🔬 materials science

GEODE: Symmetry-Preserving Cartesian Diffusion for Crystal Generation

GEODE is a novel generative model that combines coordinate and lattice diffusion in Cartesian space with Wyckoff-constrained losses to preserve crystal symmetries, achieving state-of-the-art rates of metastable, unique, and novel structures while enabling property-guided sampling.

Original authors: Yuchen Lou, Alex M. Ganose

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuchen Lou, Alex M. Ganose

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The search for new materials often feels like looking for a needle in a haystack, but with a twist: the haystack is made of trillions of possible arrangements of atoms, and most of them are unstable or useless. For decades, scientists have relied on high-throughput screening, a method that tests existing databases of known crystals against specific criteria to find promising candidates. This approach works well for finding what is already known, but it cannot invent what has never been seen. To truly discover new materials, researchers need to generate entirely new structures from scratch. The challenge lies in the nature of crystals themselves. Unlike a random pile of sand, a crystal is defined by a rigid, repeating pattern where atoms sit in specific, symmetrical arrangements. These patterns are not arbitrary; they follow strict mathematical rules called space groups, which dictate how the atoms can move and rotate while keeping the structure intact. Most real-world crystals, from table salt to complex semiconductors, obey these rules. However, when scientists try to use artificial intelligence to invent new crystals, the models often fail to respect these symmetries, producing structures that look like crystals but fall apart or simply do not exist in nature.

A team of researchers at Imperial College London has developed a new artificial intelligence system called GEODE to solve this specific problem. Instead of trying to force a general-purpose AI to learn symmetry rules on the fly, they built the rules directly into the machine's architecture. The system operates in two distinct stages. First, it selects a "symmetry template," which is essentially a blueprint specifying the type of repeating pattern and the allowed positions for atoms within that pattern. Once a blueprint is chosen, the system generates the full crystal structure, including the size of the repeating unit, the exact positions of every atom, and the type of atom at each position. Crucially, the system ensures that every step of this generation process respects the chosen blueprint. If the blueprint says an atom can only move along a specific line, the system never allows it to drift off that line. This approach stands in contrast to previous methods that treated symmetry as an optional condition or tried to fix broken structures after they were generated. By treating symmetry as a fundamental constraint from the very beginning, the researchers found that the AI could produce stable, unique, and novel materials much more reliably than before.

The researchers tested their system against a standard set of over 45,000 known inorganic crystals. They asked the AI to generate thousands of new structures and then checked how many of these were valid, unique, and stable enough to potentially exist. The results showed a significant improvement over other symmetry-aware models. Without any extra adjustments, the system produced a rate of successful, novel crystals that was nearly double that of the next best model in its category. When the researchers added a simple filtering step to the process—selecting only the most promising blueprints before generation began—the success rate jumped even higher, surpassing even some of the most advanced models that ignore symmetry rules entirely. This improvement was achieved without retraining the model, simply by being more selective about the starting templates. The system also demonstrated the ability to guide the generation process toward specific physical properties. For example, the researchers showed they could direct the AI to create cubic crystals with a specific ability to store electrical energy, all while ensuring the material remained perfectly symmetrical.

One of the key innovations in this work was how the system handled the physical scale of the atoms. Many previous models added random noise to the positions of atoms in a way that depended on the size of the crystal cell, which made the learning process inconsistent. GEODE instead added noise in a way that represented a fixed physical distance, regardless of the crystal's size. This small change made the training process more stable and the final results more accurate. The researchers also introduced a method to handle the fact that identical atoms in a crystal can swap places without changing the structure. By allowing the system to recognize these swaps during training, they prevented the AI from getting confused by arbitrary labels, leading to cleaner and more accurate predictions. The system was tested on two different datasets, and in both cases, it outperformed its competitors in generating structures that were not only mathematically valid but also physically plausible.

The study highlights a shift in how artificial intelligence approaches materials discovery. Rather than relying on massive amounts of data to guess the rules of symmetry, the researchers embedded those rules directly into the model's design. This approach proved to be more efficient and effective, allowing the system to navigate the vast space of possible crystal structures with greater precision. While the system is not yet a perfect solution for all material discovery challenges, it represents a significant step forward in generating stable and novel crystals. The ability to control specific properties, such as electrical behavior, while maintaining strict structural symmetry suggests a path toward designing materials with tailored functions. The researchers noted that the process remains computationally expensive, requiring significant time to generate each structure, but the quality of the results suggests that the investment is worthwhile. By combining a deep understanding of crystal symmetry with modern generative techniques, GEODE offers a powerful new tool for scientists looking to expand the boundaries of what materials are possible.

Technical Summary: GEODE – Symmetry-Preserving Cartesian Diffusion for Crystal Generation

Problem Statement

Computational materials discovery has traditionally relied on high-throughput screening of existing databases, which limits the discovery of truly novel materials. Generative models offer a path to sampling new structures directly; however, existing models struggle to reproduce the non-trivial space group symmetries observed in most known inorganic crystals. While some models condition on space group labels, they often fail to recover the target symmetry (recovering it only 10–20% of the time) and disproportionately generate structures in the trivial space group $P1$ .

Furthermore, current symmetry-aware models face architectural and training challenges:

Coordinate Representation: Most models diffuse fractional coordinates, where the physical scale of noise depends on the cell size and shape, complicating the learning process.
Lattice-Coupling: Models using Cartesian diffusion have historically kept the lattice fixed to avoid the complexity of coupled updates, missing the opportunity to generate diverse unit cells.
Uniqueness and Novelty: High-symmetry crystals often share limited structural prototypes, leading generative models to produce repetitive structures or those already present in databases, reducing uniqueness and novelty metrics.

Existing approaches that enforce symmetry often perform worse than symmetry-agnostic models on standard benchmarks for stability, uniqueness, and novelty.

Methodology: GEODE

The authors introduce the Generative Equivariant Orbit Diffusion Engine (GEODE), a symmetry-aware generative model that operates in two distinct stages:

1. Wyckoff Template Sampling (VAE)

GEODE first samples a symmetry template $\tau = (G, W)$ , consisting of a space group $G$ and a list of Wyckoff orbits $W$ .

Architecture: An $E(3)$ -equivariant message-passing encoder processes atom types and Wyckoff site embeddings to parameterize a Gaussian latent vector $z$ . An MLP decoder predicts the space group distribution, while an autoregressive transformer generates the sequence of Wyckoff orbits conditioned on $z$ and the space group.
Training: The Variational Autoencoder (VAE) is trained to reconstruct the space group and orbit sequence from the training data, regularized by a KL divergence term.

2. Joint Cartesian Diffusion

Given a sampled template, a diffusion model jointly generates the crystal structure $C = (A, X, L)$ (atom types, coordinates, and lattice) while strictly preserving the template's symmetry.

Cartesian Diffusion: Unlike prior works using fractional coordinates, GEODE diffuses atomic coordinates in Cartesian space. This ensures the noise scale represents a consistent physical distance (Ångströms) regardless of cell dimensions.
Lattice Diffusion: The model diffuses the six cell parameters ( $a, b, c, \alpha, \beta, \gamma$ ) rather than the lattice matrix. A projection operator $P_G$ ensures that noise added to the lattice respects the constraints of the specific crystal system (e.g., equal lengths for cubic systems, fixed angles for orthorhombic systems).
Symmetry-Preserving Noise: Coordinate noise is constructed to be correlated within each Wyckoff orbit. A random vector is sampled for a representative atom, averaged over its site-symmetry group to remove components outside its degrees of freedom, and then transformed to all equivalent atoms.
Wyckoff-Constrained Loss:
- Reachable Copies: The training target (wrapped-normal score) is restricted to periodic images that are reachable within the atom's allowed degrees of freedom, avoiding inconsistent gradients.
- Permutation Remapping: Before calculating the loss, the model matches perturbed atoms to reference sites within the same Wyckoff orbit using minimum-cost linear assignment (Hungarian algorithm). This prevents the loss from depending on arbitrary atom labeling when equivalent atoms swap positions.
Atomic Type Diffusion: Atom types are diffused using an absorbing-state process over Wyckoff orbits, ensuring all atoms in an orbit share the same species.

3. Sampling Time Template Filtering

To enhance novelty without retraining, the authors propose filtering the distribution of sampled templates. Templates are retained only if they occur within a specific frequency range in the training set (1–1,000 times) and contain a specific number of orbits (4–6). This biases the generation toward under-represented but feasible structural prototypes.

4. Property Guidance

The model supports classifier-free guidance to control properties (e.g., permittivity) while maintaining symmetry. By fine-tuning the diffusion model on a property dataset and using Feature-wise Linear Modulation (FiLM), the model can generate cubic crystals with target dielectric constants, ensuring isotropic responses.

Key Contributions

Joint Cartesian Diffusion: GEODE is the first model to combine coordinate and lattice diffusion in Cartesian space, providing a consistent physical noise scale and enabling isotropic updates across different cell dimensions.
Wyckoff-Constrained Loss: The introduction of a loss function that accounts for permutation remapping and restricts targets to reachable periodic copies, ensuring the model respects coordinate constraints and reduces training noise.
Template Filtering: A novel sampling-time strategy that increases the novelty of generated structures by filtering the template distribution, improving performance without additional model training.
Joint Guidance: Demonstration of controlling both symmetry (via template selection) and physical properties (via classifier-free guidance) simultaneously.

Results

Evaluated on the MP-20 and Alex-MP-20 datasets using the LeMat-GenBench framework:

Performance: GEODE achieves a metastable, unique, and novel (mSUN) rate of 11.92% on MP-20 without structure relaxation. This outperforms the next best symmetry-aware model (SGFM at 7.68%) and approaches the performance of leading symmetry-agnostic models.
Relaxed Performance: After relaxation with EquiformerV3, GEODE reaches an mSUN of 16.40% on MP-20.
Impact of Filtering: Applying the sampling-time template filter increases the relaxed mSUN to 23.80% on MP-20, surpassing all other entries on the public leaderboard as of September 2026.
Symmetry Fidelity: 98.4% of generated structures are assigned exactly the space group specified by their template.
Ablation Studies: Replacing Cartesian diffusion with fractional coordinates lowers mSUN from 11.9% to 9.3%. The permutation remapping and reachable-copy restrictions contribute significantly to the final performance.
Property Control: The model successfully generates cubic crystals with target permittivities (e.g., median predicted values of 2.1, 6.2, 10.5, and 12.2 for targets of 2, 5, 10, and 15), maintaining cubic symmetry throughout.

Significance and Limitations

The paper claims that GEODE demonstrates that explicit symmetry treatment in generative models can yield competitive results in stability, uniqueness, and novelty, challenging the notion that symmetry-agnostic models are superior. By separating template generation from crystal diffusion, the model allows for post-hoc optimization of the generation process (via filtering) without retraining.

Limitations acknowledged by the authors:

Two-Stage Training: The template generator and diffusion model are trained independently, meaning the template generator receives no feedback on the stability of the resulting crystals. This can lead to the proposal of templates that are difficult for the diffusion model to realize.
Fixed Symmetry: The space group remains fixed during the diffusion process; the model cannot resolve an unsuitable template by changing its symmetry.
Computational Cost: Sampling requires 1,000 reverse steps, each evaluating an equivariant network over the full cell, making the process computationally expensive.
Scope of Guidance: Property guidance was demonstrated only for a scalar property (permittivity) within a single crystal system (cubic). Broader applicability across properties and symmetry classes requires further testing.