Deep learning in-depth analysis of crystal graph convolutional neural networks: A new era in materials discovery and its applications

1 Introduction

The field of material engineering and science is the confluence point of processing, structure, properties, and performance of materials [1]. The efficacy of a material in specific tasks depends on its properties, such as hardness, density, and thermal expansion, which, in turn, are influenced by its structure. Materials processing converts raw materials into usable forms through precise adjustment of the structure and properties of materials to achieve desired outcomes. These interconnected aspects yield significant results across diverse fields such as medicine, energy, manufacturing, and biotechnology, demonstrating the continuous evolution of material engineering and science and its potential for societal and environmental progress. Like many other fields, computational methods have arisen as propitious solutions to tackle the questions inherently posed in material analysis, right from processing to their applications [2,3,4,5]. High-throughput computational techniques enrooted in quantum mechanics enable predictive modeling of material properties crucial for advancing materials design across applications from electronics to energy storage [1,6]. However, there exists a notable gap in the seamless integration of high-throughput computational techniques, particularly those rooted in quantum mechanics, with material informatics science. Specifically, there is a need to enhance the synergy between computational methods and data-driven approaches to accelerate the discovery, design, and optimization of novel materials across various applications. In this connection, material informatics science is a rapidly evolving interdisciplinary field at the intersection of materials science, data science, and machine learning (ML). It focuses on developing computational tools and methods to hasten the discovery, design, and optimization of newfangled materials. One of the central goals of materials informatics science is to predict material properties with high accuracy and efficiency, thereby reducing the time and cost associated with experimental trial-and-error approaches. Predicting material properties is of paramount importance in various scientific and engineering domains, including semiconductor devices, energy storage systems, catalysis, and structural materials. Researchers like Hill et al. [7] and Saad et al. [8] illustrated the application of data mining systems in envisaging properties of AB compounds, underscoring the pivotal role of computational methods in materials science. Jain et al. emphasized the significance of computational materials science in accelerating materials innovation, particularly through predictive approaches [4]. Traditional methods for predicting material properties often rely on theoretical models, empirical correlations, or costly experimental measurements. Commonly employed ab initio methods such as density-functional theory (DFT) and density functional perturbation theory (DFPT) and wave function theories have aided in predicting material properties [9,10,11]. However, such traditional techniques are often time-consuming, resource-intensive, and limited in their ability to capture complex structure–property relationships [6,12]. With the accumulation of vast experimental and simulation-based datasets, there is a growing need for modern tools capable of automatically exploring big data. Artificial intelligence (AI) systems, a ML algorithm, and particularly deep learning (DL) have emerged as transformative forces for material engineering and science, offering automated learning from past experiences to enhance performance without explicit programming [13] and have emerged as powerful tools for materials discovery and design. A schematic of AI–ML–DL context and subdivision of DL and their application in the material engineering and science fields (are shown in Figure 1). ML algorithms, which model material structure and atomic interactions, provide comparable accuracy to DFT calculations with significantly reduced computation time [13]. Yet, challenges persist, especially in representing variable data such as crystal systems as fixed-length vectors for ML compatibility [14,15,16,17,18].

Figure 1

Overview of the hierarchical structure linking AI, ML, and DL, highlighting key subcategories and a range of real-world applications.

This article aims to address the limitations of traditional and ab initio methods in predicting material properties, particularly in capturing complex structure–property relationships and handling large datasets. Furthermore, it seeks to explore the potential of AI, ML, and DL algorithms in material engineering and science. Specifically, the study aims to investigate how DL algorithms can model material structure and atomic interactions accurately with reduced computation time compared to traditional methods. In addition to DL frameworks like crystal graph convolutional neural network (CGCNN), recent studies have also explored hybrid learning approaches and fuzzy logic-based ML models to improve prediction accuracy and handle uncertainties in material property data. Hybrid models, which combine traditional physical models with ML algorithms, have shown promise in enhancing generalizability across different material classes. These alternative approaches contribute to the diversification of methodologies within materials informatics and highlight the importance of interpretability and flexibility in modeling strategies [19,20]. Moreover, recent research has focused on developing approaches to represent complex materials data such as crystal systems in formats suitable for DL algorithms, thereby improving both model interpretability and applicability in materials science. In particular, efforts have been made to address the interpretability challenges associated with graph-based DL models like CGCNN. Several studies [21,22] have proposed techniques such as attention mechanisms and feature attribution to identify the most influential nodes and edges in crystal graph structures, helping to clarify how specific atomic configurations influence property predictions. Additionally, visualization tools have been introduced to interpret complex graph embeddings, offering more intuitive insights into model behavior. These advancements mark significant progress toward making graph-based models more transparent and trustworthy, which is critical for their widespread adoption in materials informatics. Supported by large-scale materials datasets from high-throughput simulations and combinatorial experiments, such models continue to serve as powerful tools for accelerating materials discovery and design [4,23,24,25,26,27,28,29]. Models like the atomistic line graph neural network and convolutional graph neural networks (CGCNN) have demonstrated promising results, particularly in predicting the structural and electronic characteristics of various materials, especially crystalline materials, due to their tailored exploitation of the inherent structure of crystal lattices [22,30]. Despite these advancements, existing DL models face challenges, including their exclusive development for molecular or crystal datasets and the lack of global state descriptions necessary for predicting state-dependent properties [31,32]. These limitations underscore the necessity for a more unified, adaptable, and critically assessed approach to model development. The novelty of this article lies in its dedicated and systematic focus on CGCNN, aiming to consolidate scattered insights into a cohesive framework that evaluates its theoretical underpinnings, architectural evolutions, application domains, and practical limitations. A comprehensive review enables in signifying valued perceptions about the state-of-the-art techniques, benchmarking protocols, best practices, and future research directions for advancing CGCNN and its applications in predicting material properties such as prediction of energies of formation [33–35], bandgaps [16,36,37], melting temperatures [38–40], thermal conductivity [14,39], and mechanical behavior of materials [41–43]. The subsequent investigation focuses on leveraging DL technologies, specifically graph-based models, to efficiently predict material properties while addressing challenges related to variable inputs and the quantity of the dataset. Performance evaluations and comparison of graph-based models like CGCNN with traditional approaches are undertaken across various material properties. Integration of domain knowledge into DL models is explored to enhance both performance and interpretability with special emphasis on the convergence of materials science and DL to facilitate efficient prediction of material properties. Benchmarking against ab initio methods like DFT [9,11] and assessing the effectiveness of graph-based models like CGCNN and its variants in capturing atomic interactions are also central to this study. The motivation for conducting a comprehensive review of CGCNN stems from the growing interest and significance of this approach in materials informatics research. While CGCNN has shown promising results in various applications, there remains a need for a systematic and in-depth analysis of its principles, capabilities, limitations, and potential advancements. Therefore, this research aims to fill the existing gap in the literature by providing a thorough evaluation of CGCNN and proposing strategies to overcome its limitations, ultimately advancing the field of materials informatics. Through the integration of valued perceptions about state-of-the-art techniques, benchmarking protocols, best practices, and future research directions. Importantly, this article addresses a gap in current literature by critically assessing CGCNN’s contributions and challenges and proposing strategic research directions for its future advancement. Through this, it contributes to the broader convergence of materials science and ML, fostering more efficient and intelligent material property prediction workflows.

2 Fundamentals of CGCNNs

2.1 Explanation of CNNs for crystal graphs

Data-driven approaches have revolutionized material research, replacing traditional trial-and-error methods. Neural networks, at the forefront of DL with the rise of big data and AI, impart a lasting impact in fitting probability distributions of samples [44,45]. In the context of materials science, the challenge lies in extracting features from crystal structures for regression results. Graph neural networks (GNNs) appeared as a solution, particularly in expressing crystal structures using graph data structures [46–48]. Here, nodes represent atoms, and edges signify chemical bonds. The universal properties of crystals, like atomic density and volume, are considered, as crystal structures fundamentally govern such properties [33]. A specific adaptation for crystal graph representation is the CGCNN [22]. CGCNN employs atomic coordinates to create crystal graphs, defining bonds based on proximity and Voronoi faces, which are the boundaries between regions in a Voronoi diagram that define the closest areas around each atom [49]. The connection between atoms is established when they possess a common Voronoi face where the distance lies inside the Cordero covalent bond length, referring to the standard covalent bond distances between atoms as defined by Cordero et al., with a tolerance of 0.25 Å [16]. Notably, the studies illustrate that connecting only to the nearest 12 neighbors yields effective results. Representation of crystal structures is pivotal for accurate property prediction. For GNNs, suitable graph design is essential to express the original crystal structures [31]. In contrast to traditional DL algorithms focusing on chemical properties, recent graph models like CGCNN emphasize the crystal structure distribution of materials. Nodes are encoded via one-hot encoding, and edge features comprise atom information and bond weight [16]. The effectiveness of GNN models lies in their ability to cluster data in a high-dimensional space, feature calculation, and map generation to target areas. Figure 2 illustrates the full CGCNN workflow, starting from different types of material structures (e.g., bulk, MOF, 2D, polymer), which are transformed into node–edge graphs.

Figure 2

General workflow schematic of the procedure of CGCNN.

These graphs are input into a CGCNN architecture that performs feature extraction via convolution and pooling operations. Various hyperparameters are tuned to optimizing learning. The final output is the predicted material property, such as formation energy or bandgap, with high accuracy (R ² = 0.91).

2.2 Overview of different graph architectures

Graphs represent a sort of data structure that can model object sets (nodes) along with their association sets (edges). In contemporary research, investigation of graphs through DL approaches is gathering immense pondering owing to the significant expressive capability of graphs and thus can be employed in denoting varied research schemes spanning diverse fields like social sciences [50], natural sciences [51–53], humanity [54], and others [55]. Graph analysis on account of being a typical non-Euclidean data structure on DL emphasizes node classification, link prediction, and clustering. GNNs belong to techniques based on DL functioning in graph domains. In the modern-day scenario, GNN emerges as a widely applied graph analysis system owing to its resounding performance. In this section, an overview deliberation on different architectures used in GNNs [56–58], specifically focusing on graph convolutional networks (GCNs) [59], graph attention networks (GATs) [60], and other relevant architectures is provided, highlighting their key characteristics and references to seminal works in the field. In addition to GCNs and GATs, there exist several other architectures in the domain of GNNs, each with its own unique characteristics and applications. Some examples include Graph Sample and Aggregation (GraphSAGE) [61], gated graph neural networks [62], and message passing neural networks [63]. These architectures differ in terms of their mechanisms for passing messages, aggregation strategies, and ways of incorporating node features. Figure 3 presents the conceptual evolution of graph learning architectures, tracing the progression from CNNs to GNNs and their specialized derivatives. Initially, CNNs were developed to handle regular grid structures, such as 1D sequences (e.g., text) and 2D grids (e.g., images), by applying convolutional filters across structured inputs. However, CNNs are inherently limited to Euclidean domains, where data points are regularly spaced. To overcome this limitation, GNNs generalize convolution operations to irregular graph structures by enabling nodes to aggregate information from their neighbors based on graph connectivity.

Figure 3

Overview of different architectures of GNN, CNN, GCN, and GAT.

2.2.1 GCNs

GCNs are one of the foundational designs in the domain of GNNs. They are designed to extend the concept of CNNs to irregular graph structures. CNNs revolutionized DL due to their ability to extract multi-scale localized spatial features and generate decidedly communicative representations, primarily through shared weights, multiple layers, and local connections [64]. Such features prove to be decisive in resolving graph-related problems. Nevertheless, CNNs capability of operation on regular Euclidean data like texts (1D sequences) and images (2D grids) only, though such data structures are considered as instances of graphs, pose certain problems. Thus, GCNs are considered a generalization of CNNs to data on graph structures. Introduced by Thomas Kipf and Max Welling in 2017 [59], GCNs aggregate information from neighboring nodes iteratively using convolution operations, allowing them to learn representations of nodes within a graph. GCNs have been widely used in various graph-associated events, namely, link prediction, node classification, and graph classification. The underlying idea in the context of GCNs is the generalization of the convolution operation from regular grids (like images) to graph-structured data, enabling effective information propagation and feature learning in graph domains. Thus, GCNs have the ability to model input and/or output comprising elements as well as interdependence. Numerous all-encompassing reviews of GCNs are available. Bronstein et al. [65] thoroughly reviewed the geometric DL and deliberated not only on the associated difficulties and problems but also on viable solutions, real-time applications, and finally the guidelines for its future. Another extensive review on GCNs is presented by Zhang et al. [66]. These reviews reveal that GCNs are one of the fundamental architectures in GNNs. GCNs extend the concept of convolutional layers from regular grids to irregular graph structures. A GCN layer defines the first-order approximation of a localized spectral filter on graphs, and we can describe the propagation rule in GCNs as shown in Figure 3.

2.2.2 GATs

GAT incorporates the attention mechanism into the propagation step of graph-based learning [60] and represents a recent advancement in the field of GNNs. The distinctions between GCNs and GATs in terms of architecture, aggregation, and feature learning mechanisms are systematically outlined in Table 1.

Table 1

GCNs and GATs

Feature	GCNs	GATs
Year introduced	2017 (Thomas Kipf & Max Welling)	2017 (Petar Veličković et al.)
Main idea	Extends CNN convolution operations to graph data, aggregating information from neighboring nodes	Uses attention mechanism to dynamically determine the importance of neighboring nodes during aggregation
Aggregation method	Aggregates information from all neighboring nodes with equal weights	Dynamically assigns weights to neighboring nodes, giving more importance to relevant nodes
Information propagation	Propagates information between nodes using convolution with fixed weights	Propagates information between nodes using self-attention mechanism
Key advantages	Simple structure and efficient learning process	More capable of handling complex graph structures with flexible weight assignment
Disadvantages	Treats all neighboring nodes equally, unable to assign more importance to critical nodes	Higher computational cost compared to GCN, can become costly on large graphs
Key feature learning	Learn features based on fixed weights between nodes and their neighbors	Learning varying weights between nodes, capturing more complex relationships
Mechanism utilized	Convolution operation	Attention mechanism

2.3 Graph convolution and CGCNN: Unveiling structural information in materials

Graph convolution is a key operation in graph-based neural networks. It involves aggregating feature vectors from neighboring nodes based on the connections in the graph, as illustrated in Figure 4.

Figure 4

Assorted architectures of (a) GCNN and (b) CGNN models.

In the context of crystal structures, where atoms form bonds by connecting through chemical bonds, this operation becomes crucial for capturing structural information. In the standard representation, a graph is denoted by ( V , E , A , or u ), where V represents the set of nodes (atoms), E represents the set of edges (bonds), and A or u symbolizes the adjacent matrix or interface defining the connections between nodes. This representation forms the basis for CGCNN [22]. The idea of GNNs is contemplated by Battaglia et al. [31] and provides a modular framework for DL supporting relational reasoning and combinatorial generalization. In a crystal graph, atoms connected by the same bond are considered neighbors. To enhance the influence of bonds in CGCNN, features of nodes are concatenated with features of bonds, as shown in Figure 4. Designing suitable graphs for expressing original crystal structures is crucial for accurate material property predictions using GNNs. Unlike traditional DL algorithms that focus on chemical properties, CGCNN represents nodes using one-hot encoding, and edge features incorporate information about two connected atoms and bond weight. Xie and Grossman introduced CGCNN, uniquely considering crystal lattice periodicity and space group symmetries. It addresses the challenge of representing crystal structures effectively [22]. CGCNN utilizes superpositions of local features to distinguish between different crystal structures. The main idea revolves around utilizing CNNs to map data to a target distribution. This involves updating atom features based on surrounding atoms and bonds, and then computing overall atom and bond features using a pooling function. Pooling layers are used in neural networks to reduce the dimensionality of the data while retaining essential features, which helps to decrease computational load and improve the model’s ability to generalize. In this context, the pooling layer combines node and edge feature vectors, aiming to minimize losses and align with the target distribution defined by DFT. In experiments, CGCNN has demonstrated outstanding prediction performance in capturing the properties of crystals, showcasing its efficacy in uncovering structural information in materials [31].

3 Methodologies and architectures

4 Graph-based DL in materials data science

At present, techniques like GCNN, belonging to the deep neural network type, have exhibited noteworthy potential in the enhancement of proficiency in extracting characteristics for multifunctional materials [73]. Despite the availability of different methods for gathering the physicochemical properties related to the estimation of the materials property, the strong contender in this direction emerged is GNN. Graph-based DL, particularly exemplified by the CGCNN model, appears as a prominent feature in materials information science. CGCNN leverages the inherent graph structure of materials, representing atoms as nodes and bonds as edges, to effectively capture the complex relationships between crystal structures and material properties (Figure 5).

Figure 5

CGCNN model representing a crystal as graph G, atoms in the crystal cell as nodes ( v _i, v _j), and any interatomic information as edges ( u _i,j ). Global state of the graph, edges, and nodes is entrenched into vectors based on manifold information levels extracted from the materials.

By applying graph convolutional operations, CGCNN effectively extracts features from the graph representation, incorporating spatial relationships between atoms and capturing the local environment of each atom. This approach allows CGCNN to learn rich and informative representations of materials, enabling accurate predictions of various material properties. The efficacy of CGCNN in materials information science has been demonstrated in several studies. Although DL models like SchNet, a model developed to predict the physical properties of chemical substances and solid materials by effectively learning atomic interactions in both molecular and solid-state materials, have incorporated periodic boundary conditions for convolution and are applicable for solid-state materials and molecules alike, the majority of DL models applicable to molecules fail in the case of solid-state systems owing to their dissimilar characteristics [73]. However, recent advancements have seen the progression of multiple graphs related DL models focused on predicting material properties of crystal systems, representing a fundamental step toward widespread adoption of GNNs in inorganic crystalline materials research [74]. The assignment of material property prediction is formulated like understanding a mapping from the graph to the target property. Notable examples include the CGCNN model first proposed in 2018 by Xie and Grossman, which considers information from the crystal structure graph as an input for a CNN. Likewise, Xie and Grossman applied CGCNN to predict material properties such as electronic conductivity, thermal conductivity, and bandgap energies with high accuracy and interpretability. Similarly, Louis et al. analyzed novel graph neural network (GATGNN) in the material property forecasting like formation and absolute energy, bandgap and elastic properties, and bulk and shear moduli of inorganic materials [75]. Park and Wolverton have developed the upgraded variant of the CGCNN model, namely iCGCNN, for the accuracy of two distinctive illustrations [76]. In the first illustration, iCGCNN is shown to realize a drastically enhanced prognostic accuracy in the case of training/validation on 180,000/20,000 thermodynamic stability entries calculated from DFT that were, in turn, obtained from the open quantum materials database (OQMD) and appraised using 230,000 entries as a separate test set [24]. As a second illustration, iCGCNN achieved a 31% success rate that is 2.4 times greater in comparison to the original CGCNN and whopping 310 times higher when compared to an undirected high-throughput search in the assistance of superior output search of materials in case of ThCr₂Si₂ structure-type [76]. Screening of 132,600 compounds having elemental decorations belonging to the ThCr₂Si₂ prototype crystal structure is undertaken by them, and this exercise has resulted in the identification of 97 novel distinctive compounds possessing high stability. In the process, 757 DFT calculations are performed by them, leading to a 130 times acceleration of the computational time required for the high-throughput search. Thus, iCGCNN speeds up the detections of high-performance novel materials via swift and precise identification of crystalline compounds possessing the desired properties [76]. Wang et al. studied a modified version of the CGCNN algorithm, namely mCGCNN, which effectively amalgamates materials’ physical and/or chemical properties with their crystal structure attributes. It solves problems like data fusion associated with the contemporary generic models for materials. Substantial improvement in the proficient and accurate estimation of gravimetric capacity is achieved by them for lithium-ion batteries (LIBs) [77]. Many open-source data such as Materials Project database, etc. are used to collate all the crystalline material properties already employed in CGCNN models. As shown in Table 3, CGCNN and its advanced variants leverage diverse datasets to predict material properties ranging from bandgaps to gravimetric capacity in LIBs, yielding significant performance improvements across multiple domains.

So far, 2,440 data crystal structures related to LIBs materials are generated from the Materials Project database by employing MP-API. Six gravimetric capacity-related properties representing the capacity per unit mass of battery that are utilized as labels in the prediction process are also obtained. The crystallography information file format (found from the Materials Project database) tracks a key-value pair structure, where each explicit data item is denoted by a key with the conforming value as the data [4]. Such data may include crystal nomenclature, its elemental constitution, cell parameters, symmetry, atomic coordinates, associated references, etc. [78]. This standardization of format enables the DL algorithms to read and analyze the data with ease. In addition, Wang et al. have developed the algorithms for envisaging the methane adsorption behavior of metal−organic frameworks (MOFs) and utilized them for predicting the methane adsorption volume of a few randomly chosen zeolitic imidazolate frameworks and covalent organic framework materials [79]. The findings indicate the model’s effectiveness in forecasting more porous substances. The same model was used to conduct a tiered analysis of a theoretical MOFs database (around 330,000 MOFs), with four theoretical MOFs showing the highest methane absorption efficiency. Additional investigation of all the MOFs screened evinces the interplay between working capacity with certain structural features. The results deduce that the unearth of innovative materials not only related to gas adsorption but also to other fields that include materials and molecules interactions can be accelerated through the integration of DL and hierarchical screening [79]. Furthermore, CGCNN is exploited efficaciously in several material science activities, like materials discovery, property forecast, and materials design. Its ability to seamlessly integrate the characteristics associated with material crystal structure and its physical as well as chemical properties makes it a versatile tool for advancing materials information science [17,80–84]. Overall, CGCNN represents a significant advancement in graph-based DL for materials information science, offering a powerful framework for predicting material properties and accelerating materials discovery and design.

5 Other variants and improvements in the CGCNN model

Graph-based representations naturally capture the connectivity between entities, making them intuitive and computationally feasible for expressing atomic structures. Traditional DL methods often flatten these representations into lower-dimensional vectors, leading to the loss of spatial and structural information. CNNs address this limitation by allowing hierarchical information processing while preserving spatial characteristics. Thus, employing graph representations and GCNs has emerged as an effective approach for classifying and predicting properties of atomic systems. GCNs have demonstrated strong effectiveness in node-representation learning tasks [59,66,78]. They have been extended to predict material properties by representing the atom as a graph structure, where nodes represent atoms/particles and edges represent interatomic connections. This approach laid the foundation for the CGCNN model [22]. This structure is easily understandable because it allows for the extraction of local contributions of chemical atmosphere, which significantly influences the overall properties of crystals. Insights gained from CGCNN regarding chemical characteristics aid in refining the exploration area for identifying new materials. To quantitatively assess the performance of CGCNN and its variants, we summarize the reported mean absolute errors (MAEs) across several material properties in Table 4.

Park and Wolverton introduced an enhanced version of the CGCNN model known as the improved variant of the CGCNN model (iCGCNN) [76]. This adaptation surpasses the original by integrating information regarding 3-body correlations between nearby atoms, Voronoi decomposition of the crystal structure, and enhancing the depiction of interatomic bonds in the graphs [22,35]. iCGCNN demonstrates approximately 20% higher accuracy compared to CGCNN. However, iCGCNN struggles with predicting thermal stability due to its oversight of the geometric structure of crystals, despite considering the atom placement within them. A recent advancement in the field introduced the orbital graph convolutional neural network (OGCNN) [87]. This model builds upon the CGCNN architecture by incorporating atomic orbital information through the orbital field matrix (OFM) representation. The OFM includes bonding details between atomic orbitals, utilizing electron configurations and solid angles of neighboring atoms for enhanced information [88]. By integrating larger-dimensional edge feature vectors compared to CGCNN, OGCNN exhibits improved learning and prediction capabilities for both formation energetics and bandgaps. Additionally, enhanced bonding information facilitates better communication between nodes in the network. Chen et al. introduced the materials graph network (MEGNET) [73], which predicts molecular properties by integrating global state variables. MEGNET forms a unified framework for correlated properties, with each block capturing interactions among atoms and their local environments. However, increasing the number of blocks may reduce the contrast of atoms’ embeddings. In the context of the MEGNET model, transfer learning involves leveraging learned element embeddings and model parameters from a pre-trained model to enhance the performance of a model trained on a smaller or different dataset [73]. This approach aims to improve prediction accuracy by transferring knowledge encoded in the embeddings, enabling better generalization even with limited data. Schütt et al. introduced the SchNet model [89], which employs continuous filter convolutional layers to predict properties of molecules based on atom-wise renditions. Unlike traditional models, the output layers in the SchNet model are not tailored to individual properties. Sanyal et al. [70] introduced the multi-task crystal graph convolutional network (MTCGCNN), which combines CGCNN with multi-task learning (MTL) to predict multiple material properties. In MTCGCNN, tasks within MTL share the crystal representation to predict material properties, allowing for the transfer of learning from one characteristic to another. While MTL accelerates the learning rate of the crystal embedding space, it may not necessarily improve prediction accuracy for all properties. For instance, some properties, like formation energy, may be excellently predicted using a simple CGCNN model. To accurately compare MTL and MTCGCNN models, it's essential to understand the specific limitations of MTL models, such as scalability and interpretability. While MTL models offer benefits in jointly learning multiple tasks, MTCGCNN may address some of these limitations by employing specialized architectures tailored for material science tasks, like CGCNNs. This tailored approach in MTCGCNN to handle material data and tasks could potentially result in improved performance, scalability, and interpretability compared to traditional MTL models [70]. Additionally, Louis et al. introduced the global attention graph neural network (GATGNN) [75] framework, aimed at enhancing the understanding of internal relationships within crystal graphs and their properties. This framework incorporates augmented-GAT layers, which include local attention layers for observing the environment and a global attention layer for learning relationships among adjacent particles and overall crystal structure properties. The global attention layer aggregates specific environment vectors to provide a comprehensive vision of the entire crystal graph. While CGCNN and MEGNET models excel in predicting certain material properties, GATGNNs also demonstrate strong potential in capturing features for effective material property prediction. In GNNs, similar to traditional CNNs, the convolution and pooling functions are utilized. Pooling function mechanisms can be systematically classified based on their features [60,75,90]. Though the pooling function can occur at both local levels and global levels, research suggests that the local pooling function alone does not fully account for the efficiency of GNNs on broadly used benchmarks [91]. Nevertheless, pooling function operators are typically constructed to be appropriate across both local and global contexts. The Eigen Pooling operator, utilizing the graph Fourier transform, pools features based on structural data from the graph signal, preserving the original graph structure [92]. Enhancing the pooling mechanism in the CGCNN model (Figure 6) with several variants presents a significant opportunity for improvement, often overlooked yet crucial for model performance. As depicted in Figure 6, the integration of advanced pooling strategies into the CGCNN framework including ReLU-activated addition characteristics and global eigen pooling demonstrates potential for notable improvements in predictive accuracy and model generalization.

Figure 6

A few variants of the CGCNN model. (a) Conventional architecture design of CGCNN. (b) Alteration architecture design of CGCNN.

Notably, the CGCNN modified version, which represents the local surroundings of crystals using composition and structure-dependent vectors rather than human-designed features, shows promise in predicting properties like number group, element blocks, radius, and electronegativity in perovskites and elemental boron [22,68,93]. However, its performance in predicting the local energy of inorganic materials lags [46].

Recent advancements in DL models have shown promise in predicting multiple material properties within a single framework. For instance, the MatErials Graph Network (MEGNet), an advanced version of the GNN, has been leveraged to forecast internal energy across various temperatures and properties such as entropy, enthalpy, and energy-free Gibbs within the same model [73]. MEGNet also demonstrates notable precision in predicting electrical properties by consolidating various free energy models into a unified framework, incorporating global state variables like temperature, pressure, and entropy. Another notable model, HydraGNN, investigated by Lupo Pasini et al. [94], employs multitask learning to predict mixing enthalpy, atomic magnetic moment, and charge transfer, simultaneously. With architecture comprising two sets of layers, HydraGNN learns both common and specific features of each material property. In HydraGNN’s convolutional layers, a variant of GCNN known as principal neighborhood aggregation facilitates the classification of different graphs. While HydraGNN is primarily designed for predicting molecular atomization energies [92], it also demonstrates flexibility for applications across various materials science properties. To facilitate a clearer comparison among leading graph-based models, Figure 7 presents the MAE values of CGCNN, SchNet, and MEGNET in predicting formation energies, adapted from a widely cited benchmark study [2,68,71].

Figure 7

Comparative MAE for formation energy prediction using CGCNN, SchNet, and MEGNet models.

This visual representation highlights the relative predictive accuracies of these models and offers valuable insight into their practical utility in materials informatics.

6 Real-world applications of CGCNN in material discovery

While CGCNN has shown great promise in academic settings, its practical impact in real-world materials discovery is equally significant and increasingly evident through various case studies. In this section, we highlight concrete examples demonstrating how CGCNN-based approaches have directly contributed to experimental materials design and selection. One such example is the adsorption proficiency of Nb₂CT _x toward the heavy metals with the help of a CGCNN model has been predicted by Jaffari et al. [85]. Their results specify that Pb(ii) ions exhibiting a MAE and root mean squared error (RMSE) >0.09 and >0.16 eV, respectively, possess larger adsorption energy as compared to Cd(ii) ions. In a nutshell, their investigation reveals the great efficacy of the proposed technique in comparison to the adsorption behavior of various materials. The study can be a guiding light in designing aqueous environmental experiments for the elimination of other hazardous pollutants. Figure 8 depicts the schematic design of the complete research undertaken by Jaffari et al. [85].

Figure 8

Schematic illustration of the CGCNNs for the estimation of adsorption capability of Nb₂CT _x for Cd(ii) and Pb(ii) ions. (Reproduced (or adapted) with permission. [85] Copyright 2013, Royal Society of Chemistry.).

The CGCNN framework offers advantages over traditional ML operations by eliminating the necessity for costly and error-prone extensive data labeling. This feature enables CGCNN to avoid recomputing the heat of adsorption or geometric properties common in traditional ML models. Recent applications of CGCNN include predictions in various materials domains such as intermetallic alloys [95], electroreduction catalysts [96], and photoanode materials [97]. In the gas storage sector, Wang et al. [79] demonstrated CGCNN’s effectiveness in predicting methane adsorption in metal-organic frameworks (MOFs) as illustrated in Figure 9.

Figure 9

Representation of hypothetical MOFs for methane adsorption using the CGCNN model and four hypothetical MOFs (a)–(d) of crystal structures adsorption optimized at 298 K and under 35 bar, and for quantified adsorption isotherms. (Reproduced (or adapted) with permission. [79] Copyright@ 2020, American Chemical Society.).

Gu et al. introduced a facile yet adaptable illustration to accelerate catalyst screening using the inclusive deep-learning models for binding energy estimation [98]. The model, trained on a dataset including methane working capacity (N^wc), accurately screened potential MOFs from a database of approximately 330,000 structures [79,99]. An all-through DL classification model founded on CGCNN achieved an AUC of 0.930 for methane uptake prediction, facilitating rapid evaluation of a colossal, large number of hypothetical MOFs. This approach holds promise for developing materials tailored to specific gas storage or separation applications, with potential industrial significance. For energy storage applications, Wang et al. introduced the mCGCNN (modified CGCNN) model, which effectively integrates physical as well as chemical properties of materials with crystal structure features and thereby improves the precision and proficiency of estimation of gravimetric capacity for lithium-ion batteries [77]. This model addresses the data fusion challenge in materials science and achieves high accuracy in gravimetric capacity prediction and classification tasks, as shown in Figure 10.

Figure 10

Modified CGCNN model diagram for prediction of gravimetric capacity LIBs such as (a) model structure and (b) and (c) CGCNN and mCGCNN model test set. (Reproduced (or adapted) with permission. [77] Copyright 2023 published by Elsevier Ltd.).

The model achieved an AUC of 0.99 and classification accuracy of 95.5%, demonstrating its utility for real-time battery material screening. mCGCNN provides enhanced suppleness and can be employed as an invaluable tool in swift and accurate screening of highly persuasive batteries through the unification of numerical data with the crystal structure of the materials. Moreover, its potential extends to the detection of many additional multi-functional materials, thus exhibiting its wide applicability across various domains of materials science [77].

In the field of catalysis, researchers applied an ensemble CGCNN model to a dataset of over 40,000 unrelaxed alloy structures for CO₂ reduction catalysts (Figure 11).

Figure 11

(a) The complete roadmap of the estimation model for binding energies and (b) a typical bar plot of predicted binding energies of In₃Cu₇ using 5 different models for LS-CGCNN, LS-SchNet, representing their bias. (*) denotes the application of the original CGCNN (binding site) substitution is carried out. (Reproduced with permission [98]. Copyright 2020 American Chemical Society).

This modified model is applied to unrelaxed structures realizing the MAEs (0.085 and 0.116 eV) for binding energies of H and CO, respectively. These results of an economical model outperformed the best reported method that needs costly geometry relaxations yet can achieve errors of 0.13 eV for both H and CO binding energies. Analysis of the model parameters discloses its effective learning of the binding site related to chemical information. By exploiting the graph structure of materials, CGCNN effectively models the interactions between atoms, facilitating precise electronic property predictions. In addition to electronic properties, CGCNN is adept at predicting mechanical properties like elastic constants, fracture toughness, and bulk modulus. Its ability to consider atomic arrangements and their connectivity enables accurate modeling of material behaviors under mechanical stress, crucial for designing materials with desired mechanical properties. Case studies demonstrate CGCNN’s effectiveness in real-world applications. Researchers have leveraged the CGCNN in various applications. As an instance, CGCNN has been used to predict band structures of new materials for photovoltaic applications, contributing to the discovery of efficient solar cell materials [100]. Beyond property prediction, CGCNN has facilitated targeted materials design in photovoltaics by Xie and Grossman [22], Takatsu et al. [93], and Zhan et al. [100] used CGCNN to screen a large scale of double-perovskite chemical space, identifying promising perovskite candidates for solar cells [22,93,100]. In another application, a CGCNN model was designed to learn per-site properties of materials for assessing local information such as atomic vibration frequency, magnetic moment, Bader charge, and site-projected d-band and O2p-band centers. Such pre-site properties have an extensive arena of applicability in electronics, spintronics, thermodynamics, and catalysis [101]. These capabilities make CGCNN suitable for practical applications in electronics, spintronics, thermoelectrics, and catalysis. ML methods, including DL methods like CGCNN, are increasingly used in predicting material characteristics, demonstrating their usefulness in gathering complex associations within the data and exhibiting valuable perceptions about the material performance. These real-world case studies highlight CGCNN’s practical contributions across multiple domains, ranging from environmental remediation and energy storage to catalysis and photovoltaics. Its ability to learn from complex structural data and deliver reliable predictions has made it a critical tool in reducing experimental workloads, optimizing materials selection, and accelerating innovation in materials science.

7 Advancements in graph representation learning: Evaluation and challenges

8 Future directions and integration with other computational methods

As the field of materials informatics continues to evolve, several potential avenues for future research in CGCNN and material property prediction are emerging. First, there is a growing interest in enhancing the interpretability of CGCNN models. Incorporating interpretable AI methods, such as attention mechanisms or saliency maps, can offer insights into the learned features and the fundamental physics influencing material properties [125]. In the domain of crystalline materials, a persistent challenge is the accurate prediction of electronic structures, elastic properties, and related physical quantities. Existing graph learning frameworks often fall short in reliably predicting material properties such as bandgaps and bulk/shear moduli. Further advancements in graph-based DL frameworks necessitate innovative architecture designs that transcend real space, incorporating multiple tiers of material information, including local and/or global symmetries, topologies, orbitals, and reciprocal space data information [126]. Additionally, further exploration of uncertainty quantification methods within CGCNN frameworks can enable more reliable predictions by accounting for uncertainty in input data and model parameters [114]. Moreover, extending CGCNN models to predict multi-property relationships and exploring transfer learning approaches for knowledge transfer between different materials domains represent promising directions for future research [44,127]. Recent advancements in CGCNN and material property prediction have been driven by ongoing developments in DL techniques, data availability, and computational resources. One emerging trend is the integration of domain knowledge into CGCNN models through the incorporation of physical constraints and principles. This hybrid approach, often referred to as physics-informed ML, leverages domain-specific knowledge to guide model learning and improve prediction accuracy [128]. Another emerging trend is the utilization of GNNs for highly productive screening of material properties, enabling the rapid detection of advanced materials with desired functionalities [129]. Furthermore, the democratization of materials data through open-access databases and collaborative initiatives is facilitating broader access to training datasets and fostering interdisciplinary collaborations in materials informatics research (Materials Project). Although large databases such as the Materials Project and OQMD have significantly advanced data-driven materials discovery, they exhibit notable biases that limit model generalizability. These include overrepresentation of certain material classes (e.g., oxides), underrepresentation of others (e.g., nitrides, halides), and a focus on equilibrium phases while neglecting metastable yet experimentally relevant materials. Additionally, structural simplifications such as the omission of defects, disorder, and surface effects – reduce the realism of the data. Similar dataset limitations are seen in other materials research domains. For instance, in fused silica, DL models for detecting subsurface defects are constrained by datasets that fail to capture microstructural complexity [130]. Likewise, in lithium-rich manganese oxides, understanding oxygen-redox mechanisms is hindered by the lack of structurally diverse data reflecting real degradation pathways [131]. Addressing these gaps through better-curated, diverse datasets and integration of experimental data is essential for improving the reliability and transferability of models like CGCNN. For instance, combining CGCNN with DFT calculations can enhance prediction accuracy by incorporating quantum mechanical insights into the model. Integrating CGCNN with other computational techniques presents promising opportunities to enhance material property prediction and discovery. For example, combining CGCNN with ab initio methods such as DFT can incorporate quantum mechanical insights, improving prediction accuracy and reliability. Similarly, coupling CGCNN with molecular dynamics (MD) simulations allows for the exploration of dynamic properties and phase transitions in complex materials systems. Despite its successes, several limitations of CGCNN must be acknowledged. A primary concern is overfitting, particularly when models are trained on small or unbalanced datasets. A lack of diversity in crystal structures or material types can limit the model’s ability to generalize to unseen systems. Interpretability also remains a challenge. While recent advances in explainability offer some insights, the graph-based architecture of CGCNN often results in opaque decision-making processes, which may hinder its use in critical or high-stakes applications. Moreover, the computational intensity of CGCNN, especially when handling large-scale graphs or high-dimensional inputs, can limit scalability. However, by integrating CGCNN with complementary computational frameworks like DFT and MD, many of these challenges can be mitigated. These hybrid approaches promise not only improved accuracy but also deeper physical insight, making them valuable for accelerating materials discovery. Furthermore, integrating CGCNN with uncertainty quantification methods, such as Bayesian inference or ensemble learning, can provide probabilistic predictions and quantify the confidence of model predictions [112]. An essential feature in the deployment of graph-based DL in the case of material science is the practicability of extending learning models that are established on trivial and regular graphs to bigger and complex material systems [132]. These systems include molecule–solid hybrid systems, surface and interface structures, solids with point and line defects, among others. Technologies like transferring learning and graph attention, actively developed in other DL fields, warrant attention to ascertain their applicability in this evolving domain of research. Overall, the future of CGCNN and material property prediction lies in enhancing model interpretability, leveraging domain knowledge, and integrating with other computational methods for a more holistic understanding of material behavior. Ongoing developments and emerging trends in DL, data availability, and collaborative research efforts are expected to drive further advancements in the field.

9 Conclusions

This comprehensive review offers deliberations into the capabilities and limitations of CGCNN in the prediction of material properties, contextualizing its performance within the broader landscape of materials informatics and DL methodologies. Throughout the analysis, it becomes evident that the CGCNN model, tailored for crystals with atoms possessing lattice periodicity, has demonstrated remarkable versatility and effectiveness in various materials science applications. Owing to this, graph-based illustrations for materials science applications are increasingly leveraged for the evolution of CGCNN architectures. To date, CGCNN models have successfully predicted electronic properties such as bandgaps, carrier mobility, and electronic band structures, and their effectiveness is established in practical scenarios, signifying an ever-evolving role in accelerating materials discovery and design processes. Among the key advantages of CGCNN are its ability to directly learn from crystal structures without manual feature engineering, its scalability to large datasets, and its capacity to capture complex atomistic interactions using graph representations. These characteristics have made CGCNN a benchmark model in materials property prediction tasks. However, several limitations persist. CGCNN models may suffer from overfitting when trained on limited or imbalanced datasets, and their interpretability remains a challenge due to the black-box nature of DL models. Additionally, they can be computationally demanding and may require significant resources for training and hyperparameter tuning. Data heterogeneity and the lack of universal representations for diverse material systems further limit model generalizability. Despite these challenges, future expansion of CGCNN in material property prediction hinges on improving model interpretability, integrating with other computational frameworks, and leveraging increasingly diverse and high-quality datasets from interdisciplinary fields in materials science. It is heartening to note that contemporary developments and emerging trends in DL, data availability, and collaborative research efforts are poised to drive progressions in this field. By capitalizing on these advancements and fostering interdisciplinary collaboration, the future of CGCNN and its role in accelerating materials discovery and design processes appears promising. Ultimately, CGCNN stands as a testament to the power of DL in revolutionizing materials engineering and science, paving the way for innovative solutions and breakthroughs in diverse applications.