PDDFormer: Pairwise Distance Distribution Graph Transformer for Crystal Material Property Prediction

## Abstract

Abstract To date, the International Zeolite Association Structure Commission (IZA-SC) has cataloged merely 255 distinct zeolite structures, with millions of theoretically possible structures yet to be discovered.
The synthesis of a specific zeolite typically necessitates the use of an organic structure-directing agent (OSDA), since the selectivity for a particular zeolite is largely determined by the affinity between the OSDA and the zeolite.
Therefore, finding the best affinity OSDA-zeolite pair is the key to the synthesis of targeted zeolite.
However, OSDA-zeolite pairs frequently exhibit complex geometric structures, i.e., a complex crystal structure formed by a large number of atoms.
This geometric structure exhibits coarse-grained crystal periodicity and fine-grained local variability:
(1) Fine-grained local variability: From the perspective of fine-grained structure, each unit cell shares the same crystal lattice, but there are subtle structural differences, such as the different poses of OSDA in an OSDA-zeolite pair, as shown in figure 1 (c).
(2) Coarse-grained periodicity: Despite the subtle differences within the unit cell, the overall structure of an OSDA-zeolite pair can still be viewed as the periodic repetition of the unit cell in space from a coarse-grained structural perspective.
Although some existing machine learning methods can represent the periodicity of crystals, they cannot accurately represent crystal structures with local variability.
To address this issue, we propose a novel approach called Zeoformer, which can effectively represent coarse-grained crystal periodicity and fine-grained local variability.
Zeoformer reconstructs the unit cell centered around each atom and encodes the pairwise distances between this central atom and other atoms within the reconstructed unit cell.
The introduction of pairwise distances within the reconstructed unit cell more effectively represents the overall structure of the unit cell and the differences between different unit cells, enabling the model to more accurately and efficiently predict the properties of OSDA-zeolite pairs and general crystal structures.
Through comprehensive evaluation, our Zeoformer model demonstrates the best performance on OSDA-zeolite pair datasets and two types of crystal material datasets.

## 1 Introduction

Figure: Figure 1: This example depicts that a zeolite (e.g. SOV in (b)) is combined (docked) with an OSDA (e.g. $C_{6}H_{1}5NO_{2}$ in (a)) in various poses, to form an OSDA-zeolite pair ($C_{6}H_{1}5NO_{2}$-SOV in (c)). In (b) and (c), the blue lines are artificial boundaries used to form a possible unit cell, which repeats in infinite space to create the given structures. The black lines represent the supercell formed by repeating the unit cell in the XOY plane.
Refer to caption: x1.png

Figure: Figure 2: Comparison of different methods for constructing periodic patterns. The blue dashed lines indicate artificial boundaries for defining a possible unit cell, i.e., lattice parameters. In Figure (c), self-connecting edges are generated based on lattice parameters, whereas in Figure (d), self-connecting edges are computed according to the spatial distances between atoms and their copies. As shown in Figures (a) and (b), there are subtle differences between unit cells. In Figure (a), the green dashed line outlines the reconstructed unit cell centered around atoms. The subtle differences between unit cells can be reflected in the pairwise distances between atoms (such as $d_{1}$ and $d_{2}$). $d_{1}$ and $d_{2}$ represent different distances for the same pair of atoms, thereby indicating the positional differences of the same atom (in red) in different unit cells. In OSDA-zeolite structures with a large number of atoms, the differences in this pairwise distance encoding become more pronounced.
Refer to caption: x2.png

Zeolite (a type of porous crystal) materials have garnered widespread attention for their applications in gas adsorption  Pérez-Botella et al. (); Ackley et al. (), catalysis  Weitkamp (); Chai et al. (), and drug delivery  Servatan et al. (); Serati-Nouri et al. ().
Zeolite structures are entirely composed of a periodic repeating structure of $TO_{4}$ tetrahedra (where $T$ represents $Si$, $Al$, or $P$), which form three-dimensional (3D) networks by sharing vertex oxygen atoms $O$ Li and Yu ().
The diversity in how these periodic tetrahedra connect results in a wide variety of zeolite structure types, theoretically allowing for millions of possible configurations Schmidt et al. ().
However, designing synthetic routes for target zeolite topologies is a labor-intensive task.
To date, the International Zeolite Association Structure Commission (IZA-SC) Baerlocher and McCusker () has cataloged only $255$ distinct zeolite structure types.
In chemical reactions, zeolites typically crystallize under hydrothermal conditions in the presence of inorganic precursors and organic templates, forming as metastable phases that require organic structure-directing agents (OSDAs) to compensate for energy loss and provide additional energy to stabilize these phases and enable diversification Shin et al. (); Moliner et al. (); Dusselier and Davis ().

Specifically, synthesizing a specific zeolite usually requires an OSDA, since the selectivity for a particular zeolite is largely determined by the affinity between the OSDA and the zeolite  Shin et al. ().
Therefore, finding the best affinity OSDA-zeolite pair is the key to the synthesis of targeted zeolite.
An example in Figure 1 shows the combination of zeolite and OSDA forms the structure of the OSDA-zeolite pair.
However, the OSDA-zeolite pair often has a complex geometric structure, i.e., a complex crystal structure formed by a large number of atoms.
This geometric structure exhibits coarse-grained crystal periodicity and fine-grained local variability:
(1) Fine-grained local variability: From the perspective of fine-grained structure, each unit cell shares the same crystal lattice, but there are subtle structural differences, such as the different poses of OSDA in an OSDA-zeolite pair, as shown in 1(c).
(2) Coarse-grained periodicity: Despite the subtle differences within the unit cell, the overall structure of an OSDA-zeolite pair can still be viewed as the periodic repetition of the unit cell in space from a coarse-grained structural perspective.
Therefore, this kind of crystal structure cannot be accurately represented by simply repeating a unit cell according to lattice parameters.
This complex geometric structure challenges most existing deep-learning methods.

Based on the assumption that the relative coordinates of atoms between different unit cells in a crystal remain unchanged, many deep learning methods Yan et al. (); Lin et al. (); Yan et al. () represent crystal structures by modeling the crystal graph of a single unit cell along with its periodic pattern.
For example, Matformer Yan et al. () directly encodes periodic patterns by assigning six self-connected edges determined by a manually defined lattice matrix $\mathcal{L}=\left[l_{1},l_{2},l_{3}\right]^{T}\in\mathbb{R}^{3\times 3}$ for each atom within the unit cell, as shown by the six solid blue lines in Figure 2(c). In Figure 2(a), the black arrows show the periodic encoding of an atom in a two-dimensional structure based on lattice parameters, which may point to non-existent atomic positions.
Conformer Yan et al. () encodes periodicity by incorporating the spatial distances between different replicas of each atom along three directions as edge information, as shown in Figure 2(d). As shown by the black arrows in Figure 2(b), the periodic encoding based on this method reflects the differences for specific atoms. However, for unit cells with a large number of atoms, this method may not adequately capture the local differences between different unit cells.
Although these periodic encoding methods achieve periodicity, they overlook fine-grained local variability.
Specifically, in OSDA-zeolite structures, there are differences in the relative coordinates of atoms between different unit cells.
Conformer and Matformer encode periodicity at the level of each individual atom, making them ineffective at capturing local variability caused by changes in atomic positions across different unit cells.
More related methods and detailed discussions are presented in Appendix D.

To effectively capture the differences between different unit cells, we need to encode the coarse-grained periodic pattern from the perspective of the entire unit cell, rather than considering each atom individually.
Therefore, we reconstruct the unit cell centered around each atom in the original unit cell and encode the pairwise distances between the central atom and other atoms within the reconstructed unit cell. This method encodes the spatial distances between a given atom and the nearest replicas of other atoms within the unit cell, rather than only considering the replicas of the atom itself.
When overlapping the reconstructed unit cells for all atoms, the extent of the atom-reconstructed unit cells in two-dimensional space, as shown in Figure 2(e) (indicated by the green box), can extend up to four times the range of the original unit cell (indicated by the black box), and up to eight times in three-dimensional space, forming a supercell.
This encoding method better identifies subtle differences between different unit cells, allowing the model to more accurately represent the geometric structure of OSDA-zeolite pairs, specifically the coarse-grained crystal periodicity and fine-grained local variability.
The green box in Figure 2(a) provides a simplified illustration. For general crystal materials, this periodic encoding also offers significant advantages. For a detailed analysis, please refer to Appendix D.
Finally, these encodings are integrated into the Transformer model, Zeoformer, achieving the best results in predicting OSDA-zeolite affinity and crystal material properties.

The main contributions are as follows:

- •
To the best of our knowledge, we are the first to conduct the prediction of the affinity between OSDAs and target zeolite topologies in the field of machine learning, which is crucial for guiding zeolite synthesis by screening OSDAs.
- •
In order to well represent the geometric structure of OSDA-zeolite pairs, i.e., the coarse-grained crystal periodicity with fine-grained local variability, we propose a novel deep learning model, called Zeoformer, which introduces pairwise distance encoding in the reconstructed unit cell.
- •
We conducted experiments on a zeolite-molecule pairing dataset (OSDB Jensen et al. ()) and two commonly used materials benchmark datasets (Materials Project Chen et al. () and JARVIS Choudhary et al. ()). The experiments validated the importance of our components and achieved better results.

## 2 Related Work

### 2.1 OSDA-guided Zeolite Synthesis

Some studies Moliner et al. (); Daeyaert et al. () propose candidate OSDAs for specific zeolite structures using computational methods, including density functional theory and molecular dynamics. Schwalbe and Gomze Schwalbe-Koda and Gómez-Bombarelli () address biases in simulation methods by comparing different approaches to calculate the interaction energy between zeolites and OSDAs, providing robust guidelines for computing binding energies in zeolite-OSDA pairs. Jensen et al. Jensen et al. () employ data-driven approaches to study the relationship between OSDAs, qualitative gel chemistry, and resulting zeolite structures. Machine learning has been applied in studies that do not explicitly model OSDA-zeolite interactions, including the investigation of specific zeolites within finite chemical spaces Corma et al. (); Manuel Serra et al. (), OSDA-free zeolite systems Muraoka et al. (), and zeolite phase transformations Schwalbe-Koda et al. (). However, existing research often simplifies OSDA representation to basic attributes such as molecular volume Jensen et al. () or is limited to single zeolite structures Daeyaert et al. (). In contrast, we adopt the concept of graph neural networks to model OSDA-zeolite pairs as graphs, aiming to predict their affinities for screening suitable OSDAs to guide zeolite synthesis.

### 2.2 Crystal Property Prediction with Deep Learning

CGCNN Xie and Grossman () learns material properties from the connections between atoms in crystals. MegNet Chen et al. () is a graph network with global state attributes. GATGNN Louis et al. () uses multiple graph attention layers (GAT) to learn properties of local neighborhoods and global attention layers to weight global atomic features. ALIGNN Choudhary and DeCost () and M3GNet Chen and Ong () incorporate angle information into message passing to produce richer and more discriminative representations. CrysMMNet Das et al. () uses a multimodal framework that integrates graph and text representations to generate joint multimodal representations of crystalline materials. CrysDiff Song et al. () is a pretraining-finetuning framework based on diffusion models. However, these aforementioned methods struggle with accurately representing lattice parameters and crystal periodicity.
Matformer Yan et al. () encodes periodic patterns through self-connecting edges of atoms. PotNet Lin et al. () considers the infinite summation of pairwise atomic interactions. Crystalformer Taniai et al. () considers infinitely connected attention, thereby performing infinite summations of interatomic potentials. ComFormer Yan et al. () represents the geometric information of crystals using equivariant vector representations and invariant geometric descriptors of Euclidean distances and angles.

## 3 Preliminaries

This section provides a brief introduction to crystal structures and OSDA-zeolite structures.
For detailed information on OSDA-zeolite structures and the periodic invariance of crystals, please refer to Appendix C.

#### The Structure of Crystals.

The crystal is a substance with a highly ordered structure. By selecting an appropriate structural unit, the entire crystal can be viewed as the periodic repetition of this unit in space.
This property of atoms within a crystal repeating
in 3D space according to a specific pattern is called periodicity, with the smallest repeatable structural unit being the unit cell of the crystal.
The unit cell fills the crystal space when the structural unit is translated through lattice vectors.
Thus, the unit cell can be defined as $\mathcal{U}=\left(\mathcal{X},\mathcal{P}\right)$, where $\mathcal{X}$ and $\mathcal{Z}$ can be represented in matrix form.
Typically, $\mathcal{X}=\left[x_{1},x_{2}\cdots x_{n-1},x_{n}\right]^{T}\in\mathbb{R}^{n
\times 1}$, where
$n$ represents the number of atoms and ${x_{i}}\in\mathbb{R}^{1}$ represents the atomic type of atom ${i}$ in the unit cell.
$\mathcal{P}=\left[p_{1},p_{2}\cdots p_{n-1},p_{n}\right]^{T}\in\mathbb{R}^{n
\times 3}$ is the atomic position matrix, where ${p_{i}}\in\mathbb{R}^{3}$ represents the Cartesian coordinates of the atom ${i}$ in the unit cell in 3D space.
To further encode the periodic pattern, additional lattice vectors $\mathcal{L}=\left[l_{1},l_{2},l_{3}\right]^{T}\in\mathbb{R}^{3\times 3}$ are used to describe how the unit cell repeats itself in three directions.
Therefore, in 3D space, we can represent the crystal as $\left(\mathcal{X},\mathcal{P},\mathcal{L}\right)$.
Definition 1 (Unit Cell $E(3)$ Invariance): Following Matformer Yan et al. (), A function $\mathit{f}:\left(\mathcal{X},\mathcal{P},\mathcal{L}\right)\to\mathcal{Y}^{\prime}$ is unit cell $E(3)$ invariant if, for all $Q\in\mathbb{R}^{3\times 3}$, where $|Q|=1$, and $b\in\mathbb{R}^{3}$, we have $\mathit{f}\left(\mathcal{X},\mathcal{P},\mathcal{L}\right)=\mathit{f}\left(
\mathcal{X},Q\mathcal{P}+b,Q\mathcal{L}\right)$.

In other words, the crystal structure remains unchanged when the position matrix $\mathcal{P}$ of the unit cell structure undergoes rotation, translation, or reflection.

#### The Structure of OSDA-zeolite.

OSDA-zeolite is a variant of crystal structure. It still adheres to the geometric constraints of crystals, but there are local differences between different unit cells in the OSDA-zeolite structure.
Therefore, we can still represent the OSDA-zeolite structure as a crystal graph, but this representation needs to reflect the differences between different unit cells.

## 4 The Proposed Zeoformer

In this section, we propose Zeoformer, with a key focus on encoding the coarse-grained Periodic Pattern Encoding of crystal structures, referred to as $\mathcal{PPE}$, as illustrated by the Update $\mathcal{PPE}$ layer in Fig 5.
We first demonstrate how to construct coarse-grained periodic patterns that reflect differences between different unit cells, and then explain how the crystal graphs are built upon periodic patterns to ensure the $E(3)$ invariance of the crystal graphs.
Finally, we present the overall framework of Zeoformer.

Figure: Figure 3: The schematic diagram illustrates periodic pattern encoding. Light blue dashed lines represent artificial boundaries forming possible unit cells that repeat the given OSDA-zeolite structure in infinite space. Here, zeolite atoms are marked in red, and OSDA atoms are represented in other colors. Figure (a) shows the schematic representation of periodic patterns in 2D space outlined by the black box. At this point, the periodic encoding is confined to the unit cell where the atoms are located, treating the crystal as a finite structure. Figure (b) demonstrates our periodic pattern construction by building unit cells centered around each atom. This approach allows us to capture differences between different unit cells, thereby better representing coarse-grained periodicity.
Refer to caption: x3.png

Figure: Figure 4: We present an example of a 3D crystal structure along with a corresponding simplified schematic. For the 2D schematic, we use light blue dashed lines to represent the periodic unit cell, and black solid lines to indicate the atoms included in the periodic encoding centered on a specific atom. We demonstrate the transformations of the unit cell under translation, rotation, and reflection conditions.
Refer to caption: x4.png

### 4.1 Encoding of Coarse-grained Periodic Pattern

Previous methods Yan et al. (); Lin et al. (); Yan et al. () for constructing periodic patterns typically rely on the assumption that the relative coordinates of atoms between different unit cells in a crystal remain unchanged. However, in OSDA-zeolite structures, the atomic coordinates can vary between different unit cells.
As a result, we cannot encode periodic structures independently for each atom, as was done in the past using lattice parameters.
Instead, we need to encode periodicity from the perspective of the entire unit cell.
This means that when constructing periodicity, the focus should be on the overall structure of all atoms within the unit cell, rather than treating each atom as an independent entity for encoding.
We achieve this by reconstructing the unit cell centered around each atom and encoding the pairwise distances between this central atom and other atoms within the reconstructed unit cell.

Formally, given the position $p_{i}$ of the atom $i$ and the modified positions of other atoms within the reconstructed unit cell $\mathcal{P}_{j}=p_{j}+k_{1}l_{1}+k_{2}l_{2}+k_{3}l_{3}$, where $k_{1},k_{2},k_{3}$ belong to $\left[-1,0,1\right]$.
These positions represent the nearest copies of the central atom $i$ and the remaining atoms in its reconstructed unit cell, as shown in Figure 3.
The following equation provides the periodic pattern encoding, denoted as $\mathcal{PPE}$:
$\begin{array}[]{@{}c@{}c@{}}\mathcal{D}_{i}=\left\{{\textstyle\bigcup_{j}^{n}}
\sqrt{\left(\overline{p_{i}}-\overline{p_{j}}\right)^{2}}\mid i,j\in\mathcal{Z
},\overline{p_{i}},\overline{p_{j}}\in\overline{\mathcal{P}},1\leq j\leq n
\right\}\\
d_{i}=\left(\overline{p_{i}}-\overline{p_{j}}\right)^{2}=\left(\overline{x_{i}
}-\overline{x_{j}}\right)^{2}+\left(\overline{y_{i}}-\overline{y_{j}}\right)^{
2}+\left(\overline{z_{i}}-\overline{z_{j}}\right)^{2},\\
\mathcal{PPE}=\left\{\frac{1}{\bigcup_{i}^{n}\mathcal{D}_{i}}\mid i\in\mathcal
{Z},1\leq i\leq n,d_{i}\ni\mathcal{D}_{i}\neq 0\right\}\end{array}$
where $\mathcal{D}_{i}$ represents the union of feature vectors of distances between the atom $i$ and other atoms within the unit cell centered on the atom $i$, with $\mathcal{D}_{i}\in\mathbb{R}^{n}$, and $n$representing the number of atoms in the unit cell.
$\mathcal{PPE}\in\mathbb{R}^{n\times n}$ represents the union of distance features between all atoms.
Since the interaction energy between an atom and its neighboring atoms is usually inversely proportional to the distance, we take the reciprocal of the distance feature after removing zeros.

Figure: Figure 5: Description of the Zeoformer Architecture. (a) Overview of the Zeoformer Workflow. (b) Detailed Description of the Update Node (from (a)). (c) Detailed Description of the Update PPE (from (a)).
Refer to caption: x5.png

### 4.2 Crystal Graph Construction

We represent our crystal graph based on the method of constructing multi-edge crystal graphs.
In the graph, each node represents an atom $i$ and all its infinite duplicates in 3D space,
with positions $\{\hat{p}_{i}|\hat{p}_{i}=p_{i}+k_{1}l_{1}+k_{2}l_{2}+k_{3}l_{3},k_{1},k_{2},k
_{3}\in\mathcal{Z}\}$, and node features $x_{i}$.
An edge is established from node $j$ to node $i$ when the Euclidean distance $\left|e_{j^{\prime}i}\right|^{2}$ between a duplicate of $j$ and $i$ satisfies $\left|e_{j^{\prime}i}\right|^{2}=\left|p_{j}+k^{\prime}_{1}l_{1}+k^{\prime}_{2
}l_{2}+k^{\prime}_{3}l_{3}-p_{i}\right|^{2}\leq r$,
where $r\in\mathbb{R}$ is the cutoff radius.
We select the nearest $t$ edges within the cutoff radius, each with a corresponding edge feature $\left|e_{j^{\prime}i}\right|^{2}$.
As shown in Figure 3(b), the blue circles represent the region defined by the cutoff radius.
At the same time, the black boxes indicate the atoms included in our periodic pattern encoding.
Formally, we represent the constructed crystal graph as $\mathcal{G}=(\mathcal{X},\mathcal{XI},\mathcal{E},\mathcal{PPE})$.
Therein, $x_{i}\in\mathcal{X}$ is the feature vector of the atom $i$, $e_{ij}^{h}\in\mathcal{E}$ is the feature vector of the $h$-th edge between nodes $i$ and $j$, and we denote $\mathcal{XI}$ as the indices of the nodes $i$ and $j$ that form the edge.
$\mathcal{PPE}\in\mathbb{R}^{n\times n}$ is the periodic pattern encoding we introduced above, where $n$ is the number of atoms in the constructed graph.

#### Proof of the E ⁢ ( 3 ) 𝐸 3 E(3) italic_E ( 3 ) Invariance of the Constructed Crystal Graphs.

As shown in Figure 4, applying $E(3)$ transformations alters the absolute positions of atoms within the unit cell but does not change the types of atoms, their relative positions within the crystal, or the lattice parameters; in other words, it changes the spatial distances between atoms, but the Euclidean distances remain invariant. When reconstructing the lattice parameters centered on each atom, the spatial distances between different replicas of that atom change, resulting in a change in the orientation of the reconstructed unit cell, while its size and shape remain unchanged. The atoms selected within this unit cell and their nearest replicas maintain their Euclidean distances relative to the central atom. Consequently, the constructed periodic pattern encoding remains invariant. Additionally, since we construct multi-edge crystal graphs solely based on the Euclidean distances between atoms in the crystal, the constructed crystal graphs remain invariant under $E(3)$ transformations.

**Table 1: Comparison of test MAE on the OSDB dataset. To ensure clarity and fairness of comparison, we present results from models retrained using the same training, validation, and test sets. The best result is shown in bold.**
|  | Competition Total | Binding Total | Directivity Energy | Templating |
| --- | --- | --- | --- | --- |
| Method | kJ/mol OSDA | kJ/mol OSDA | kJ/mol Si | kJ/mol |
| CGCNN | 23.54 | 17.48 | 0.330 | 0.283 |
| GATGNN | 53.84 | 46.28 | 0.502 | 0.423 |
| Matformer | 15.61 | 12.45 | 0.243 | 0.214 |
| iComFormer | 10.52 | 6.534 | 0.132 | 0.168 |
| Zeoformer | 9.62 | 5.996 | 0.115 | 0.129 |

**Table 2: Comparison between Zeoformer and other baselines in terms of test MAE on JARVIS dataset. The best results are shown in bold and the second best results are shown with underlines.**
|  | Formation Energy | Bandgap(OPT) | Total Energy | Ehull | Bandgap(MBJ) |
| --- | --- | --- | --- | --- | --- |
| Method | eV/atom | eV | eV/atom | eV | eV |
| CFID (2018) | 14 | 0.30 | 240 | 220 | 0.53 |
| CGCNN (2018) | 63 | 0.20 | 78 | 170 | 0.41 |
| SchNet (2018) | 45 | 0.19 | 47 | 140 | 0.43 |
| MEGNET (2019) | 47 | 0.145 | 58 | 84 | 0.34 |
| GATGNN (2020) | 47 | 0.17 | 56 | 120 | 0.51 |
| ALIGNN (2021) | 33.1 | 0.142 | 37 | 76 | 0.31 |
| M3GNet (2022) | 39.0 | 0.145 | 41 | 95 | 0.36 |
| Matformer (2022) | 32.5 | 0.137 | 35 | 64 | 0.30 |
| PotNet (2023) | 29.4 | 0.127 | 32 | 55 | 0.27 |
| CrysMMNet (2023) | 28.0 | 0.128 | 34 | – | 0.278 |
| CrysDiff (2024) | 29.0 | 0.131 | 34 | 62 | 0.287 |
| Crystalformer (2024) | 30.6 | 0.128 | 32 | 46 | 0.274 |
| eComFormer (2024) | 28.4 | 0.124 | 32 | 44 | 0.28 |
| iComFormer (2024) | 27.2 | 0.122 | 28.8 | 47 | 0.26 |
| Zeoformer | 26.9 | 0.120 | 28.0 | 33.5 | 0.25 |

**Table 3: Comparison between Zeoformer and other baselines in terms of test MAE on MP dataset.**
|  | Formation Energy | Band Gap | Bulk Moduli | Shear Moduli |
| --- | --- | --- | --- | --- |
| Method | eV/atom | eV | log(GPa) | log(GPa) |
| CGCNN | 31 | 0.292 | 0.047 | 0.077 |
| SchNet | 33 | 0.345 | 0.066 | 0.099 |
| MEGNET | 30 | 0.307 | 0.060 | 0.099 |
| GATGNN | 33 | 0.280 | 0.045 | 0.075 |
| ALIGNN | 22 | 0.218 | 0.051 | 0.078 |
| M3GNet | 24 | 0.247 | 0.050 | 0.087 |
| Matformer | 21.0 | 0.211 | 0.043 | 0.073 |
| PotNet | 18.8 | 0.204 | 0.040 | 0.065 |
| CrysMMNet | 20.0 | 0.197 | 0.038 | 0.062 |
| Crystalformer | 18.6 | 0.198 | 0.0377 | 0.0689 |
| eComFormer | 18.16 | 0.202 | 0.0417 | 0.0729 |
| iComFormer | 18.26 | 0.193 | 0.0380 | 0.0637 |
| Zeoformer | 17.59 | 0.190 | 0.0388 | 0.0697 |

### 4.3 The Architecture of Zeoformer

In this section, we introduce the information propagation scheme of Zeoformer. The information propagation scheme of Zeoformer consists of four parts: the graph embedding layer, node-wise transformer layer, inspired by the ConformerYan et al. (), $\mathcal{PPE}$ message passing layer, and output layer. Figure 5 illustrates the overall framework architecture of Zeoformer.

#### Feature Embedding Layer.

First, let’s introduce the construction of the graph embedding layer.
We use the atomic encoding from CGCNN for embedding. For the edge information $e_{ij}^{h}$, we use radial basis functions to encode the distance between two adjacent nodes in the graph, represented by Equation 1, where $\gamma$ and $\mu$ are hyperparameters. For the periodic pattern encoding, we perform matrix multiplication on $\mathcal{PPE}$ to carry out the initial aggregation of atomic information, obtaining information for the $\mathcal{PPE}$ message passing layer. Thus, we obtain the graph embedding as

$$ $\begin{array}[]{@{}c@{}c@{}}\mathcal{A}={\mathcal{CGCNN}}(\mathcal{X}),\>e_{ij }^{h}=\exp\left(-\gamma\left(\frac{\left\|p_{i}-p_{j}\right\|^{2}}{\mu}\right) \right),\\ \mathcal{PPE}=\mathcal{PPE}\ast\mathcal{A}\end{array}$ (1) $$

where $\otimes$ denotes the Hadamard product.

#### Node-wise Transformer Layer.

Building upon the constructed graph, we aggregate the node information. Let $a_{i}^{l}$ represent the input feature vector of node $i$ at layer $l$ in Zeoformer.
The information propagation scheme for layer $l$ is described as follows.
We use $\oplus$ and $\odot$ to denote concatenation and element-wise product.
The message from node $j$ to $i$ is formed by the corresponding query $q_{ij}^{l}$, key $k_{ij}^{l}$, and value features $v_{ij}^{l}$ as follows :

$$ $\begin{array}[]{@{}c@{}c@{}c@{}}q_{ij}^{l}=LQ\left(a_{i}^{l}\right),\>k_{ij}^{ l}=\left(LK\left(a_{i}^{l}\right)\oplus LK\left(a_{j}^{l}\right)\oplus LE\left (e_{ij}^{h}\right)\right),\\ v=\left(LV\left(a_{i}^{l}\right)\oplus LV\left(a_{j}^{l}\right)\oplus LE\left( e_{ij}^{h}\right)\right),\\ v_{ij}^{l}=v\odot sigmoid\left(LNorm\left(LN_{update}\left(v\right)\right) \right),\\ att^{l}=\frac{q_{ij}^{l}\odot LN_{K}\left(k_{ij}^{l}\right)}{\sqrt{d_{q_{ij}^{ l}}}},\\ m_{ij}^{h}=q_{ij}^{l}+sigmoid\left(BN\left(att^{l}\right)\right)\odot LN_{V} \left(v_{ij}^{l}\right)\end{array}$ (2) $$

where LQ, LK, LV, and LE are the linear transformations for query, key, value, and edge features. $LN_{K}$, $LN_{V}$ are the nonlinear transformations for key and value, including two linear layers and an activation layer in between. $LE_{update}$ represents the linear transformation for updating messages, and $LNorm$ denotes the layer normalization operation. BN denotes the batch normalization layer, and $d_{q_{ij}^{l}}$ is the dimension of $q_{ij}^{l}$.

Then, we obtain the message $M_{i}^{l}$ by aggregating the information from the neighborhood of node $i$ over multiple edges, and $A_{i}^{l+1}$ is realized as follows :

$$ $\begin{array}[]{@{}c@{}c@{}}M_{i}^{l}=\sum_{j\in A_{i}}\sum_{h}LNmsg\left(m_{ ij}^{h}\right),\\ A_{i}^{l+1}=softplus\left(a_{i}^{l}+BN\left(M_{i}^{l}\right)\right)\end{array}$ (3) $$

where $LN_{msg}$ is the linear transformation used for updating the edge messages.

#### 𝒫 ⁢ 𝒫 ⁢ ℰ 𝒫 𝒫 ℰ \mathcal{PPE} caligraphic_P caligraphic_P caligraphic_E Message Passing Layer.

$\mathcal{A}^{l}$ and $\mathcal{PPE}^{l}$ represent the atomic features and 3D periodic pattern encoding at layer $l$, respectively. Its message-passing mechanism is as follows:

$$ $\begin{array}[]{@{}c@{}c@{}c@{}}\mathcal{PPE}^{l+1}=\mathcal{PPE}^{l}+\mathcal {A}^{l+1},\\ A_{1},A_{2}=LN_{PPE}\left(BN\left(PPE^{l+1}\right)\right),\\ A^{l+1}=A^{l}+LN_{update}\\ \left(LN_{update}\left(A_{1}\right)\odot Drop\left(GELU\left(A_{2}\right) \right)\right)\end{array}$ (4) $$

In this process, we update $A^{l+1}$ using residual connections He et al. ().

Finally, we use average pooling to aggregate the features of all nodes in the graph, followed by a nonlinear layer, and then a linear layer to obtain the scalar output of the graph as described above. A detailed description of the Zeoformer architecture can be found in Appendix F.

## 5 Experimental Studies

### 5.1 Experimental Setup

We conducted experiments on one OSDA-zeolite pair dataset (referred to as the OSDB Schwalbe-Koda et al. () dataset) and two material benchmark datasets, namely the Materials Project Chen et al. () and Jarvis Choudhary et al. (). Detailed descriptions of the three datasets can be found in the Appendix A. Baseline methods include CFID Choudhary et al. (), CGCNN Xie and Grossman (), SchNet Schütt et al. (), MEGNET Chen et al. (), GATGNN Louis et al. (), ALIGNN Choudhary and DeCost (), M3GNet Chen and Ong (), Matformer Yan et al. (), PotNetLin et al. (), CrysMMNet Das et al. (), CrysDiff Song et al. (), Crystalformer Taniai et al. (), and ComFormer Yan et al. (). For all baseline methods on the material datasets, we report the results provided in the cited papers. Results for all baseline models on the OSDB dataset are based on experimental replications using their open-source code. Detailed Zeoformer configurations for different tasks are provided in the Appendix G.

### 5.2 Experimental Results

**Table 4: Num. Layer represents the number of Node-wise transformer layers and $\mathcal{PPE}$ message passing layers.**
| Method | Num. Layer | Ehull | Banding |
| --- | --- | --- | --- |
| Zeoformer | 4,0 | 40.6 | 8.351 |
| Zeoformer w $\mathcal{PPE}$ | 3,2 | 36.3 | 6.113 |
| Zeoformer w/o $\mathcal{PPE}$ | 4,3 | 36.1 | 7.416 |
| Zeoformer w $\mathcal{PPE}$ | 4,3 | 33.5 | 5.996 |

OSDB Dataset.
The dataset can be accessed through Schwalbe-Koda et al. (), comprising $112,424$ OSDA-zeolite pairs. We retrained all baseline models. To ensure a fair comparison, all baseline models were retrained under identical conditions, ensuring identical training, validation, and test sets, and reported the results for the prediction of four properties in Table 1. Please refer to Appendix I for the other four properties. Detailed configuration information for the retrained baseline models is provided in Appendix H. The metric used is the testing mean absolute error (MAE). Compared to previous studies, Zeoformer achieved the best results across all $8$ prediction tasks, demonstrating significant improvements over other baseline models.
JARVIS. The quantitative results for JARVIS Choudhary et al. () are shown in Table 2. Zeoformer achieves superior performance across all tasks. Notably, Zeoformer demonstrates over $25$% improvement compared to the ComFormer variants in the Ehull task. Additionally, we present results for another $4$ tasks in Appendix I, where Zeoformer also achieves better performance.
The Materials Project (MP). The experimental results are shown in Table 3. It can be seen that Zeoformer outperforms other baseline models across tasks using $69,239$ training samples. Additionally, Zeoformer achieves competitive results in the bulk moduli and shear moduli tasks with only $4,664$ training samples.
Efficiency.
We compared ZeoFormer on the formation energy task in the JARVIS dataset against Matformer and variants of ComFormer. The results show that Zeoformer significantly outperforms in both training and inference speed while consuming fewer computational resources. For more details, please refer to Appendix I.

Overall, Zeoformer not only achieves excellent results on the OSDA-zeolite dataset but also performs well on benchmark material datasets, demonstrating the model’s outstanding generalization performance and robustness.

### 5.3 Ablation Studies

In this section, we demonstrate the impact of periodic pattern encoding on the representation learning of OSDA-zeolite and crystal materials through ablation studies. Specifically, we conducted experiments on the OSDB and JARVIS datasets by comparing the results for Banding total and Ehull attributes, using testing MAE as the quantitative evaluation metric.

By comparing models with different layers of periodic pattern encoding against models without periodic pattern encoding but retaining the $\mathcal{PPE}$ message passing layer, we validate the importance of periodic pattern encoding. The results show that compared to networks without periodic pattern encoding, we achieved improvements of $28.2$% and $17.5$% in the Banding total and Ehull tasks, respectively. Compared to the models that retain only the $\mathcal{PPE}$ message passing layer but lack periodic pattern encoding, we achieved improvements of $18.9$% and $7.2$% in these two tasks, respectively.

## 6 Conclusions

In order to predict the OSDA-Zeolite Affinity, we proposed a novel model called Zeoformer.
By reconstructing the unit cell centered on each atom and encoding the pairwise distances between this central atom and other atoms, Zeoformer better represents the geometric structure of OSDA-zeolite pairs, capturing both coarse-grained crystal periodicity and fine-grained local variability.
Experimental results on three datasets demonstrate that our Zeoformer significantly outperforms baseline methods in various property prediction tasks.
Our model helps narrow down the search space for candidate OSDAs, demonstrating significant potential in the field of directed zeolite synthesis.
Future work will explore more intrinsic structures of the OSDA-Zeolite pair and generate reasonable OSDA-Zeolite structures.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: See weaknesses.
soundness: 3
strengths: 1. The WPDDFormer model consistently outperforms other state-of-the-art methods across multiple tasks on the JARVIS and Materials Project datasets, demonstrating its effectiveness.
2. The authors provide theoretical guarantees on the continuity and completeness of the WPDD-based graphs, ensuring robust performance under minor structural perturbations.
summary: The paper introduces PDDFormer using the Pairwise Distance Distribution (PDD) as an invariant representation for crystal property prediction and enhances it with atom-specific weights (WPDD) and intra-unit cell structures (UPDD) to better capture atomic interactions and maintain computational efficiency. Experiments on datasets (Materials Project & JARVIS) show that WPDDFormer outperforms existing methods in predictive accuracy and computational efficiency.
weaknesses: I have no major concerns with this paper; however, an ablation study on the effect of varying the radius on WPDDFormer’s performance and efficiency would provide valuable insights into its scalability. Additionally, a time comparison between WPDDFormer and UPDDFormer would strengthen the evidence supporting UPDDFormer’s efficiency claims.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: See Weaknesses part.
soundness: 3
strengths: Originality:  
This work makes very significant novelty contributions in combining pairwise distance distribution with machine learning based crystal property prediction.

Quality:  
The quality of this work is evidenced by good theoretic analysis and strong experiment results.

Clarity:  
The writing of this work is overall good and clear.

Significance:  
The idea of efficiently integrating pairwise distance distribution into transformer model proposed by this work is insightful and enlightning for researchers in borad AI for science community.
summary: This work proposes PDDFormer, a novel framework for crystal material property prediction. The proposed PDDFormer uses insights about pairwise distance distribution impacts on crystal properties from material science community, designs a novel way to integrate pairwise distance distance distribution into crystal graph representations and a novel transformer model. PDDFormer is shown to achieve state-of-the-art performance on various crystal material property prediction benchmarks.
weaknesses: (1) A remarkable motivation of this work is described in Abstract "However, in reality, atoms in material always vibrate above absolute zero, causing continuous fluctuations in their positions" (line 15-17). But it seems there is no clear discussion why the use of pairwise distance distribution resolves this atom position fluctuation issue? Authors are encouraged to give detailed clarification or discussions to this question.

(2) Generally, two novelty contributions are proposed in this work, including the use of pairwise distance distribution in graph features and a novel transformer architecture. It would make this work more solid if authors could conduct more ablation studies to study which novelty contributes better to good performance, such as applying pairwise distance distribution to other model architecture or removing it from the proposed PDDFormer model.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: As listed above in weaknesses.
soundness: 2
strengths: 1. The proposed crystal representation demonstrates continuity under small distortions and perturbations, which can be advantageous for robustness.
summary: This paper introduces PDDFormer, a model designed to construct geometrically complete and invariant representations of crystals for the task of crystal property prediction. However, the completeness claim of the proposed representation is problematic, as it relies on the assumption that "the PDD is a generally complete invariant" (citing Widdowson & Kurlin, 2022), asserting that distinct crystal structures yield distinct PDD representations. This claim is not accurate. The PDD matrix does not ensure completeness for unstable crystal structures and fails to distinguish between chiral crystal structures. Furthermore, the proposed UPDD crystal graph construction method was previously introduced in "Zeoformer: Coarse-Grained Periodic Graph Transformer for OSDA-Zeolite Affinity Prediction," and the message-passing layers draw inspiration from ComFormer. Given these issues, the paper has limited technical contributions and several potentially misleading claims.
weaknesses: 1. Incomplete and Potentially Misleading Claims about Completeness

The completeness proof in the paper relies on the assumption that the PDD matrix acts as a generally complete invariant for distinguishing different crystal structures. However, this is inaccurate. The PDD matrix is not guaranteed to distinguish unstable crystal structures and cannot differentiate between chiral crystal structures, indicating limitations in the claimed theoretical foundation.  Action suggested: reorganize the proof writing section or reorganize the completeness claims.

2. Limited Novelty in Crystal Graph Construction

The proposed UPDD crystal graph construction method is not novel and has been previously introduced in the work "Zeoformer: Coarse-Grained Periodic Graph Transformer for OSDA-Zeolite Affinity Prediction." This overlap with prior work limits the novelty and originality of the proposed approach. Action suggested: check this previous work and discuss similarity and difference with it.

3. Message Passing Layers Largely Derived from ComFormer

The architecture of the message passing layers appears to be inspired by ComFormer, which further limits the technical novelty and contribution of the proposed method. Could you highlight any novel aspects or improvements in your approach compared to ComFormer?

In summary, while this paper proposes a method for generating geometrically complete and invariant crystal representations, its theoretical claims regarding completeness are flawed, and the approach itself relies heavily on prior work, both in graph construction and in network architecture. The combination of these issues limits the paper’s contributions and justifies a rejection at this stage.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 4
questions: - UPDD has multiple drawbacks compared to WPDD and shows inferior performance in experiments. What’s its advantage and when is it preferred over WPDD?
- The source of improvement by incorporating PDD is unclear. Is there a way to investigate whether it is because of accounting for periodicity, or because of input being more informative in other ways?
- Would the authors consider evaluating the interpretability of (W/U)PDD crystal graphs and/or the proposed transformer models?
soundness: 3
strengths: - The proposed (W/U)PDDs incorporate atomic information into PDD and address its high computational cost.
- The paper is overall well-written with nice flow, clarity, and illustrations.
summary: This paper presents atom-Weighted Pairwise Distance Distribution (WPDD) and Unit cell Pairwise Distance Distribution (UPDD) to characterize periodic crystal structure while accounting for atomic information. The continuity and geometrical completeness of these PDDs are theoretically evaluated. It then incorporates them into crystal graphs and develops graph transformer architectures for crystal property prediction. Comparative experiments show their better accuracy and efficiency over previous ML models, and ablation studies validate the importance of (W/U)PDD.
weaknesses: - The experimentation focuses on relatively simple scalar properties, some of which do not physically depend on crystal periodicity.
- Incorporating PDD into crystal graphs makes the data physics-informed, however, the proposed model does not seem to consider interpretability.
- Minor issues
  - In Definition 1, I suggest using boldface or other methods to help distinguish scalar, vector, and matrix.
  - Some languages in Definitions are vague: (1) Line 133, the range of what “crystal graph” refers to in this context should be specified. (2) Line 137, the description of geometrical completeness is unclear.
  - Typos, e.g., Line 129 “and If” and Line 512, “However”.