WyckoffDiff-Adaptor: Symmetry-Aware Generation

• The paper shows that embedding conditional properties via adaptor modules in a Wyckoff-position-based framework ensures precise symmetry enforcement.
• It uses a discrete diffusion model on a protostructure representation to integrate chemical and energetic constraints while preserving space-group symmetry.
• Empirical results on Li–O and Ti–O systems highlight up to 6× faster CIF generation and improved target space-group preservation compared to standard generators.

WyckoffDiff-Adaptor is a symmetry-aware conditional generation method for crystal structures that embeds conditional generation within a WyckoffDiff architecture, so that targeted conditions such as a chemical system, an energy-above-hull target, and a specified space group can be imposed without relinquishing precise symmetry control (Ishii et al., 13 Jan 2026). It was introduced to address a specific limitation in conditional crystal generation: simply supplying a space-group label to a standard conditional generator is not sufficient when the underlying model is not symmetry-native, because generated structures may fail to preserve the requested symmetry, especially after structural relaxation. In WyckoffDiff-Adaptor, space group is treated as a structural input inside a Wyckoff-position-based representation, while chemical system and energetic constraints are embedded through adaptor modules (Ishii et al., 13 Jan 2026).

1. Research problem and motivating gap

Crystal structure prediction aims to find plausible or stable crystal structures for a given chemical composition. In materials discovery, the objective is typically conditional rather than unconditional: one seeks structures satisfying a chemical system such as Li–O or Ti–O, a stability target such as energy above hull $=0$, and often a desired space group. This setting is difficult because the crystal search space simultaneously involves composition, atom counts, lattice, atomic coordinates, and symmetry among the 230 space groups (Ishii et al., 13 Jan 2026).

The central problem addressed by WyckoffDiff-Adaptor is the mismatch between property conditioning and symmetry-native structure representation. Coordinate-based generators can accept a space-group label as an input condition, but nothing in that representation guarantees that the generated atom positions and lattice obey the corresponding symmetry operations. Even approximate symmetry can be broken during geometry optimization. The paper therefore argues that a space-group label functions as a soft prompt in such systems rather than as a hard generative scaffold, and concludes that conditioning alone is not enough if the base architecture does not explicitly encode Wyckoff symmetry structure (Ishii et al., 13 Jan 2026).

This framing places WyckoffDiff-Adaptor within a broader line of work on symmetry-aware crystal generation. WyckoffDiff had already shifted crystal generation from full coordinate space to a discrete symmetry-complete protostructure representation, with diffusion performed over symmetry-aware tokens rather than atomic coordinates (Kelvinius et al., 10 Feb 2025). WyckoffDiff-Adaptor extends that foundation to the conditional regime.

2. Protostructure representation and symmetry-native state space

The defining representational object is a protostructure

$$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$

Here $s$ is the space group, $\boldsymbol{z}^\infty$ denotes unconstrained Wyckoff positions, and $\boldsymbol{z}^0$ denotes restricted Wyckoff positions (Ishii et al., 13 Jan 2026). The intended distinction is between general positions, whose coordinates contain free internal parameters, and special positions, whose coordinates are constrained by symmetry.

Wyckoff positions are defined relative to the symmetry operations of a space group. Once $s$ is fixed, allowable atomic placements are partitioned into symmetry-equivalent position classes. WyckoffDiff-Adaptor therefore does not generate arbitrary coordinates directly. Instead, it generates a discrete Wyckoff element occupancy representation in which the learned variables are occupancy-type assignments over symmetry-defined positions. For general Wyckoff positions, the model generates a matrix $\boldsymbol{z}^\infty$ representing atom species $\times$ count. For restricted Wyckoff positions, it generates a vector $\boldsymbol{z}^0$ representing atom species $+0$ for vacant sites (Ishii et al., 13 Jan 2026).

The conversion pipeline is explicit: $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$0 This matters because the generated object already lies in a space indexed by $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$1, so symmetry preservation is supported by the state space itself rather than imposed only after sampling (Ishii et al., 13 Jan 2026).

The representation is inherited from WyckoffDiff, whose formalism writes the protostructure as $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$2, with discrete occupancies on constrained and unconstrained Wyckoff positions (Kelvinius et al., 10 Feb 2025). In that framework, the space group determines the legal Wyckoff set $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$3, multiplicities are implicit in the chosen position, and the core generated variables are entirely discrete. A major caveat, preserved in WyckoffDiff-Adaptor, is that internal continuous parameters of general Wyckoff positions are not generated directly; they are assigned later by PyXtal, semi-randomly (Ishii et al., 13 Jan 2026).

3. Architecture, conditioning mechanism, and training configuration

WyckoffDiff-Adaptor takes the base WyckoffDiff model and incorporates adaptor modules inspired by MatterGen so that property conditions can be injected into a symmetry-native generator (Ishii et al., 13 Jan 2026). The backbone network is WyckoffGNN, the graph neural architecture originally introduced for WyckoffDiff to denoise the discrete Wyckoff occupancy representation (Kelvinius et al., 10 Feb 2025). The conditional mechanism is described in the paper as follows: “The conditional properties are embedded by the Adaptor modules during the $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$4 layers of the original WyckoffGNN layer connections” (Ishii et al., 13 Jan 2026).

The conditioning variables used in the experiments are chemical system or atomic species, energy above hull, and space group. The crucial distinction is that space group is already native to the representation through $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$5, whereas chemical system and energetic constraints are injected through adaptor modules. In this design, space group is structural input rather than side information (Ishii et al., 13 Jan 2026).

The paper does not provide explicit adaptor equations. It states that the model is based on the MatterGen codebase, that adaptor modules are inserted during the $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$6 WyckoffGNN layer connections, and that these modules embed conditional properties into the main network. It does not provide formulas such as affine feature modulation, residual adapter equations, or classifier-free guidance rules. It also does not mention classifier-free guidance, conditional dropout, conditional likelihood weighting, or guidance scales at sampling time (Ishii et al., 13 Jan 2026).

The reported implementation settings are the following.

Component	Value
Maximum diffusion time step $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$7	1000
Number of chemical elements	100
Maximum number of atoms per element	54
Number of GNN layers $$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$8	3
Hidden dimension	256
Position/space-group embedding size	16
Activation	SiLU
Adaptor hidden dimension	64
Base learning rate	$$\boldsymbol{M} = (s, \boldsymbol{z}^\infty, \boldsymbol{z}^0).$$9
Fine-tuning learning rate	$s$0
Batch size	32
Base model epochs	900
Fine-tuning epochs	200

These are the main concrete architectural and optimization details reported for WyckoffDiff-Adaptor (Ishii et al., 13 Jan 2026). They closely track the underlying WyckoffDiff backbone, which likewise used a 3-layer WyckoffGNN with hidden dimension 256, positional embedding dimension 16, and SiLU activations (Kelvinius et al., 10 Feb 2025).

4. Diffusion process, decoding pipeline, and symmetry enforcement

The paper repeatedly identifies WyckoffDiff-Adaptor as a discrete diffusion model and states that it performs “discrete diffusion inverse generation,” but it does not print the actual diffusion equations, transition kernels, reverse parameterization, or training loss formulas (Ishii et al., 13 Jan 2026). The underlying base model, WyckoffDiff, uses the D3PM framework, factorizes the joint model as

$s$1

and parameterizes reverse transitions through predicted clean-token distributions combined with the exact D3PM posterior (Kelvinius et al., 10 Feb 2025). WyckoffDiff-Adaptor inherits the discrete denoising setting but does not restate those equations.

What is explicitly stated for WyckoffDiff-Adaptor is that the denoiser predicts the matrix $s$2 for general Wyckoff positions and the vector $s$3 for restricted Wyckoff positions, given the space group $s$4 and conditional embeddings (Ishii et al., 13 Jan 2026). The output is therefore still a protostructure rather than a full continuous geometry.

Symmetry preservation is enforced at two levels. First, there is representation-level enforcement through $s$5: the generated object is a protostructure consistent with the target group by construction. Second, there is post-processing validation in the PyXtal conversion pipeline. After denoising, the generated Wyckoff element matrix is converted to a protostructure label and then to a full crystal structure with PyXtal. The paper specifies a maximum of 100 post-processing attempts with PyXtal, and if the generated crystal’s protostructure does not match the input protostructure, generation is rejected (Ishii et al., 13 Jan 2026).

This two-stage design explains why the paper presents space-group control as more precise than in coordinate-based conditional generators. The model does not begin in unconstrained coordinate space and then attempt to recover symmetry; it generates within a symmetry-defined orbit space and only later realizes the continuous geometry.

5. Dataset, evaluation protocol, and empirical results

The model is trained on the MP-20 dataset, which is also used in earlier crystal generation work including CDVAE and WyckoffDiff (Ishii et al., 13 Jan 2026). Both MatterGen and WyckoffDiff-Adaptor are trained using three conditions: space group, energy above hull, and atomic species. The main binary test systems are Li–O and Ti–O, with additional larger-element tests on ternary Al–Mg–O and quaternary Ba–Ta–In–O (Ishii et al., 13 Jan 2026).

The key evaluation task is conditional generation of stable structures under the condition

$s$6

with a specified chemical system and specified space groups. The input space groups are selected from the 20 most frequent space groups in MP-20. Generated structures are optimized with MatterSim and then evaluated in formation-energy phase diagrams using Pymatgen. For Li–O, the paper reports generating 1280 structures for each model, MatterGen and WyckoffDiff-Adaptor (Ishii et al., 13 Jan 2026).

The most prominent empirical result is the symmetry heatmap comparison. MatterGen shows poor preservation of the specified space group or crystal system after optimization, with only partial preservation for some trigonal and monoclinic systems and many cases in which optimization moves the generated structure into a different, often lower-symmetry crystal system. WyckoffDiff-Adaptor shows much stronger diagonal concentration in the heatmap, meaning that optimized structures remain within the target crystal system more often (Ishii et al., 13 Jan 2026).

For Li–O, the paper states that both MatterGen and WyckoffDiff-Adaptor show good agreement with the Materials Project convex hull, but WyckoffDiff-Adaptor succeeds in exploring a broad range of stable structures under the conditions. It emphasizes that WyckoffDiff-Adaptor can generate structures reproducing the same configurations as structures in the Materials Project; in particular, a generated low-energy $s$7 resembles mp-1960 (Ishii et al., 13 Jan 2026). For Ti–O, both models can recover the convex-hull region, but the stable structures generated are not identical to those in the Materials Project. A highlighted limitation is that a generated stable $s$8 is not the standard rutile structure corresponding to mp-390 (Ishii et al., 13 Jan 2026).

The paper also reports an efficiency comparison for 16 Li–O samples on a V100 16GB GPU: MatterGen requires 366 s to generate CIF files, whereas WyckoffDiff requires 60 s total, split into 34 s for prototype generation and 26 s for prototype-to-structure conversion. The paper interprets this as a strong efficiency advantage, with WyckoffDiff-Adaptor roughly $s$9 faster in CIF generation (Ishii et al., 13 Jan 2026).

A notable omission is that the paper does not report standard crystal-generation metrics such as validity, uniqueness, or novelty percentages. Instead, it focuses on symmetry-preservation heatmaps, phase-diagram recovery, qualitative structure inspection, and runtime (Ishii et al., 13 Jan 2026).

6. Limitations, failure modes, and broader significance

The paper identifies several limitations. First, chemically implausible local bonding can occur even when a structure is symmetry-valid. The Ba–Ta–In–O example shows unreasonable O placements near Ta, indicating that the representation captures symmetry occupancy better than local chemistry and interatomic interaction realism (Ishii et al., 13 Jan 2026). Second, the omission of internal parameters for general Wyckoff positions is a major modeling constraint. WyckoffDiff-Adaptor generates only protostructures based on the Wyckoff element matrix and does not contain internal-parameter information; PyXtal assigns these semi-randomly. As a result, structures with important general-position degrees of freedom are difficult to generate reliably, and the same protostructure can map to multiple distinct geometric crystals. The MgAl$\boldsymbol{z}^\infty$0O$\boldsymbol{z}^\infty$1 spinel example is presented as an illustration of this ambiguity (Ishii et al., 13 Jan 2026).

Third, lower-symmetry drift can still occur after relaxation when the initial generated geometry is chemically unrealistic. Fourth, convex-hull coverage remains incomplete: in Ti–O, the model finds low-energy regions without exactly recovering known stable polymorphs (Ishii et al., 13 Jan 2026). The empirical evidence is also limited in ablation terms. There is no formal ablation table isolating adaptor on/off within WyckoffDiff, energy conditioning versus combined conditioning, post-processing rejection on/off, or different adaptor depths and dimensions. The main comparative evidence is the architectural contrast between MatterGen with space-group conditioning and WyckoffDiff-Adaptor with a symmetry-aware base representation (Ishii et al., 13 Jan 2026).

These limitations clarify the method’s conceptual contribution. The broader implication drawn in the paper is that symmetry-aware conditional generation should not be treated as property conditioning plus a space-group token. A plausible implication is that targeted crystal design may require hard or native symmetry parameterization during generation, with property conditioning layered on top rather than substituted for it. In that sense, WyckoffDiff-Adaptor extends the central lesson of WyckoffDiff—that symmetry-aware discrete protostructure generation can be an effective basis for materials generation—into the conditional regime (Kelvinius et al., 10 Feb 2025).

The term “adaptor” in WyckoffDiff-Adaptor refers to conditional-property modules embedded in WyckoffGNN layer connections. In the wider diffusion literature, adaptor modules have also been analyzed in other settings, including low-rank residual updates and token-based consistency modules for DDIM-based video editing (Song et al., 22 Apr 2025). That usage is conceptually broader than the crystal-specific role of adaptors in WyckoffDiff-Adaptor, where the decisive issue is not temporal consistency but the integration of property conditioning into a symmetry-native protostructure generator.