Wyckoff Transformer: Generation of Symmetric Crystals [ICML 2025]

Crystal symmetry plays a fundamental role in determining its physical, chemical, and electronic properties such as electrical and thermal conductivity, optical and polarization behavior, and mechanical strength. Almost all known crystalline materials have internal symmetry. However, this is often inadequately addressed by existing generative models, making the consistent generation of stable and symmetrically valid crystal structures a significant challenge. We introduce WyFormer, a generative model that directly tackles this by formally conditioning on space group symmetry. It achieves this by using Wyckoff positions as the basis for an elegant, compressed, and discrete structure representation. To model the distribution, we develop a permutation-invariant autoregressive model based on the Transformer encoder and an absence of positional encoding. Extensive experimentation demonstrates WyFormer's compelling combination of attributes: it achieves best-in-class symmetry-conditioned generation, incorporates a physics-motivated inductive bias, produces structures with competitive stability, predicts material properties with competitive accuracy even without atomic coordinates, and exhibits unparalleled inference speed.

WyFormer Generated Datasets

If you just need the generated datasets for benchmarking, they are available at Figshare, including both original and DFT: WyFormer; WyFormer, DiffCSP(++), SymmCD, MiAD, WyCryst, CrystalFormer.

Installation

1. Clone the repository
2. Copy pyproject.toml.zeus to pyproject.toml
3. Edit pyproject.toml as nesessary for your CUDA environment. The file won't work out-of-the-box. The simplest way is to remove the local torch wheels from pyproject.toml, and then use the instructions from the pytorch website for your CUDA version. Here are poetry-specific instructions. Optionally, pytorch-sparse and pytorch-scatter are needed to run the CDVAE property prediction model, but otherwise can be skipped.
4. Run poetry install.
5. Log into Wandb, and configure your entity; or disable Wandb. Internally, we used symmetry-advantage. It can be configured in poetry:

poetry self add poetry-dotenv-plugin
echo "WANDB_ENTITY=symmetry-advantage" > .env

5. Preprocess the data on Wychoff positions:

python scripts/preprocess_wychoffs.py

Running a pilot model

To veryfy that the installation is working, run a pilot model. Next token prediction:

python scripts/cache_a_dataset.py mp_20
python scripts/tokenise_a_dataset.py mp_20 yamls/tokenisers/mp_20_sg_multiplicity.yaml --new-tokenizer
python scripts/train.py yamls/models/NextToken/v6/base_sg.yaml mp_20 cuda --pilot

This will train a model, and save the results in the runs folder. The files are:

• best_model_params.pt - the model weights chosen by the validation loss
• generated_wp_no_calibration.json.gz Wyckoff representation of the generated structures
• generated_wp_temperature_calibration.json.gz Wyckoff representation of the generated structures, with the temperature calibration applied
• token_engineers.pkl.gz - preprocessing information, e. g. multiplicity and spherical harmonics
• tokenizers.pkl.gz - tokenizers

Training Data Preprocessing

The available datasets correspond to the folders in data and cdvae/data. Dataset idetifiers are the folder names, they are used throught the project. Note that some of the folders are symlinks. Available datasets (in GitHub): mp_20, mp_20_biternary (binary and ternary structures from MP-20), mpts_52, carbon_24, perov_5. It is also possible to download and use matbench_discovery_mp_2022 notebook and matbench_discovery_mp_trj_full notebook.

For any data to be used for training, we need to do two preprocessing steps.

Compute and cache symmetry information

python scripts/cache_a_dataset.py <dataset-name>

This will create a pickled representaiton of the dataset in cache/<dataset-name>/data.pkl.gz. The script supports setting symmetry tolerance and this is not done automatically, the datasets which include tolerance in their name were obtained by manually using the command-line option.

Tokenization

The tokenization script serves two purposes: it produces the mapping from the real data to token ids, and saves the resulting tensors. To produce a new tokenizer:

python scripts/tokenise_a_dataset.py <dataset-name> <path-to-tokenizer-yaml> --new-tokenizer

Tokenizer configs are stored in yamls/tokenisers. The tokeniser is saved to cache/<dataset-names>/tokenisers/**.pkl.gz, preserving the folder structure of the config.

Alternatively, you can use a cached tokeniser. This is important when a model that was trained on one dataset is applied to a different dataset.

python scripts/tokenise_a_dataset.py <dataset-name> <path-to-tokenizer-yaml> --tokenizer-path cache/<dataset-names>/tokenisers/<tokenizer-name>.pkl.gz

Training

python scripts/train.py <path-to-model-yaml> <dataset-name> <device>

The model weights are saved to runs/<run-id>, and to WanDB, along with the tokenizer. See here for the list of configs. Adding --pilot will run the model for a small number of epochs.

Generating structures

Wyckoff representations

Wyckoff representations are produced and stored in WanDB during model training. They can be generated by:

scripts/generate.py --wandb-run <wandb-id> <output-file> --use-cached-tensors

Note that the code does not automatically download Wandb artifacts, you need to do it manually, and place them in the runs folder.

3D Structures

There are two ways to generate 3D structures from Wyckoff representations: DiffCSP++ and CHGNet. They later can be relaxed with CHGNet and/or DFT.

DiffCSP++

Wyckoffs can be relaxed with modified DiffCSP++ code

CrySPR + CHGNet

CrySPR scheme that combines pyxtal with CHGNet (or any other machine-learning interatomic potentials)

$ cp scripts/cryspr_pyxtal_chgnet.py mp_20/WyckoffTransformer/WyckoffTransformer_mp_20.json.gz /your/working/dir/
$ cd /your/working/dir/
$ gzip -d WyckoffTransformer_mp_20.json.gz
$ python ./cryspr_pyxtal_chgnet.py 0 -1 ./WyckoffTransformer_mp_20.json model_name
# model_name is optional, default is "model_name"
$ head -40 model_name_id_formula_energy_0-1000.csv
model,id,formula,energy
......
model_name,35,H6O8Si2,-97.98080444335938
model_name,36,CdDy,-6.170680522918701
model_name,37,Ba8Sr4As4O24,-264.228572845459
model_name,38,Co2Ni2F10O2,-81.85555267333984
......
$ ls 35/
min_e_strc.cif  trial-0  trial-1  trial-2  trial-3  trial-4  trial-5  trial-lowest

• There are n_trial_each_wyckoff_gene=6 trial calculations under each id directory.
• trial-lowest and min_e_strc.cif are the symbolic links to the one with the lowest energy out of all trials.
• Under each trial, ${reduced_formula}_${full_formula}_cell+pos.cif is the CHGNet relaxed structure.
• ${reduced_formula}_${full_formula}_2_cell+pos_symmetrized.cif is symmetrized structure by a certain symmetry precision (symprec), see each log file for details.
• This script can also be used as a module,

from scripts.cryspr_pyxtal_chgnet import single_run, single_pyxtal
import json
# from chgnet.model import CHGNetCalculator
# calculator = CHGNetCalculator(use_device="cpu")
# you could use other calculator
# the following codes do not take effect to set OMP_NUM_THREADS,
# please set this env var in your terminal, like in Bash
# by `export OMP_NUM_THREADS=1` before running Python
# otherwise you will get much slower CHGNet run!
import os
os.environ["OMP_NUM_THREADS"] = "1"

with open("./WyckoffTransformer_mp_20.json") as f:
    data = json.load(f)

gene = data[99]
atoms_in = single_pyxtal(gene)
atoms_relaxed = single_run(atoms_in)

CHGNet relaxation

The structures from all models can be optionally relaxed with CHGNet.

$ cp scripts/cryspr_chgnet.py mp_20/WyckoffLLM-naive/DiffCSP++/parsed_materials_10000_pyxtal.json_structures.json.gz /your/working/dir/
$ cd /your/working/dir/
$ gzip -d parsed_materials_10000_pyxtal.json_structures.json.gz
$ python ./cryspr_chgnet.py 0 -1 ./parsed_materials_10000_pyxtal.json_structures.json wylm-dcpp
$ head wylm-dcpp_id_formula_energy.csv
model,id,formula,energy
wylm-dcpp,0,Y4Ho4Ir8,-128.26178
wylm-dcpp,1,La4Cu4Si8,-89.13443
wylm-dcpp,2,Rb3Br3,-20.12497
wylm-dcpp,3,Gd1Ni2,-26.83080
wylm-dcpp,4,Pb3O18,-96.37995
wylm-dcpp,5,Tl2Au2S4Br4,-41.40260
wylm-dcpp,6,Ru1As2In3Ce3,-49.81073
wylm-dcpp,7,Pd4As4Ni4Se4,-73.34074
wylm-dcpp,8,Na4Lu4F16,-149.18192

• Output files are in the same format as above (CrySPR + CHGNet).
• ${reduced_formula}_${full_formula}_cell+pos.cif is the CHGNet relaxed structure.

DFT relaxation

We followed the Materials Project protocol, atomate2.vasp.flows.mp.MPGGADoubleRelaxStaticMaker. There isn't much to add, as the rest of the details of running DFT, unfortunately, depend on the HPC setup, and VASP is not open source. Here is the code to run at ASPIRE2.

Generated Data Analysis

Storage

Public Figshare

Most analyzed datasets in a uniform format are available at Figshare.

Private Dropbox

The raw files are stored in a private Dropbox. To pull:

rclone copy "NUS_Dropbox:/Nikita Kazeev/Wyckoff Transformer data/generated.tar.gz" . --progress
tar -xvf generated.tar.gz

To push:

tar --use-compress-program=pigz -cvf generated.tar.gz generated
rclone copy generated.tar.gz "NUS_Dropbox:/Nikita Kazeev/Wyckoff Transformer data/" --progress

Tar is used to handle the large number of small files, and pigz is used to speed up the compression. The Dropbox folder is private, if you are a collaborator, please contact for access.

Preprocessing

In order to be analyzed the data must be preprocessed and cached. To preprocess all generated datasets in generated/datasets.yaml:

poetry run python scripts/cache_generated_datasets.py

It supports filtering by dataset and transformations, e. g.:

poetry run python scripts/cache_generated_datasets.py --dataset mp_20 --transformations DiffCSP++ DFT

Completing this step will enable loading the data with evaluation.generated_dataset.GeneratedDataset.from_cache

Metric computation

The ICML 2025 results were computed by the notebooks in ICML_eval. They include, but not limited to the following metrics:

1. S.U.N. - the fraction of stable, unique, and novel structures.
2. S.S.U.N. - the fraction of symmetric, stable, unique, and novel structures.
3. Space Group $\chi^2$ - the $\chi^2$ statistic of the space group distribution between the generated and the test set.
4. P1 - the fraction of generated structures that lack internal symmetries, i. e. belong to space group P1.
5. Property similarity and naive validity from Xie et al.