Quotient Complex Transformer (QCformer) for Perovskite Data Analysis

TL;DR

The paper introduces Quotient Complex Transformer (QCformer), a representation and learning framework for material data that encodes periodic crystal structure and higher-order interactions via quotient complexes. It uses Simplex Transformer blocks to perform higher-order message passing among simplices, enabling effective integration of 0-, 1-, and 2-simplices and their cofaces. Pretraining on large inorganic datasets (Materials Project, JARVIS) and fine-tuning on HOIP data yields state-of-the-art predictions for multiple properties and accurate bandgap predictions for new perovskites, with clear gains when including three-body (triangle) interactions. The results highlight the value of topological representations and higher-order attention for accelerating discovery of HOIPs and other functional materials, and the framework lays groundwork for integrating persistent-homology features in future work.

Abstract

The discovery of novel functional materials is crucial in addressing the challenges of sustainable energy generation and climate change. Hybrid organic-inorganic perovskites (HOIPs) have gained attention for their exceptional optoelectronic properties in photovoltaics. Recently, geometric deep learning, particularly graph neural networks (GNNs), has shown strong potential in predicting material properties and guiding material design. However, traditional GNNs often struggle to capture the periodic structures and higher-order interactions prevalent in such systems. To address these limitations, we propose a novel representation based on quotient complexes (QCs) and introduce the Quotient Complex Transformer (QCformer) for material property prediction. A material structure is modeled as a quotient complex, which encodes both pairwise and many-body interactions via simplices of varying dimensions and captures material periodicity through a quotient operation. Our model leverages higher-order features defined on simplices and processes them using a simplex-based Transformer module. We pretrain QCformer on benchmark datasets such as the Materials Project and JARVIS, and fine-tune it on HOIP datasets. The results show that QCformer outperforms state-of-the-art models in HOIP property prediction, demonstrating its effectiveness. The quotient complex representation and QCformer model together contribute a powerful new tool for predictive modeling of perovskite materials.

Figures (7)

• Figure 1: Illustration of a finite crystal structure extended by relative topological representations. Starting from the CaTiO$_3$ crystal (a), we transform it into a graph (b), then into a simplicial complex (c), and finally into their respective quotient versions (d)(e) by merging nodes of the same color. The finite cell structure is visualized using the VESTA program momma2011vesta. (f) The resulting quotient complex decomposes into 0‐simplices (nodes), 1‐simplices (edges), and 2‐simplices (triangles), each with its associated features.
• Figure 2: Detailed illustration of the QCformer architecture.(a): The overall framework of QCformer, where quotient complexes (QCs) are constructed and processed through feature embedding, message passing, and pooling modules. The final prediction is obtained via an MLP layer. (b):Feature embedding block in (a). The 0-simplex feature is embedded as a one-hot vector, following the approach in CGCNN xie2018crystal. The features of 1-simplices and 2-simplices are embedded using radial basis functions (RBF). Finally, simplices of all dimensions are projected into the same hidden dimension through an MLP layer. (c): Detailed Simplex Transformer (Sformer) block in (a). Note that each $n$-simplex $\sigma_n$ is updated by aggregating information from its neighbors ($n$-simplices) $\tau_n$ and cofaces ($(n+1)$-simplices) $(\sigma_n,\tau_n)_s$ and higher dimensional simplices are updated to be inputs for the updating of lower dimensional simplices.
• Figure S1: Illustration of simplicial complexes and their corresponding quotient complexes. In these examples, equivalent vertices are indicated by the same color. The quotient operation merges equivalent vertices in the simplicial complexes, resulting in the derived quotient complexes.
• Figure S2: An illustrative example depicting the construction of the simplicial complex $\widetilde{\mathcal{K}}$, which is homotopy equivalent to the quotient complex $\overline{\mathcal{K}}$. In this example, $\mathcal{K}$ denotes a simplicial complex with 6 vertices., 7 edges, and 1 triangle. The vertex set $V$ is divided into three groups: $V_1$, $V_2$, and $V_3$. By adding star-shaped sets $S_1$, $S_2$, and $S_3$ to the simplicial complex $\mathcal{K}$, we obtain the simplicial complex $\widetilde{\mathcal{K}}$, which is homotopy equivalent to the quotient complex $\overline{\mathcal{K}}$.
• Figure S3: Illustration of the crystal quotient complex construction with three nearest neighbors. First, the atoms within the unit cell are selected as nodes, with directed edges added from the three nearest neighbors to each target node. Next, triangles are formed based on edge directions, constructing the simplicial complex. Finally, nodes of the same color are merged to form the quotient complex.

Theorems & Definitions (9)

• Definition 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.4
• Theorem 4.5
• Proposition S1.1: brown2006topology
• Theorem S1.2: brown2006topology
• Corollary S1.3
• Theorem S1.4: hu2025quotient