Materials science is undergoing a data-driven transformation, with mathematics playing a key role. Driven by the rapid development of big data and artificial intelligence, a central challenge in contemporary materials research is to extract stable and essential structural features from complex, high-dimensional, and noisy data, and to use these features to understand, predict, and design material properties. Topological Data Analysis (TDA) offers a powerful mathematical approach to addressing this challenge. At its core, TDA employs persistent homology to analyze how geometric structures evolve across scales, tracking the birth, persistence, and death of hidden global features such as connected components, loops, holes, and channels. These features serve as qualitative, quantitative, and stable structural fingerprints of complex data.

In 2012, Grigor’yan, Lin, Muranov and Yau introduced the path homology theory for digraphs (Now it is renamed as GLMY homology theory). This theory not only enables the topological characterization of directional interactions, asymmetric relations, and higher-order structures, but also overcomes the face-closure restriction of traditional simplicial complexes, thereby extending the scope of classical homology theory and enhancing its applicability to broader classes of problems. Traditional TDA methods are usually based on simplicial complexes constructed from point cloud data and are well suited for describing undirected geometric structures. However, in many real materials and chemical systems, interactions often exhibit directionality and asymmetry, such as directional features of chemical bonds and multicomponent synergistic effects in complex catalytic systems. In such cases, simplicial complexes are often insufficient for capturing asymmetric interactions and complex structural relationships, whereas GLMY homology offers an effective topological framework for addressing this limitation. In recent years, it has become an important topological tool in materials science and has shown broad applicability in related fields, including biological systems and complex networks.

Recently, the review article “Topological Data Analysis in Materials Science: Principles, Machine Learning Integration, and Application Landscapes” was published in Chemical Reviews, an internationally recognized and highly influential journal in chemistry and materials-related research. The article was coauthored by Prof. Shing-Tung Yau, President of the Beijing Institute of Mathematical Sciences and Applications (BIMSA), Prof. Jie Wu’s team, together with Prof. Feng Pan’s team from the School of Advanced Materials at Peking University Shenzhen Graduate School. This publication highlights TDA and AI-driven materials research as an emerging frontier in materials science and materials informatics, and further underscores the value of GLMY theory and modern topological methods in advancing a new paradigm for materials research.

This review provides a systematic overview of the mathematical foundations, machine-learning integration strategies, and applications of TDA in materials science. Taking GLMY homology theory, introduced by Prof. Yau and coauthors, as one of its central themes, the article surveys key topological tools, including persistent homology, GLMY homology, generalized GLMY homology, and Euler characteristic curve. It shows how TDA extracts multiscale, stable, and interpretable topological features from complex materials data and uses these features to characterize diverse materials systems, while also presenting representative applications of these methods in materials research. Building on these ideas, the review further introduces the framework of topological learning and discusses strategies for integrating TDA with artificial intelligence. Representative applications are then surveyed across key areas of materials science, including polymeric materials, glasses, porous crystalline materials, heterogeneous catalysis, and superionic conductors. It also illustrates the practical value of TDA in analyzing atomic structures, molecular dynamics trajectories, imaging data, and potential energy surfaces, and extends its scope to molecular science and biochemical systems.

Overall, this work presents a coherent picture of “TDA + AI + materials science,” covering theoretical foundations, methodological integration, and practical applications. It highlights the distinctive strengths of modern topological methods in materials science and their potential for deep integration with artificial intelligence, providing important methodological guidance for data-driven materials discovery and intelligent materials design.

Example 1–Polymer Materials: From Microscopic Ring Structures to Macroscopic Mechanical Response

Research object: Topological constraints (threading) between rings in ring polymer systems.

Method: Using persistent homology to compute persistence diagrams of individual rings (PD(i), PD(j)) and that of the merged two-ring configuration (PD(i ∪ j)). By comparing these diagrams, vanishing features (i.e., topological features present in individual PDs but absent in the merged PD) are identified.

Key findings: When two rings are not threaded, their respective features are almost completely preserved in the merged diagram. When threading occurs, a topological feature associated with one ring disappears in the merged diagram, forming a vanishing feature, indicating a topological change induced by the threading of the other ring.

The number of threading events increases with chain length, revealing the critical role of threading in regulating polymer diffusion and dynamics.

Significance: This method overcomes the limitations of traditional heuristic approaches, providing a rigorous, parameter-free criterion for detecting threading events. It offers a novel topological perspective for understanding the entangled dynamics and glassy behavior of ring polymers.

Example 2–Porous Crystalline Materials: Topological Fingerprints Enable Precise Quantification of Pore Geometry

Research object: Macroscopic properties such as adsorption and diffusion in porous crystalline materials, particularly metal-organic frameworks (MOFs).

Method: Wei et al. proposed the Category-Specific Topological Learning (CSTL) approach. Its core innovation lies in explicitly embedding chemical information into the topological representation. All elements are partitioned into eight categories according to their valence-electron configurations and chemical functionalities. For each element category, a separate simplicial complex is constructed, and persistent homology features are computed.

Key findings: CSTL not only quantifies the overall pore morphology and topological architecture of MOFs, but also enables the synergistic capture of multiscale geometric-chemical information through element-specific classification, including local coordination environments, ring motifs, and cavity distributions. This overcomes a key limitation of conventional TDA, which relies solely on geometric coordinates and neglects chemical heterogeneity among atomic species.

Significance: When combined with a gradient boosting regression model, CSTL outperforms state-of-the-art deep learning methods across multiple MOF property datasets. Furthermore, feature importance analysis reveals the distinct contributions of different element categories and topological dimensions to target properties. This significantly enhances the physical interpretability of machine learning models in materials science and provides explicit chemical-topological guidance for rational MOF design.