PLaID++: Targeted Crystal Generation

• The paper demonstrates that coupling a symmetry-aware Wyckoff representation with iterative DPO significantly improves the stability, uniqueness, and novelty of generated crystal structures.
• PLaID++ encodes crystal structures compactly using space group, lattice parameters, and site-specific information to enforce crystallographic symmetry during generation.
• Empirical evaluations on MP-20, supported by DFT validation, show that PLaID++ achieves up to a 50% improvement in generation quality compared to prior methods.

PLaID++ is a large-language-model-based system for targeted inorganic crystal generation that combines a symmetry-aware text representation with preference alignment via Direct Preference Optimization (DPO). It fine-tunes Qwen-2.5 7B to generate crystal structures, uses a novel Wyckoff-based serialization to encode symmetry constraints directly into text, and applies iterative preference optimization to steer generation toward structures that are thermodynamically stable, unique, and novel while also supporting space-group-conditioned generation (Xu et al., 8 Sep 2025).

1. Conceptual basis and scope

PLaID++ is motivated by a standard bottleneck in inorganic materials discovery: the search space of possible compounds and crystal structures is large, while experimental and density-functional-theory screening are slow and expensive. The system treats crystal generation as a language modeling problem, but it departs from generic text-based crystal generators in two specific ways. First, it replaces unconstrained coordinate-heavy serializations with a compact symmetry-aware representation based on Wyckoff positions. Second, it adds post-training preference alignment so that the model is not optimized only for likelihood on the training corpus, but also for chemically useful outputs as judged by stability, novelty, and, in the conditional setting, space-group correctness (Xu et al., 8 Sep 2025).

The paper frames this alignment stage as Reinforcement Learning from Interatomic Potentials (RLIP). In that formulation, preference labels are not obtained from human annotators, but from machine-learned interatomic potentials (MLIPs). This makes PLaID++ a post-trained crystal generator rather than merely an instruction-tuned LLM for CIF-like strings. The target regime includes both unconditional generation and space-group-conditioned generation, with prompt-based conditioning rather than a separate architecture for each task.

A common misunderstanding is to reduce PLaID++ to “an LLM for CIF generation.” The underlying claim is narrower and more technical: generation quality improves when the text representation respects crystallographic symmetry and when post-training explicitly biases the policy toward desirable regions of chemical space. This suggests that the system’s novelty lies in the interaction between representation and alignment, not in the use of a foundation model alone.

2. Symmetry-aware text representation

The representation layer is centered on the asymmetric unit and its symmetry metadata. In the paper’s formal notation, a crystal can be written as

$$C = (l_1, l_2, l_3, \theta_1, \theta_2, \theta_3, e_1, x_1, y_1, z_1, \dots, e_N, x_N, y_N, z_N).$$

Wyckoff positions for space group $G$ are defined by

$W = \{g x \mid g \in G\},$

with symmetry action

$x' = Rx + t,$

where $R$ is a rotation, reflection, or inversion matrix and $t$ is a translation vector. PLaID++ serializes a structure using this symmetry-aware viewpoint rather than enumerating all atomic positions independently (Xu et al., 8 Sep 2025).

Each crystal string contains the chemical formula, space group number, lattice parameters $a,b,c,\alpha,\beta,\gamma$, and, for each site, the element type, fractional coordinates, Wyckoff site label, and site multiplicity. The paper places the chemical formula first “to ensure self-consistency during generation.” An example given in the appendix is:

$R$4

This encoding is explicitly more compact than the coordinate text baseline used for comparison. On MP-20, the average sequence length is 185.5 tokens/crystal with the Wyckoff representation versus 214.7 tokens/crystal with the standard representation from CrystalLLM, a 14% reduction. The paper interprets that reduction as both a compression benefit and a structural prior: one text decision can imply a symmetry orbit rather than a single free coordinate.

The representational choice is also tied to model stability under post-training. In the ablations, the 3D coordinate representation deteriorates substantially under iterative DPO, whereas the Wyckoff representation remains effective. This suggests that the symmetry-aware text acts as a regularizer on the model’s generative search space.

3. Training pipeline and preference alignment

PLaID++ is trained in two stages: supervised fine-tuning (SFT) followed by iterative DPO. The base model is Qwen-2.5 7B, adapted with LoRA rather than full-parameter updating. During SFT, the model is trained on crystal strings in either the coordinate or Wyckoff format using standard next-token likelihood. The appendix gives the SFT objective as

$$\mathcal{L}_{SFT} = -\sum_{b=1}^B \log \pi_0(\hat{y}_b \mid x).$$

The reported SFT setup uses AdamW, batch size 16, learning rate $10^{-5}$, fp-4 mixed precision, LoRA rank 8, alpha 32, dropout 0.05, 4-bit quantization, and 10 epochs. During fine-tuning, one-third of the time the model performs infilling and two-thirds of the time it performs de novo generation (Xu et al., 8 Sep 2025).

The alignment stage constructs pairwise preferences from generated crystals. The paper first introduces the KL-regularized RL objective

$$\max_{\pi_\theta} \mathbb{E}_{x\sim D, y\sim \pi_\theta(y\mid x)}\left[ r_\phi(x, y) \right] - \beta D_{\mathrm{KL}}\left[\pi_\theta(y\mid x) \| \pi_{\text{ref}}(y\mid x)\right],$$

and then specializes it to DPO. For preference pairs $G$0, the Bradley–Terry model is

$G$1

with DPO loss

$G$2

Preference construction is stratified by stability. Using eqV2 (EquiformerV2 86M) during training, samples are grouped as stable if $G$3 eV/atom, metastable if $G$4 eV/atom, and unstable if $G$5 eV/atom. The preference dataset includes $G$6, $G$7, and $G$8 pairs. Novelty is added by distinguishing stable and novel from stable and not novel, using Pymatgen’s StructureMatcher. For conditional generation, additional preferences separate crystals with the correct target space group from otherwise comparable crystals with the wrong space group.

The DPO stage is iterative: at iteration $G$9, the reference model is the previous policy, $W = \{g x \mid g \in G\},$0, new structures are sampled from $W = \{g x \mid g \in G\},$1, and a fresh preference dataset is built. The reported DPO setup uses the TRL DPO Trainer, Adam, batch size 16, learning rate $W = \{g x \mid g \in G\},$2, fp-4/bfloat-16, $W = \{g x \mid g \in G\},$3, and 1 epoch per DPO dataset. The appendix reports that a 1:2 accept/reject ratio performed best. A dynamic temperature schedule is used across DPO iterations to preserve diversity, since uniqueness is a set-level property that is not directly encoded by pairwise preferences.

4. Evaluation protocol and metrics

The training and evaluation corpus is MP-20, containing 45,231 inorganic crystalline materials from the Materials Project, restricted to structures with up to 20 atoms. Unconditional evaluation samples 10,000 structures from each model. Conditional evaluation samples 1,000 structures for each of 7 space groups: $W = \{g x \mid g \in G\},$4, $W = \{g x \mid g \in G\},$5, $W = \{g x \mid g \in G\},$6, $W = \{g x \mid g \in G\},$7, $W = \{g x \mid g \in G\},$8, $W = \{g x \mid g \in G\},$9, and $x' = Rx + t,$0. Space-group verification uses PyXtal, and novelty and uniqueness use Pymatgen StructureMatcher (Xu et al., 8 Sep 2025).

The principal unconditional metric is the S.U.N. rate, the fraction of generated structures that are Stable, Unique, and Novel. The appendix gives

$x' = Rx + t,$1

For space-group-conditioned generation, the paper uses S.S.U.N., inherited from MatterGen: correct Symmetry group, Stable/metastable, Unique, and Novel. In this conditional setting, the acceptable energy threshold is $x' = Rx + t,$2 eV/atom.

Evaluation intentionally separates training-time and test-time oracles. eqV2 is used to construct preference pairs during post-training, while eSEN is used for stability evaluation “to avoid overfitting to the same oracle used in training.” Both MLIPs perform structure relaxation with 500 relaxation steps and maximum force 0.02. The paper also includes DFT/VASP validation on a subset of generated structures, with VASP 6.3.2, 520 eV cutoff, a $x' = Rx + t,$3-centered mesh of 64 k-points per Å$x' = Rx + t,$4, and force tolerance $x' = Rx + t,$5 eV/Å.

5. Empirical performance and ablations

The main unconditional results show that the flagship PLaID++ model achieves the best reported combination of stability and S.U.N. among the compared methods. The paper’s key table is:

Method	Stability	S.U.N.
FlowLLM	13.9	4.7
Jointly-trained ADiT	15.4	5.3
PLaID++ Wyckoff (non-DPO)	7.17	3.58
PLaID++ Iterative SFT	9.85	4.13
PLaID++ Wyckoff + DPO	15.59	6.25
PLaID++	22.27	7.74

The flagship model’s 22.27% stability and 7.74% S.U.N. underwrite the paper’s claim that it generates stable, unique, and novel materials at a $x' = Rx + t,$6 greater rate than prior methods. The same section states that among stable relaxed structures, 72% are novel and 55% are unique (Xu et al., 8 Sep 2025).

The ablations isolate two dominant effects. First, the Wyckoff representation is superior to the 3D coordinate baseline even before alignment: 3.58% S.U.N. versus 2.81%. Second, iterative DPO materially exceeds fine-tuning alone. The paper summarizes this as $x' = Rx + t,$7 improvement in unconditional generation and $x' = Rx + t,$8 improvement in space-group-conditioned generation compared to fine-tuning alone. A related ablation shows that the coordinate representation combined with DPO reaches 13.94% stability but only 1.69% S.U.N., whereas the Wyckoff representation plus DPO reaches 15.59% stability and 6.25% S.U.N. This is strong evidence that alignment without a symmetry-aware representation can increase stability while collapsing diversity.

For space-group-conditioned generation, the paper reports that applying DPO to space-group preference pairs yields an average 22% increase in S.S.U.N. over the base Wyckoff model, and that adding joint unconditional preference data plus dynamic temperature yields a total 47% increase over the base model. It also notes heterogeneity across target groups: low-data groups such as $x' = Rx + t,$9, $R$0, and $R$1 show little improvement or slight degradation.

The DFT validation on 1,000 generated structures reports 19.1% stability rate and 13% S.U.N. rate, which the authors describe as broadly consistent with the MLIP-based conclusions. They also compare the training and evaluation oracles to DFT near the hull: eSEN vs DFT gives $R$2, while eqV2 vs DFT gives $R$3. The throughput result is likewise emphasized: PLaID++ generates 10,000 crystals in ~23 minutes on a single H100, corresponding to 27.17 S.U.N. crystals/minute, compared with 5.25 S.U.N. crystals/minute for FlowLLM under the reported setup.

6. Interpretation, limitations, and nomenclature

The paper’s central technical implication is that materials generation benefits from combining a symmetry-aware latent text space with post-training preference alignment. The representation appears to regularize the model’s output space, while iterative DPO biases the model toward structures with better stability and novelty properties. A further implication is methodological: prompt-based crystal generation can support both unconditional discovery and constrained generation without changing the model class.

The limitations are explicit. PLaID++ is trained and evaluated only on MP-20, which contains about 45K structures and only crystals with up to 20 atoms. The conditional experiments focus on space group rather than a broader portfolio of target properties such as band gap or ionic conductivity. The post-training signal depends on surrogate reward models, specifically eqV2 and eSEN; although DFT validation is included, thermodynamic stability is not equivalent to synthesizability. The paper also explores only DPO, leaving alternatives such as PPO or GRPO for future work. In low-data space groups, alignment shows weaker gains, which the authors interpret as evidence that post-training mainly elicits capabilities already weakly present in the SFT model rather than creating them ex nihilo (Xu et al., 8 Sep 2025).

Because the term “Plaid” is heavily overloaded, PLaID++ should also be distinguished from unrelated systems in other fields. In arXiv usage, Plaid can refer to a likelihood-based diffusion LLM (Gulrajani et al., 2023), a ColBERT retrieval engine and its descendants (MacAvaney et al., 2024, Nardini et al., 2024), and several other domain-specific frameworks. The explicit title “PLaID++: A Preference Aligned LLM for Targeted Inorganic Materials Design” identifies the materials-design system discussed here (Xu et al., 8 Sep 2025).