Symmetry-based analysis is a method that utilizes the crystal symmetries of materials to diagnose topological invariants, such as the Chern number, the Chern-Simons term [4] and the Z2 invariant first proposed by Kane and Mele [22]. In earlier studies, only a few hundred materials were confirmed to possess nontrivial topological properties in the prediction and discovery of TIs and TSMs. Bradlyn et al. [9] proposed the TQC framework, a comprehensive electronic band theory [1] that combines a graph-theoretical description in momentum space with a dual group-theoretical description in real space, thereby revealing the relationship between topology and localized chemical bonding. This theory systematically classifies the band structures arising from localized atomic orbitals across all 230 crystal symmetry groups, identifying nontrivial topological bands. It not only provides new interpretations for known TIs but also demonstrates a powerful capability to predict a large number of previously unknown TIs. Po et al. [23] developed a systematic approach to reveal symmetry-based TIs in all 230 space groups. This method represents band structures in terms of (elementary band representations) EBRs and then isolates topological bands by removing subsets of atomic insulators defined through localized symmetric Wannier functions. The approach uncovers previously hidden topological band symmetries, providing a systematic tool for the search for topological materials.

Subsequently, Elcoro et al. [28] extended the TQC framework to magnetic space groups, establishing a comprehensive real-space band topology theory applicable to both magnetic and nonmagnetic crystalline solids, known as Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, they derived a complete set of symmetry indicators for electronic band topology, enabling the identification of symmetry-allowed bulk states, as well as anomalous surface and hinge states.

Building on this foundation, Xu et al. [29] performed the first high throughput search for magnetic topological materials using MTQC derived indicators. They developed a complete framework including magnetic indicators, co representations, and compatibility relations, along with code to calculate them from first principles. Applying this to 549 magnetic structures with DFT plus Hubbard U, they predicted several novel magnetic topological phases. The significance is clear. This work transformed the theoretical MTQC framework into a practical screening tool. Prior to this, magnetic topological materials could only be identified through manual analysis.

However, three caveats should be considered. The Hubbard U values must be chosen empirically, and sensitivity to this choice was not explored. The predictions require experimental verification. The computational cost is substantial, which may limit practical applicability. Nevertheless, this work marked an important milestone. Theoretical advances like MTQC gain real power only when paired with computational implementations that enable large scale screening. Table 1 provides a comparison between the TQC formalism and the MTQC framework.

Based on the recently developed TQC framework, Vergniory et al. [30] performed the first large scale high throughput search for topological materials. Using their own code, they calculated symmetry indicators for 26,938 compounds from the ICSD and identified thousands of topological insulators and semimetals. Over 27% of the analyzed compounds exhibited nontrivial topology. The source code and data were released publicly. The significance is substantial. This work transformed the field from case by case discovery to database driven screening. Prior to this study, only a few hundred topological materials were known. However, three caveats should be noted. The study covered only non magnetic materials. Spin orbit coupling was not included for all calculations. The 27% figure is an upper bound; some identified materials may have band gaps too small for experimental observation. Despite these caveats, this work marked a turning point. The methodology became the standard, and the public code release demonstrated that large scale reproducibility is possible in computational materials science.

Topological materials exhibit unconventional linear responses in the bulk and anomalous gapless boundary states at their edges. These materials are not only of fundamental scientific interest but also hold great potential for applications in advanced low-power, high-speed spintronic, magnetoelectric, and optoelectronic devices, as well as quantum computing. However, due to the computational complexity of topological invariants or topological nodes, the identification of topological materials has historically relied heavily on materials science expertise and advanced theoretical tools, which has hindered their efficient discovery. To address this, Zhang et al. [31] developed a highly efficient, fully automated algorithm capable of diagnosing nontrivial band topology for the majority of nonmagnetic materials. The algorithm is based on the established complete correspondence between the symmetry representations of occupied bands and topological invariants. By systematically scanning nearly 40,000 materials in crystal structure databases, the algorithm identified over 8000 nontrivial topological materials. The results have been made publicly accessible through an interactive user interface within the database.

Although the diversity of spatial symmetries leads to a rapid increase in the possible topological crystalline (TC) phases of electrons, including topological crystalline insulators (TCIs), their physical realization has progressed slowly due to the practical difficulties in determining band topology computationally. The algorithm developed by Tang et al. [32] is based on the comprehensive symmetry indicators framework introduced in [23]. Analogous to the well-known Fu–Kane parity criterion for TIs, this framework enables the reliable identification of topological materials even in the absence of prior knowledge about the precise topological nature of the bands involved [33–36]. They applied their program to search for materials within a database [37] containing previously synthesized compounds, focusing on 8 of the 230 space groups. Their approach revealed topological materials with a wide range of distinct topological properties, spanning weak and strong TIs as well as the recently proposed higher-order topological insulators (HOTIs) [12–15,35,36]. In addition, their algorithm successfully captured TSMs. The general high-throughput discovery workflow combining databases, machine learning screening, first-principles calculations, and topology diagnosis is outlined in Fig. 2.

We should note that the practical reproducibility of the aforementioned high-throughput workflows is influenced by several factors. First, the consistency between different databases. The computed properties of the same material (e.g., formation energy, band gap) can vary across databases due to differences in DFT parameters, pseudopotentials, or post-processing conventions. High-throughput screening workflows that combine multi-source data without cross-validation may introduce systematic biases. Second, the treatment of symmetry tolerances can subtly alter material classification. DFT codes typically identify crystal symmetries by applying numerical thresholds. Minor variations in these tolerances can change whether a structure is classified as cubic or tetragonal, thereby affecting symmetry-derived properties and the assignment of topological invariants. Third, numerical convergence in DFT calculations directly determines the reliability of predicted properties. Parameters such as plane-wave cutoff energy, k-point sampling density, and force convergence criteria must be systematically tested for convergence. Insufficient convergence may lead to false positives, meaning materials that appear promising in screening but fail under stricter thresholds.

Therefore, to enhance the reproducibility and reliability of high-throughput screening studies, the following practices should be adopted: (i) cross-validation when using multi-source data, (ii) explicit reporting of symmetry detection tolerances, and (iii) convergence tests for DFT calculations with well-defined numerical thresholds.

Significant progress has been made in predicting novel topological materials using high-throughput, symmetry-based descriptors derived from empirical rules. To date, these methods have primarily been applied to materials within existing databases, but they are severely limited to systems with well-defined symmetries, leaving a vast portion of the materials space unexplored.

Cao et al. [24] developed an artificial intelligence based descriptor for topological insulators using the SISSO (Sure Independence Screening and Sparsifying Operator) method. The descriptor uses only atomic numbers and electronegativities. It does not require symmetry information or detailed band structures. Applied to chalcogenide alloys, nearly half were identified as topological insulators. The significance is twofold. First, expensive band structure calculations are not always necessary. A simple elemental descriptor can be surprisingly effective. Second, the descriptor works without symmetry information, making it applicable to low symmetry or disordered systems. However, three limitations should be noted. The descriptor was only tested on chalcogenides. Its transferability to other material families is unknown. It also provides only a binary prediction without identifying the specific topological invariant. The nearly 50% hit rate is impressive but applies only within this specific chemical space. The broader implication is important. The SISSO method produces interpretable descriptors rather than black box predictions. This is valuable for materials discovery, where understanding why a material is topological is as important as knowing that it is. Future work should test transferability to other families and extend the approach to predict specific invariants.

Andrejevic et al. [38] took a different approach, training a neural network on computed X-ray absorption near-edge structure (XANES) spectra to classify materials as topological or trivial. Beyond achieving F1 scores of 89% and 93% for topological and non-topological materials, respectively, their work has three notable implications: (i) the model identified Weyl semimetals that TQC had missed, suggesting spectral features may capture topology beyond symmetry indicators; (ii) the method works on non-crystalline and non-cleavable materials, filling a gap in experimental characterization; and (iii) the simplicity of X-ray absorption spectroscopy (XAS) enables in situ observation of topological phase transitions.

However, the approach has limitations that the original study does not fully address. The model’s reliance on computed spectra for training means its performance on truly novel materials—those whose XANES spectra have not been computed—remains untested. Moreover, the spectral features that correlate with topological class are not explicitly identified, leaving the method as a black-box classifier rather than a tool for physical insight. Future work could combine this spectroscopic approach with interpretable machine learning techniques to extract which spectral regions carry topological information.

The topogivity rule [25] offers a composition-only heuristic. Each element is assigned a machine-learned “topologicality” value, and the weighted average of these values predicts whether a material is topologically nontrivial (positive average) or trivial (negative average), with over 80% accuracy. The rule’s importance is twofold. First, it demonstrates that topological propensity is largely encoded in elemental chemistry rather than precise atomic arrangements, corroborating findings from Claussen et al. [26]. Second, because it does not require crystal symmetry, it can screen materials that evade symmetry-based diagnosis. However, topogivity’s finite accuracy, approximately 20% error, makes it suitable only as a pre-filter in high-throughput workflows, not as a definitive classifier. The trade-off between topogivity’s broad applicability and structure-based models’ quantitative precision suggests that hybrid workflows, which use topogivity for initial screening followed by graph-based models, may be the most practical path forward.

The topologicality framework was later extended by Xu et al. [39] to address magnetic and two-dimensional materials. Three modifications were introduced. They incorporated magnetic materials into the training set, added the Hubbard U parameter to account for electron correlation, and replaced the linear weighting scheme with a convolutional neural network (CNN). These changes allowed the method to discover six classes of previously unknown Chern insulators in 2D materials, four of which exhibit a global band gap.

However, the extension raises two methodological concerns that the original study did not fully address. First, the Hubbard U parameter introduces an additional degree of freedom that must be chosen empirically. The sensitivity of the CNN’s predictions to different U values was not systematically explored, leaving uncertainty about whether the discovered Chern insulators are robust predictions or artifacts of specific U choices. Second, while the CNN improves predictive accuracy, it sacrifices interpretability. The original Topogivity’s key advantage was its simplicity and manual calculability. The CNN version loses this advantage without providing a clear mechanistic understanding of why certain compositions favor topological order.

From a broader perspective, the trade-off exemplified by this work is typical in the field. Simple heuristics like the original Topogivity offer interpretability and broad applicability but limited accuracy. Complex models like the CNN extension offer higher accuracy at the cost of transparency and generalizability across different Hubbard U regimes. For magnetic topological materials, where DFT calculations are already expensive and sensitive to U parameters, a more promising direction may be to incorporate physically informed priors rather than purely data-driven learning. The CNN approach is a step forward in predictive power, but whether it represents a step forward in physical understanding remains an open question.

Early research primarily focused on validating the feasibility of machine learning using theoretical physical models. Carrasquilla and Melko [40] showed that a neural network can identify ferromagnetic and paramagnetic phases from Ising spin configurations and locate the phase transition. The network learned to recognize magnetization, the conventional order parameter. This work opened the direction of using machine learning to identify phases without predefined order parameters. It laid the foundation for topological phase recognition, where conventional order parameters often do not exist. However, the Ising model is simple. Success here does not guarantee success on more complex systems. Verifying what the network learned was possible because the correct order parameter was known. For topological phases without such a parameter, this verification is much harder. Despite these caveats, this work was a turning point. It showed that machine learning can be a tool for discovery, not just classification.

van Nieuwenburg et al. [41] proposed learning by confusion, a method that identifies phase transitions without order parameters or prior knowledge. It assigns labels based on hypothetical boundaries and trains a neural network. When the hypothetical boundary matches the true one, validation accuracy peaks. The method successfully detected topological, thermal, and many body localization transitions. The key advantage is that it works without knowing what to look for. This is valuable for topological phases where order parameters do not exist. However, the method requires scanning over many hypothetical boundaries, which is computationally costly. It also identifies where a transition occurs but does not characterize the phases themselves. Despite these limitations, this work introduced a clever and widely applicable approach.

Zhang et al. [42] showed that a neural network can predict the winding number of one-dimensional chiral topological insulators from local Hamiltonian parameters with nearly perfect accuracy. The network trained on a subset of winding numbers generalized to larger unseen numbers, indicating that it learned the winding number formula rather than memorizing samples. Analysis of the network’s internal structure confirmed this interpretation.

The work’s significance is twofold. It provides rare evidence that neural networks can learn a mathematical invariant from data without explicit programming. It also challenges the view that deep learning models are merely interpolating black boxes. However, two limitations exist. The study was restricted to one-dimensional toy models, and the network required Hamiltonian parameters as input, which are not experimentally accessible. Extending this approach to realistic materials remains an open challenge. Nevertheless, the work established an important proof of concept for learning topological invariants from data.

A different strategy was pursued by Cheng et al. [43], who used real-space observables rather than momentum-space Hamiltonian parameters to identify Chern numbers in the Haldane model. Unlike Zhang et al. [42], who required Hamiltonian parameters as input, Cheng et al. designed a disk geometry with sixfold rotational symmetry and used physically measurable quantities as features. These included chiral edge states, where the number of edge branches gives the Chern number magnitude and the direction of edge currents gives its sign. They also used the real-space particle density distribution, where edge states concentrate at the boundary while bulk states are distributed in the interior. They further used the local density of states, where a topological phase exhibits finite edge-state local density of states within the energy gap while a trivial phase does not.

The key contribution of this work is not the specific choice of features but the demonstration that machine learning can extract topological information without access to the Hamiltonian or periodic boundary conditions. By tuning the hopping parameters and staggered flux, the method successfully distinguished trivial from nontrivial phases. Notably, when long-range hopping was introduced, the model captured phases with higher Chern numbers. This capability is particularly valuable for experimental settings where momentum-space measurements are difficult or where open boundaries and external potential traps prevent direct Chern number calculation.

However, the approach has practical limitations that deserve attention. First, the engineered disk geometry with sixfold rotational symmetry is not general. Applying the same method to arbitrary crystal structures would require redesigning the input representation for each new system. Second, the chosen features are physically motivated but still require prior knowledge of what to look for. A truly general method would learn which real-space features are relevant without human guidance. Despite these limitations, the work establishes an important principle. Topological information is encoded not only in momentum-space Hamiltonians but also in real-space observables, and neural networks can learn to decode it. This principle remains underexplored in later topological materials discovery efforts, which have largely returned to momentum-space or symmetry-based inputs.

To effectively integrate topological invariants into machine learning descriptors, it is essential to construct a formal mapping f:T→F,where T denotes the space of topological invariants and F denotes the feature space used by machine learning models. From the perspective of information retention, such mappings can be broadly categorized into the following three classes: (1)

Global Invariant Mapping. The entire topological distribution is compressed into one or a few global numerical quantities, such as the Chern number or Z2 invariant. This approach is particularly suitable for topological phase classification tasks. While it incurs the greatest information loss, it offers the lowest computational cost.

For example, to formally integrate the Berry curvature Ω(k) into the machine learning pipeline, we define the following three types of mappings f:Ω→F: (1)

Topological invariant mapping (finv). The domain is Ω(k) and the codomain is Z. The mapping rule is finv(Ω)=12π∫BZΩ(k)d2k, i.e., the Chern number. This mapping produces a single discrete feature and is suitable for topological classification tasks.

With the development of symmetry based theories such as TQC, researchers have begun using physical quantities from DFT calculations as input features for predicting topology in real materials. Sun et al. [44] extended machine learning prediction of topological invariants to more complex model systems. Using local Hamiltonian inputs without manual feature engineering, their deep neural networks predicted winding numbers and Chern numbers with high accuracy, even generalizing to unseen invariant values. Hidden layer analysis revealed that the networks learned the local winding angles and Berry curvature [45,46], confirming that they had internalized the mathematical formulas of the invariants themselves.

The key insight is that neural networks can capture global topological features from purely local inputs. This is not obvious because topological invariants are defined as integrals over the entire Brillouin zone. However, two caveats apply. First, the study used model Hamiltonians with known analytical forms; real DFT Hamiltonians may be noisier. Second, the network required Hamiltonian inputs, which are not experimentally accessible. Despite these caveats, the work provided strong evidence that deep learning can learn complex mathematical invariants and remains an important reference for theory driven machine learning in this field.

Half-Heusler compounds are promising for topological insulator discovery due to their tunable electronic structures, but screening them is computationally expensive. Liu et al. [47] developed a compressed sensing descriptor that uses only atomic numbers, electronegativities, and valence electron counts to rapidly screen potential topological insulators in this family. The descriptor successfully predicted numerous candidates, including compounds with fractional stoichiometries. The practical value is clear. A simple descriptor requiring no DFT calculations can prioritize candidates for experimental validation. However, there are still three limitations that warrant attention. The descriptor was only tested on half-Heusler compounds, so its transferability to other families is unknown. It identifies correlations but provides no causal mechanism. It also predicts only binary topology (topological or not) without specifying the invariant type.

Despite these limitations, the work demonstrates that elemental properties alone carry meaningful topological signal, consistent with the Topogivity results. A promising direction would be to combine such descriptors with uncertainty quantification, so users know when predictions are reliable and when DFT validation is needed.

CGCNN [48] pioneered the use of graph neural networks [49] for crystal property prediction. It represents crystals as graphs of atoms and bonds, uses gated convolution to learn local interactions, and achieves a mean absolute error (MAE) of 0.039 eV/atom for the formation energy on the materials project database [50,51]. Its key advantage over many later models is interpretability. By extracting contributions of local chemical environments, CGCNN can identify empirical rules, such as which elements stabilize perovskite structures. The work initiated the research direction that led to MEGNet and ALIGNN. However, two limitations are worth noting. First, CGCNN requires fully relaxed crystal structures, limiting its use for hypothetical materials. Second, it predicts ground state properties like formation energy, not topological invariants. For topological materials discovery, CGCNN is therefore a pre screening tool, not a direct classifier. Whether stability trends (like those identified for perovskites) correlate with topological nontriviality remains unknown and requires separate analysis.

Chen et al. [52] proposed MEGNet with two innovations beyond CGCNN. Global state inputs (temperature, pressure, entropy) unify internal energy, enthalpy, and Gibbs free energy into a single model. Transfer learning enables pre trained embeddings to be reused for data scarce tasks like band gap prediction. On the Materials Project dataset, MEGNet achieved a formation energy MAE of 0.028 eV/atom, outperforming CGCNN’s 0.039 eV/atom. The reported accuracy is impressive. However, two considerations arise when evaluating MEGNet for topological materials discovery. First, the improvements are demonstrated on ground state properties (formation energy, band gap), not on topological invariants. Whether MEGNet’s architectural advances translate to better topology prediction is unknown. Second, the transfer learning advantage depends on having a large source dataset. For topological materials, where labeled data is scarce, this precondition may not hold. The real value of MEGNet for topological materials research may lie not in direct prediction but in providing accurate band gaps and formation energies as features for downstream topology classifiers. In this sense, MEGNet is a powerful tool for pre screening, not a direct topology solver.

ALIGNN [53] was designed to capture bond angle information, a key limitation of CGCNN and MEGNet which only use atomic distances. By transforming the original graph into a line graph, ALIGNN encodes angular relationships and can distinguish tetrahedral from octahedral configurations. The performance gains are substantial. ALIGNN reduced formation energy MAE by 21.4% and band gap MAE by 33.9% compared to MEGNet. Ablation studies confirmed that angular information alone contributes a 29.14% error reduction.

The key insight is that bond angles are essential for accurate property prediction. However, three caveats apply for topological materials discovery. ALIGNN requires relaxed crystal structures, predicts ground state properties rather than topological invariants, and its generalization to out of distribution structures has not been tested. Nevertheless, ALIGNN represents the current state of the art for graph based property prediction. The clear message is that angular information is not a minor refinement but a critical component. Future work should explore whether higher order structural information yields further improvements.

Claussen et al. [26] used gradient boosting trees to predict topological properties from lightweight features. The model required only chemical composition and crystal symmetry, not relaxed structures. It achieved 89.7% accuracy with F1 scores of 94.0% for trivial insulators, 70% for TIs, and 92.0% for TSMs. The work revealed two important insights. First, topological properties are primarily determined by coarse grained composition and symmetry, not precise atomic positions. Second, the low TI F1 score (70%) reflects class imbalance and the intrinsic rarity of topological insulators in known databases.

The practical value is clear. The model runs much faster than DFT, and an online tool makes it accessible to experimentalists. Despite these advantages, two limitations remain. The model misses nearly one third of TIs, and its predictions inherit noise from DFT labels near phase boundaries. For large scale screening, a fast GBT model followed by targeted DFT validation may be more practical than applying expensive graph models to every candidate. This finding aligns with the Topogivity results, reinforcing the message that composition alone carries strong topological signals. Building on these developments, recent work has further extended structure- and composition-based learning to equivariant interatomic potentials such as NequIP [54] and MACE [55] as well as large-scale generative foundation models such as DiffCSP [56] and MatterGen [57], broadening the representational capacity available for downstream topological-materials screening. Table 2 provides an overview of the stages involved in applying machine learning to topological materials, along with the types of input features used at each stage.

The three graph-based models described above, CGCNN, MEGNet, and ALIGNN, have all demonstrated strong performance on their respective benchmark tests. However, when attempting to compare these models, a critical methodological issue arises: there is currently no unified benchmarking framework that evaluates them under identical conditions. As shown in Table 3, the performance reported in the original publications is based on different datasets, different training and test splits, and different evaluation metrics MAE of formation energy per atom, but using different versions of the Materials Project database and different preprocessing pipelines. These discrepancies make it impossible to determine whether the reported performance improvements arise from genuine advances in model architecture or merely from differences in evaluation protocols.

At first glance, these numbers appear to suggest a clear performance hierarchy, with ALIGNN outperforming MEGNet and MEGNet outperforming CGCNN. However, a closer examination reveals that CGCNN was evaluated using an earlier version of the Materials Project database (the 2018 release version) and a different data splitting strategy, whereas MEGNet and ALIGNN were trained and tested on a later and larger dataset with different splitting protocols. In addition, the random seeds used for data splitting are typically not reported, making it impossible to reproduce the exact training conditions. The reported 21.4% reduction in formation energy MAE of ALIGNN relative to MEGNet may indeed reflect the benefit of incorporating bond angle information. However, in the absence of controlled comparisons under identical conditions, including the same dataset version, identical data splits, fixed random seeds, and consistent evaluation protocols, the quantitative magnitude of this improvement cannot be reliably assessed. Such a lack of standardization severely hinders the community’s ability to assess genuine progress.

We believe that the field of machine learning for topological materials would greatly benefit from community accepted benchmarking frameworks. Drawing on best practices from materials informatics and the broader machine learning community, we propose the following recommendations.

Standardized datasets: All models should be evaluated on one or more publicly available, version controlled datasets. Candidate datasets include the Topological Materials Database, symmetry indicator annotations from the Materials Project, or curated subsets thereof. The database version and all preprocessing procedures, such as formation energy filtering, duplicate removal, and structural relaxation status, should be explicitly documented.

Adopting these recommendations would enable the field to move beyond the current situation of numerous incomparable performance reports toward a cumulative and quantitatively rigorous body of knowledge. Several recent efforts have already moved in this direction. For example, MatBench [58] provides standardized benchmarks for general materials property prediction, while the topological materials database offers a curated database specifically focused on topological materials. We encourage future studies in machine learning for topological materials to adopt these or similar benchmark frameworks whenever possible. Until a unified evaluation framework becomes standard practice, performance comparisons across different models should be interpreted with caution.

Inverse design is a data-driven strategy that combines desired material property data, domain knowledge, and artificial intelligence to discover new materials with specific functionalities. This approach is rapidly advancing in the field of novel materials research. Hong et al. [27] proposed the CTMT framework for the inverse design of topological insulators and semimetals. The framework integrates four components. The CDVAE (crystal diffusion variational autoencoder) generates new crystal structure candidates. Topogivity rules perform an initial screen based on chemical composition. The M3GNet interatomic potential [59] provides rapid energy, force, and stress evaluations for dynamical stability. Finally, TQC provides precise topological classification. Starting from 10,000 generated candidates, CTMT applied successive filters for novelty, validity, topogivity, thermodynamic stability, and dynamical stability. This process confirmed 20 new topological materials, including 16 TSMs and 4 TIs. Notably, the method discovered four chiral Kramers Weyl semimetals [60–63], a class of low symmetry materials that traditional symmetry based methods struggle to identify. The reported success rate was 62.5%, compared to less than 30% for conventional approaches.

The significance of CTMT is twofold. First, it demonstrates that generative models can produce genuinely new topological materials not present in existing databases. This addresses a key limitation of screening based approaches, which can only find what is already known. Second, the discovery of low symmetry chiral materials highlights an advantage of data driven generation over symmetry based diagnosis. The latter works well for high symmetry crystals but fails when symmetry is low.

The CTMT strategy can be interpreted as a closed discovery loop integrating generation, screening, validation, diagnosis, and database feedback, as shown in Fig. 3. Table 4 provides a summary of the machine learning methods discussed above for discovering topological materials.

Although the CTMT framework represents an important advance in the inverse design of topological materials, its validation protocol deserves critical examination. Before performing full topological classification, the CTMT framework employs the M3GNet interatomic potential as a preliminary filter for dynamical stability. While this screening step is computationally efficient, reliance on an approximate potential introduces the possibility of false positives. In other words, some materials that pass the M3GNet stability screening may later be found to be topologically trivial.

The original CTMT study provides sufficient data to quantify this risk. Among the 57 candidate materials that passed the DFT thermodynamic stability check, M3GNet phonon spectrum calculations identified 32 as dynamically stable, corresponding to a success rate of 56%. Subsequent TQC based topological classification confirmed that only 20 of these 32 candidates were topologically nontrivial. Therefore, among the candidates that passed the M3GNet dynamical stability screening, 12 were false positives with respect to topological nontriviality. This corresponds to a false positive rate of 37.5%.

This quantitative estimate indicates that the use of approximate interatomic potentials can indeed introduce a nonnegligible fraction of false positives into the validation workflow. Nevertheless, the CTMT framework remains highly effective. The M3GNet screening stage reduces the number of candidate materials from 57 to 32, thereby substantially decreasing the computational burden of subsequent TQC calculations. At the same time, the discovery of 20 new topological materials represents a significant achievement.

For future inverse design studies, we recommend the following practices to further strengthen the rigor of the validation process.

Explicitly report the false positive rate and false negative rate of machine learning interatomic potential screening relative to DFT validation.

It is important to note that beyond the capability of generating structures and the topological nontriviality of candidate materials, several equally critical issues must be considered when evaluating and developing generative models for topological material design. These include first structural stability, where generated crystal structures should exhibit no imaginary phonon modes indicating dynamical stability and should remain intact under finite temperature conditions indicating thermodynamic stability. Second synthesizability, referring to whether the predicted stability can be realized under actual experimental conditions. Third diversity and novelty, namely whether the generative model can produce structures that are genuinely different from those in the training set rather than minor variations. Fourth computational efficiency, measured in comparison with the cost of brute force high throughput screening. Generative models can bypass the expensive energy evaluations required in traditional crystal structure prediction methods [64]. Conditional generation frameworks have been shown to significantly improve the efficiency of target material discovery [65]. Fifth false positive rate, defined as the proportion of materials identified as stable or topological by fast screening methods but later invalidated by more rigorous DFT and TQC calculations.

Tsai et al. [66] compared three quantum information quantities as inputs for detecting topological phase transitions in the Kitaev chain [67]. The three inputs were the Majorana correlation matrix, the entanglement spectrum, and the entanglement feature vector. All could identify the phase transition point. However, the entanglement spectrum failed to distinguish different U(1) gauge phases because it compressed the information too much. The other two inputs succeeded. Gradient-weighted class activation mapping (Grad-CAM) visualization showed that the network learned to focus on the region corresponding to edge state decay.

The key lesson is that input representation choices profoundly affect what a network can learn. The failure of the entanglement spectrum was not due to a weak network but because the necessary information had already been discarded. This principle is often overlooked in materials informatics, where most effort goes into architecture rather than input construction. However, the Kitaev chain is a one dimensional toy model. Whether the same conclusions hold for real topological superconductors remains unknown.

Chung et al. [68] extended the entanglement based method to two dimensional chiral p-wave superconductors. They compared the Majorana correlation matrix, the single particle entanglement spectrum, and the entanglement feature vector. Several findings emerged. Phase information is essential. The entanglement feature vector worked only when the phase angle was included. The magnitude alone was insufficient. Geometric structure matters. The tensor format of the Majorana correlation matrix outperformed the matrix format because it preserved spatial relationships. Rescaling inputs with dimensionless coordinates improved accuracy when the superconducting gap was large.

The key contribution is systematic. Phase information matters. Geometric structure matters. Input format matters. These principles likely generalize beyond the specific model studied. Several challenges remain, among which two are particularly noteworthy. The study used clean model Hamiltonians, not realistic materials with impurities and disorder. More importantly, the quantum information quantities used as inputs are not directly measurable in experiments. Future work should focus on bridging this gap, for example by training on simulated data with realistic noise and testing on experimental measurements.

The common machine learning pipeline for topological superconductors, from model systems and input construction to predicted topological outputs, is summarized in Fig. 4.

Zhang and Kim [69] proposed quantum loop topography to address a key challenge. Topological phases lack local order parameters, so standard machine learning inputs cannot detect them. Their method encodes nonlocal topological information into images by constructing loop products of correlation functions. A simple neural network achieved over 99.9% accuracy on a Chern insulator model. A control experiment confirmed that raw particle configurations did not work. The method showed impressive generalization. A network trained on non interacting Chern insulators could identify fractional quantum Chern insulators without retraining. A network trained on a square lattice could identify phase transitions in a honeycomb lattice. The approach does not rely on translational symmetry or diagonalization and can handle strongly correlated and disordered systems.

The importance of this contribution is considerable. This work provides a general feature engineering strategy that works where conventional order parameters do not exist. The cross model generalization is particularly impressive. Nevertheless, two concerns arise. The method requires correlation functions from Monte Carlo samples. For experimental data with finite precision, the construction may be less robust. The method was also demonstrated only on model systems, not on real topological materials. Despite these limitations, the work established an important principle. Transform nonlocal topological information into images. Then standard neural networks can decode it. This principle has influenced later topological deep learning, though subsequent work has focused more on persistent homology than on loop based constructions.

Rodriguez Nieva and Scheurer [70] proposed an unsupervised method for identifying topological order from raw spin configurations without labeled data. The method constructs a similarity matrix, performs eigenvalue decomposition, uses eigenvalue degeneracy to count topological sectors, and applies k means clustering to assign samples. The method successfully identified the Berezinskii Kosterlitz Thouless transition in the XY model and the topological sectors in the Ising gauge theory. The critical temperatures matched theoretical predictions. The significance is threefold. The method requires no labeled data, which is crucial for exploring unknown phases. It does not assume knowledge of the order parameter or invariant in advance. The diffusion map automatically reveals the number of sectors without user input.

Despite its strengths, two weaknesses are evident. The similarity matrix scales quadratically with sample size, making large datasets expensive. More importantly, the method has only been tested on clean Monte Carlo data. Whether it works on noisy experimental data remains unknown. The method also detects transitions and counts sectors but does not provide the topological invariant itself, such as the Chern number. Despite these limitations, this work provides a paradigm for automated phase diagram mapping. The contrast with supervised methods is stark. Supervised methods require knowing what to look for. Unsupervised methods discover what is there. For discovering entirely new topological phases, unsupervised approaches may be the only viable path forward.

The overall procedure for identifying topological phase diagrams from physical configurations and learned low-dimensional structure is illustrated in Fig. 5.