed-balance limit ηSQ(Eg) _SQ(E_g) evaluated using the same AM1.5G spectrum. 6 First-principles validation of generated candidates across crystal systems 6.1 Uniaxial dielectric responses To test whether the dielectric responses identified by the surrogate remain valid at the first-principles level, we relaxed selected generated structures with DFT and calculated their dielectric tensors using DFPT. Candidates were selected from the generated population by requiring a uniaxial dielectric reward of runi≥0.5r_uni≥ 0.5 and a predicted Ehull≤0.10E_hull≤ 0.10 eV/atom. The reward threshold ensures that the selected candidates satisfy both the dielectric-anisotropy and finite-gap objectives defined in the reward function. The high-throughput calculations were managed using atomate2 0.1.5 and performed with VASP 6.4.2. Each candidate underwent two successive full cell-and-ion relaxations followed by a static calculation using the Materials Project-compatible PBE(+U+U) protocol implemented through MPRelaxSet and MPStaticSet. These calculations used projector augmented-wave (PAW) PBE potentials, a plane-wave cutoff of Ecut=520E_cut=520 eV, and the standard Materials Project Hubbard-U corrections for transition-metal oxides and fluorides. The relaxed geometry was then evaluated using the DFPT dielectric workflow with LEPSILON = .TRUE., IBRION = 8, the PBEsol exchange-correlation functional, and a plane-wave cutoff of Ecut=680E_cut=680 eV. The resulting dielectric tensor is denoted by 0 _0. Supplementary Figure 8(a) shows nine generated candidates whose first-principles dielectric tensors retain the targeted relation ε0,xx≈ε0,yy,ε0,zz≠ε0,xx+ε0,yy2. _0,x≈ _0,y, _0,z≠ _0,x+ _0,y2. (35) For simplicity, the subscript 00 is omitted from the individual tensor components below. The validated candidates span the hexagonal, tetragonal, orthorhombic, and monoclinic crystal systems. Their DFT band gaps range from 0.720.72 to 5.475.47 eV, and their EhullE_hull range from 00 to 6767 meV/atom. For the hexagonal and tetragonal candidates, equality between the two in-plane components is imposed by crystal symmetry. These structures include OsO4OsO X_ [t]4 in P63P6_3, ZrTi2O6ZrTi X_ [t]2O X_ [t]6 and TiOF2TiOF X_ [t]2 in P42/mnmP4_2/mnm, InPS4InPS X_ [t]4 in I4¯2mI 42m, and RbLiTeO4RbLiTeO X_ [t]4 in P63P6_3. The tetragonal P42/mnmP4_2/mnm phase of ZrTi2O6ZrTi X_ [t]2O X_ [t]6 shown here is a distinct polymorph from the trigonal P3¯1mP 31m phase highlighted in the main text. The occurrence of both polymorphs further illustrates that the targeted dielectric response is controlled by the crystal structure and symmetry, rather than by composition alone. Their DFPT tensors satisfy εxx=εyy _x= _y within the reported numerical precision, while εzz _z remains distinct. Among these candidates, InPS4InPS X_ [t]4 exhibits the largest contrast, with εxx=5.77 _x=5.77, εyy=5.78 _y=5.78, and εzz=3.57 _z=3.57. The lower-symmetry candidates show that the same numerical tensor relation can also arise without symmetry protection. Monoclinic SrCl2SrCl X_ [t]2 in C2/mC2/m and the orthorhombic structures SiTe2SiTe X_ [t]2 in P212121P2_12_12_1, TeO2TeO X_ [t]2 in P212121P2_12_12_1, and Ti(TeO3)2Ti(TeO X_ [t]3) X_ [t]2 in P21212P2_12_12 exhibit approximately equal in-plane components and a distinct response along the selected out-of-plane direction. Their space groups do not require εxx=εyy _x= _y, so the approximate equality arises from the particular relaxed structures. These responses may therefore be more sensitive to strain, defects, temperature, or further structural perturbations than the symmetry-enforced responses of the hexagonal and tetragonal candidates. Supplementary Figure 8(b) presents three comparison cases that illustrate distinct rejection mechanisms. Orthorhombic Ti2TeO6Ti X_ [t]2TeO X_ [t]6 in Pnn2Pnn2 retains an approximately uniaxial tensor, with diagonal components 8.188.18, 8.248.24, and 8.778.77, but its EhullE_hull is 105105 meV/atom, slightly exceeding the 0.100.10 eV/atom stability cutoff. Orthorhombic SrTiO3SrTiO X_ [t]3 in PnmaPnma has an almost isotropic tensor, with diagonal components 6.396.39, 6.416.41, and 6.356.35, and is therefore suppressed by the out-of-plane contrast term. Monoclinic TeI2TeI X_ [t]2 in P21/cP2_1/c has strongly unequal diagonal components, 4.264.26, 7.437.43, and 13.4413.44, and is rejected by the in-plane isotropy requirement. These examples show how stability, out-of-plane contrast, and in-plane equality contribute separately to candidate selection. The first-principles calculations confirm that the surrogate-guided search identifies finite-gap structures whose dielectric tensors reproduce the desired relation between the in-plane and out-of-plane responses. They also distinguish symmetry-enforced uniaxiality from approximate uniaxiality that occurs only for a specific relaxed configuration. The comparison cases further demonstrate that a favorable tensor contrast alone is not sufficient. A candidate must also satisfy the stability, in-plane isotropy, and finite-gap requirements of the complete selection procedure. Supplementary Figure 8: First-principles validation of generated uniaxial dielectric candidates. (a) Nine generated structures whose dielectric tensors 0 _0 retain the targeted relation εxx≈εyy _x≈ _y with a distinct out-of-plane component. Each entry shows the DFT-relaxed structure, realized space group, calculated dielectric tensor, DFT band gap EgE_g, and EhullE_hull. 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. Ti2TeO6Ti X_ [t]2TeO X_ [t]6 retains an approximately uniaxial tensor but exceeds the stability cutoff, SrTiO3SrTiO X_ [t]3 has a nearly isotropic response and is suppressed by the out-of-plane contrast term, and TeI2TeI X_ [t]2 has a biaxial response that fails the in-plane isotropy requirement. 6.2 SLME To verify that the generative loop discovers genuinely diverse semiconductors rather than repeatedly exploiting a single low-symmetry motif, we relaxed a representative set of generated structures with DFT and computed their optical responses. Because the TSENN surrogate was trained on the OptiMate dataset, the validation calculations were performed using the same high-throughput PBE independent-particle workflow used to construct the training data Grunert et al. (2024). Briefly, ground-state calculations were performed using Quantum ESPRESSO with SG15 norm-conserving pseudopotentials, and the frequency-dependent dielectric functions were calculated using Yambo over the 0–20 eV range with an energy spacing of 0.01 eV and a broadening of 0.10 eV. The plane-wave cutoff, k-point density, and number of unoccupied bands were converged following the OptiMate protocol. Candidate materials were selected according to the surrogate predictions, requiring a TSENN-predicted band gap of 0.8–1.9 eV and an SLME exceeding 25%. Of the 576 structures satisfying these criteria, 241 (42%) relaxed to gapped semiconductors with well-defined absorption spectra, whereas the remainder became metallic or nearly metallic after DFT relaxation. Because SLME depends strongly on the absorption onset, uncertainty in the predicted band gap is the dominant source of error, rather than inaccuracies in the predicted optical spectra. The eight materials shown in Supplementary Figure 9 are drawn from this DFT-confirmed semiconducting pool, and their first-principles ε2(ω) _2(ω) spectra are compared with the TSENN surrogate that guided the search. The selected materials span six of the seven crystal systems, including cubic K2TeK X_ [t]2Te (Fm3¯mFm 3m) and NaHfCu3S4NaHfCu X_ [t]3S X_ [t]4 (P4¯3mP 43m), trigonal CsGeBr3CsGeBr X_ [t]3 (R3mR3m), tetragonal SnBr2SnBr X_ [t]2 (P42/mnmP4_2/mnm), orthorhombic BiSI (PnmaPnma) and PbS (Pmn21Pmn2_1), monoclinic InGaS2InGaS X_ [t]2 (P21/cP2_1/c), and triclinic AsS4AsS X_ [t]4 (P1¯P 1). This diversity shows that the search discovers promising photovoltaic absorbers across a broad range of crystal symmetries rather than collapsing onto the triclinic P1P1/P1¯P 1 structures that dominate the raw efficiency ranking. Across this representative set, the candidates exhibit PBE indirect band gaps of 1.15–2.12 eV and SLME values of 18–32%, placing most within the desired range for single-junction photovoltaic absorbers. The TSENN surrogate closely reproduces the DFT-computed ε2(ω) _2(ω) spectra, including the absorption onset, dominant absorption peak, and higher-energy spectral features across all represented crystal systems. Spectral agreement was quantified using the spectral similarity coefficient (SC), SC[ε2TSENN(ω);ε2DFT(ω)]=1−∫|ε2DFT(ω)−ε2TSENN(ω)|ω∫|ε2DFT(ω)|ωSC[ _2^TSENN(ω); _2^DFT(ω)]=1- | _2^DFT(ω)- _2^TSENN(ω)|dω | _2^DFT(ω)|dω (36) following Ref. Grunert et al. (2024), where SC=1SC=1 indicates identical spectra. The layered orthorhombic PbS shows the closest agreement, with an SC of 0.97. The reported band gaps and optical spectra were calculated at the PBE independent-particle level and should therefore be interpreted as internally consistent screening quantities rather than quantitative quasiparticle or optical excitation energies. PBE generally underestimates band gaps, while the independent-particle approximation neglects electron–hole interactions and microscopic local-field effects. Nevertheless, the close agreement between the surrogate and the corresponding DFT calculations confirms that the reward model reproduces the level of optical physics represented in its training data across a structurally diverse set of semiconductors, supporting its use in the generative search. Supplementary Figure 9: First-principles optical response of eight generated candidates. 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