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REVIEW 3 major objections 4 minor 1 cited by

Machine Learning Free Quotients of CICYs

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A supervised classifier can predict fixed-point-freeness of a CICY group action from a compressed matrix and identifies nearly every free quotient on unseen manifolds.

desk verdict A first ML pass at CICY free-quotient detection, with a sensible encoding and a real held-out split; the missing description of how the negative examples were generated keeps the generalization claim provisional. read the letter →

arxiv 2508.19157 v1 pith:U56YNZ4I submitted 2025-08-26 hep-th

Wei Cui , Xin Gao , Mohsen Karkheiran , Juntao Wang This is my paper
classification hep-th MSC 14J3268T07
keywords Calabi-YauthreefoldsCICYsfreequotientsdiscretesymmetriescharacter-valuedindexmachinelearningmulti-headattentionstringcompactification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Calabi-Yau threefolds built as complete intersections (CICYs) are a standard set of string-compactification spaces, and their quotients by discrete symmetries are useful only when the group action has no fixed points. Traditional tests for fixed-point freeness—character-valued index calculations, fixed-locus checks, and smoothness checks—are slow, and known free quotients are rare. This paper asks whether a classifier can learn freeness directly from a compact matrix encoding of the manifold plus its symmetry action, and reports that it can for Z2, Z3, Z4 and Z2×Z2 symmetries. On CICYs excluded from training, fully connected networks identify all or almost all free quotients, and a multi-head attention model identifies every free quotient in all four symmetry classes. If this holds, machine learning could act as a fast pre-filter for free-quotient searches over datasets too large for traditional methods.

What carries the argument

The object doing the work is the compressed data matrix M: the CICY configuration matrix padded to 12×15, with the row action γ folded into two rows and the column action ρ appended as one row, giving a 15×15 input (18×15 when there are two Z2 generators). Because γ and ρ are permutations with possible phases, they are stored as signed index lists, removing the redundant zeros of the naive padded representation. The paper's two learning setups are a fully connected neural network and a transformer-style multi-head attention model that treats each (value, row, column) triple as a token and uses a CLS token to aggregate global correlations. A second load-bearing ingredient is data enhancement:

What would settle it

Build a test set of non-free actions whose fixed loci are computed by a different algorithm than the one that produced the paper's negative training examples—for example, by explicit solution of the fixed-point equations rather than by drawing from the known orientifold sample—and compare accuracy on this independent set with accuracy on the original test distribution. If the models perform at chance on the independent set while still scoring high on the original recipe, the learned signal is the label-generation signature, not fixed-point freeness.

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Extended reading notes

Core claim

The paper's central claim is that fixed-point freeness of a CICY group action is a learnable binary property of a single compressed matrix. A group action is encoded by the configuration matrix C together with a row action γ and a column action ρ; the paper compresses these into one 15×15 matrix (18×15 for Z2×Z2) by representing γ and ρ as signed permutation lists attached to a padded C. Balanced datasets are built from known free quotients and known fixed-point quotients, and equivalent representatives are generated by systematic row and column permutations. Splitting by CICY rather than by sample, the models are tested on unseen manifolds. The reported result is that per-sample accuracy is

Load-bearing premise

The negative training examples—group actions known to have fixed points—are representative of the full space of non-free actions; if they were produced by a recipe with a statistical footprint, the high accuracy could come from recognizing that footprint rather than from learning freeness.

Editorial extensions

If this is right

  • For symmetries with enough labeled examples, trained classifiers can pre-filter candidate group actions, sharply reducing the set that needs character-valued index or fixed-locus computation.
  • Because test CICYs are disjoint from training CICYs, the reported accuracy indicates transfer to new manifolds, not memorization of specific configuration matrices.
  • The same compact encoding works uniformly for cyclic groups and for the product group Z2×Z2, suggesting the representation can be reused for other abelian symmetry groups.
  • The models are balanced: in most cases they also identify all or nearly all quotients with fixed points, which matters because non-free actions vastly outnumber free ones.
  • A complete pipeline still requires a separate smoothness check for candidate free quotients; the paper does not claim to predict smoothness, only fixed-point freeness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is to generate non-free actions by a construction fully independent of the paper's negative examples and see whether accuracy survives; if it does not, the models are reading the generation recipe rather than an intrinsic freeness criterion.
  • If the compressed matrix is truly sufficient, the same classifier should transfer after fine-tuning to symmetry groups with too few known examples to train from scratch, and may uncover new free quotients in alternative CICY representations.
  • The attention model's advantage suggests that permutation-equivariant global correlations are the right inductive bias; applying similar tokenized-matrix models to toric Calabi-Yau datasets, where configuration matrices are absent, is a plausible extension, though not one the paper demonstrates.
  • The 50% voting rule effectively measures consistency across equivalent encodings; treating vote margin as a confidence score could let the method flag uncertain quotients for expensive verification, a use the paper does not spell out.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper trains fully connected and multi-head attention classifiers to distinguish free quotients of CICYs from quotients with fixed points, for symmetry groups Z2, Z3, Z4, and Z2×Z2. Inputs are compressed 15×15 or 18×15 matrices encoding the configuration matrix, row action γ, and column action ρ. Positive labels come from Braun [12] and Gray–Wang [13]; negative examples are generated fixed-point actions. Data are augmented by random row/column permutations, split by CICY size to avoid same-CICY leakage, and evaluated with a majority-vote procedure. The authors report high test accuracy and, for the attention model, perfect identification of all free quotients in their test sets.

Significance. If the generalization claim is correct, the paper provides a fast surrogate for character-valued index checks in CICY free-quotient searches, with potential application to other CICY representations and toric Calabi-Yau manifolds. Strengths include a genuine held-out split by CICY, a voting protocol over permutation representatives, comparison of two architectures, and a compact input representation that removes padding redundancies. However, the central claim depends on whether the negative examples are representative of the full space of non-free group actions; the manuscript does not currently establish this, and the positive test sets are very small. The result is plausible but needs additional validation before the broader classification claim can be accepted.

major comments (3)
  1. [§3.1 (Dataset generation) and footnote 1] The generation of label-0 (fixed-point) examples is not specified. For Z2, footnote 1 points only to the orientifold classification [17], which contains Z2 involutions of restrictive type (divisor exchange and multi-reflection) on favorable CICYs. For Z3, Z4, and Z2×Z2 no procedure or reference is given at all. The classification problem in eq. (5) is defined over arbitrary (C, γ, ρ), but the training set may only separate free quotients from one family-specific class of fixed-point actions. The reported held-out accuracy could then reflect a family signature rather than fixed-point freeness. Please document the exact generation algorithm for every group, state how fixed loci were verified, and test on an independent set of arbitrary non-free actions produced by traditional character-valued index methods.
  2. [§3.1, Tables 4–5] The positive test sets are extremely small: Z4 has only 23 known free-quotient CICYs, Z3 has 31, and Z2×Z2 voting is reported on 57 free quotients. Claims of 'perfect prediction' therefore rest on tens of original samples. No error bars, multiple seeds, or confidence intervals are reported. The generalization claim requires per-split counts of distinct CICYs and a stability analysis over random initializations.
  3. [§3.2 (Z2 voting paragraph)] It is ambiguous whether the voting results in Tables 1 and 4 are computed only on the test split or on all original samples. The text says 'The evaluation on the original samples is given by using the voting procedure' immediately after reporting test accuracy, without specifying that only test CICYs are used. If voting includes training or validation samples, the held-out claim is invalid. Please state this explicitly and report the number of original CICYs per split. Additionally, n=400 permutation representatives and the 50% voting threshold are fixed without sensitivity analysis; these are free parameters and should be justified or varied.
minor comments (4)
  1. [§2.2 (Smoothness Check)] The bullet states that 'for all quotients that are free of fixed points, one must test whether they are smooth.' For a finite group acting freely on a smooth variety, the quotient is automatically smooth. This step appears mathematically unnecessary and should be clarified or removed; the later discussion of singularity checks in §4 is also confusing for the same reason.
  2. [§3.1 (Data compression)] The compression of γ into signed integers is explained by example, but no formal definition is given for general abelian groups, including the roots of unity appearing for Z4. Since the compressed matrix is the sole model input, a precise specification would improve reproducibility.
  3. [§3.3.1 (Table 3)] The normalization Dij/20 appears ad hoc and is not motivated. The table uses the Z4 model with 15×15 input, but Z2×Z2 uses an 18×15 matrix; please clarify how the input shape and normalization vary across groups.
  4. [General / Reproducibility] No code, dataset, or trained weights are provided. Given the small number of underlying CICYs and the centrality of the label construction, releasing the compressed matrices and the generation scripts would be valuable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: supervised classifier uses external index-method labels; held-out CICY test split makes predictions genuine.

full rationale

The derivation chain is: (C,γ,ρ) → compressed matrix M → supervised classifier → binary label. The labels are not defined by the classifier; they are computed externally by the character-valued index method in Braun [12] and Gray–Wang [13]. The paper's Section 3.1 splitting procedure explicitly avoids putting the same CICY in both train and test, so the reported test accuracy and voting results are genuine out-of-sample predictions rather than re-statements of training labels. The data compression is an encoding of the group action and does not encode the target label; the label is supplied by the index calculation. The self-citation to [13] (coauthor Wang) is a data source with independent content (a prior character-index classification), not a load-bearing premise used to justify the model's success; the model's success is justified by held-out accuracy. The main weakness—unclear generation of label-0 examples, possibly from the orientifold family [17] (footnote 1, Section 3.2)—is a possible distribution-shift/external-validity limitation, not a circularity: it does not make the test prediction equal to an input by construction. No equation defines the target in terms of the input, and no result is forced by a self-citation chain.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim is an ML performance claim, so the ledger is dominated by label-source assumptions and hand-chosen hyperparameters. No new physical entities are introduced. The most consequential assumption is the completeness and correctness of the prior classifications, and the representativeness of the generated negative examples (see red flags).

free parameters (4)
  • Voting threshold = 0.50
    Section 3.1, Learning strategy: majority vote with a 50% threshold converts per-representative predictions into a final call. Shifts the precision-recall balance.
  • Number of permutation representatives per input for voting = 400
    Section 3.2, Z2 symmetry: 'we generate 400 representatives by row and column permutations' for each sample. The reported voting accuracy depends on this number.
  • Input normalization scale for attention model = Dij/20
    Table 3: 'Normalize value: Dij/20'. Hand-chosen scale; affects gradient magnitudes and training.
  • Architecture hyperparameters (hidden widths, heads, dropout, normalization) = d_model=64, heads=4, dropout=0.6, FC widths 64-256
    Chosen without reported systematic tuning; performance depends on these choices.
assumptions (3)
  • domain assumption The free-quotient classifications in [12] and [13] are correct and complete for Z2, Z3, Z4, and Z2xZ2 on the original and favorable CICY lists.
    All training and test labels are inherited from these references; any error or incompleteness in them is inherited by the ML models.
  • standard math The character-valued index criterion (Appendix B, eq. 29) correctly distinguishes free from fixed-point actions.
    Used to generate and verify the ground-truth labels, following [12,13].
  • domain assumption Random simultaneous row/column permutations of the compressed matrix M produce a valid equivalent instance of the same quotient problem.
    Section 3.1, Data enhancement: the permutation rule preserves the free/fixed-point status by construction, but this is assumed, not proven, in the paper.

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Cite this review

Pith. "Pith review of Machine Learning Free Quotients of CICYs." pith.science (2026). https://pith.science/paper/U56YNZ4I

@misc{pith2026250819157,
  author       = {Pith},
  title        = {Pith review of: Machine Learning Free Quotients of CICYs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U56YNZ4I}},
  note         = {Machine review of arXiv:2508.19157}
}
abstract

Free quotients of Calabi-Yau manifolds play an important role in string compactification. In this paper, we explore machine learning techniques, such as fully connected neural networks and multi-head attention (MHA) models, as a potential approach to detect $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathbb{Z}_2$ free quotients of CICYs. When tested on unseen examples, both models successfully identified almost all free quotients for $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. These results demonstrate that well-trained machine learning models can effectively generalize to new Calabi-Yau manifolds and may aid in the broader classification of free quotients in the future.

Figures

Figures reproduced from arXiv: 2508.19157 by the authors.

Figure 1
Figure 1. Voting for free quotients of CICYs with threshold 50%. Here [M] as an equivalent class of data while M1,M2,... are different representatives. 3.2 Free quotients from fully connected neural network We will study the free quotients of CICYs using neural network made of fully connected layers. First, we will explain how to apply it to classify Z2 quotients, followed by reporting our results for Z3 and Z4 quotients. Fin… view at source ↗
Figure 2
Figure 2. Z2 training curve for model with fully connected layers [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Z2 training curve for model with self-attention [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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Reference graph

Works this paper leans on

53 extracted references · 17 canonical work pages · cited by 1 Pith paper

  1. [17]

    A landscape of orientifold vacua,

    F. Carta, J. Moritz, and A. Westphal, “A landscape of orientifold vacua,” JHEP 05 (2020) 107, arXiv:2003.04902 [hep-th]

    arXiv 2020

  2. [12]

    On Free Quotients of Complete Intersection Calabi-Yau Manifolds,

    V. Braun, “On Free Quotients of Complete Intersection Calabi-Yau Manifolds,” JHEP 04 (2011) 005, arXiv:1003.3235 [hep-th]

    arXiv 2011

  3. [13]

    Free quotients of favorable Calabi-Yau manifolds,

    J. Gray and J. Wang, “Free quotients of favorable Calabi-Yau manifolds,” JHEP 07 (2022) 116, arXiv:2112.12683 [hep-th]

    arXiv 2022

  4. [1]

    Vacuum configurations for superstrings,

    P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, “Vacuum configurations for superstrings,” Nucl. Phys. B258 (1985) 46–74

    work page 1985

  5. [2]

    Flux compactifications in string theory: A Comprehensive review,

    M. Grana, “Flux compactifications in string theory: A Comprehensive review,” Phys. Rept. 423 (2006) 91–158, arXiv:hep-th/0509003

    arXiv 2006

  6. [3]

    Flux compactification,

    M. R. Douglas and S. Kachru, “Flux compactification,” Rev. Mod. Phys.79 (2007) 733–796, arXiv:hep-th/0610102

    arXiv 2007

  7. [4]

    On Classifying the Divisor Involutions in Calabi-Yau Threefolds,

    X. Gao and P. Shukla, “On Classifying the Divisor Involutions in Calabi-Yau Threefolds,” JHEP 11 (2013) 170, arXiv:1307.1139 [hep-th]

    arXiv 2013

  8. [5]

    Orientifold Calabi-Yau threefolds with divisor involutions and string landscape,

    R. Altman, J. Carifio, X. Gao, and B. D. Nelson, “Orientifold Calabi-Yau threefolds with divisor involutions and string landscape,” JHEP 03 (2022) 087, arXiv:2111.03078 [hep-th]

    arXiv 2022

Show all 53 references
  1. [6]

    Complete classification of reflexive polyhedra in four-dimensions,

    M. Kreuzer and H. Skarke, “Complete classification of reflexive polyhedra in four-dimensions,” Adv.Theor.Math.Phys. 4 (2002) 1209–1230, arXiv:hep-th/0002240 [hep-th]

    2002 arXiv

  2. [7]

    A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List,

    R. Altman, J. Gray, Y.-H. He, V. Jejjala, and B. D. Nelson, “A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List,” JHEP 02 (2015) 158, arXiv:1411.1418 [hep-th]

    2015 arXiv

  3. [8]

    A database of Calabi-Yau orientifolds and the size of D3-tadpoles,

    C. Crin` o, F. Quevedo, A. Schachner, and R. Valandro, “A database of Calabi-Yau orientifolds and the size of D3-tadpoles,” JHEP 08 (2022) 050, arXiv:2204.13115 [hep-th]. 22

    2022 arXiv

  4. [9]

    Orientifold Calabi-Yau threefolds: divisor exchanges and multi-reflections,

    X. Cao, H. Gao, and X. Gao, “Orientifold Calabi-Yau threefolds: divisor exchanges and multi-reflections,” JHEP 10 (2024) 188, arXiv:2407.02565 [hep-th]

    2024 arXiv

  5. [10]

    Discrete Symmetries of Calabi–Yau Hypersurfaces in Toric Four-Folds,

    A. Braun, A. Lukas, and C. Sun, “Discrete Symmetries of Calabi–Yau Hypersurfaces in Toric Four-Folds,” Commun. Math. Phys.360 no. 3, (2018) 935–984, arXiv:1704.07812 [hep-th]

    2018 arXiv

  6. [11]

    Complete Intersection Calabi-Yau Manifolds,

    P. Candelas, A. M. Dale, C. A. Lutken, and R. Schimmrigk, “Complete Intersection Calabi-Yau Manifolds,” Nucl. Phys. B298 (1988) 493

    1988

  7. [14]

    Fibrations in CICY Threefolds,

    L. B. Anderson, X. Gao, J. Gray, and S.-J. Lee, “Fibrations in CICY Threefolds,” JHEP 10 (2017) 077, arXiv:1708.07907 [hep-th]

    2017 arXiv

  8. [15]

    Hodge Numbers for CICYs with Symmetries of Order Divisible by 4,

    P. Candelas, A. Constantin, and C. Mishra, “Hodge Numbers for CICYs with Symmetries of Order Divisible by 4,” Fortsch. Phys.64 no. 6-7, (2016) 463–509, arXiv:1511.01103 [hep-th]

    2016 arXiv

  9. [16]

    Hodge Numbers for All CICY Quotients,

    A. Constantin, J. Gray, and A. Lukas, “Hodge Numbers for All CICY Quotients,” JHEP 01 (2017) 001, arXiv:1607.01830 [hep-th]

    2017 arXiv

  10. [18]

    Applying machine learning to the Calabi-Yau orientifolds with string vacua,

    X. Gao and H. Zou, “Applying machine learning to the Calabi-Yau orientifolds with string vacua,” Phys. Rev. D105 no. 4, (2022) 046017, arXiv:2112.04950 [hep-th]

    2022 arXiv

  11. [19]

    Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners,

    R. Altman, J. Carifio, J. Halverson, and B. D. Nelson, “Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners,” JHEP 03 (2019) 186, arXiv:1811.06490 [hep-th]

    2019 arXiv

  12. [20]

    Bounding the Kreuzer-Skarke Landscape,

    M. Demirtas, L. McAllister, and A. Rios-Tascon, “Bounding the Kreuzer-Skarke Landscape,” Fortsch. Phys.68 (2020) 2000086, arXiv:2008.01730 [hep-th]

    2020 arXiv

  13. [21]

    Polytopes and Machine Learning,

    J. Bao, Y.-H. He, E. Hirst, J. Hofscheier, A. Kasprzyk, and S. Majumder, “Polytopes and Machine Learning,” Math. Sci. 01 (2023) 181–211, arXiv:2109.09602 [math.CO]

    2023 arXiv

  14. [22]

    Machine learning Calabi-Yau hypersurfaces,

    D. S. Berman, Y.-H. He, and E. Hirst, “Machine learning Calabi-Yau hypersurfaces,” Phys. Rev. D105 no. 6, (2022) 066002, arXiv:2112.06350 [hep-th]

    2022 arXiv

  15. [23]

    Generating triangulations and fibrations with reinforcement learning,

    P. Berglund, G. Butbaia, Y.-H. He, E. Heyes, E. Hirst, and V. Jejjala, “Generating triangulations and fibrations with reinforcement learning,” Phys. Lett. B860 (2025) 139158, arXiv:2405.21017 [hep-th]

    2025 arXiv

  16. [24]

    Machine Learning CICY Threefolds,

    K. Bull, Y.-H. He, V. Jejjala, and C. Mishra, “Machine Learning CICY Threefolds,” Phys. Lett. B785 (2018) 65–72, arXiv:1806.03121 [hep-th]. 23

    2018 arXiv

  17. [25]

    He, The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning

    Y.-H. He, The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning. Lecture Notes in Mathematics. 5, 2021. arXiv:1812.02893 [hep-th]

    2021 arXiv

  18. [26]

    Machine Learning Calabi-Yau Four-folds,

    Y.-H. He and A. Lukas, “Machine Learning Calabi-Yau Four-folds,” Phys. Lett. B815 (2021) 136139, arXiv:2009.02544 [hep-th]

    2021 arXiv

  19. [27]

    Machine learning for complete intersection Calabi-Yau manifolds: a methodological study,

    H. Erbin and R. Finotello, “Machine learning for complete intersection Calabi-Yau manifolds: a methodological study,” Phys. Rev. D103 no. 12, (2021) 126014, arXiv:2007.15706 [hep-th]

    2021 arXiv

  20. [28]

    Machine learning complete intersection Calabi-Yau 3-folds,

    K. M. Keita, “Machine learning complete intersection Calabi-Yau 3-folds,” Phys. Rev. D 110 no. 12, (2024) 126002, arXiv:2404.11710 [hep-th]

    2024 arXiv

  21. [29]

    Distinguishing Calabi-Yau Topology using Machine Learning,

    Y.-H. He, Z.-G. Yao, and S.-T. Yau, “Distinguishing Calabi-Yau Topology using Machine Learning,” arXiv:2408.05076 [math.AG]

  22. [30]

    Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning,

    L. B. Anderson, M. Gerdes, J. Gray, S. Krippendorf, N. Raghuram, and F. Ruehle, “Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning,” JHEP 05 (2021) 013, arXiv:2012.04656 [hep-th]

    2021 arXiv

  23. [31]

    Neural network approximations for Calabi-Yau metrics,

    V. Jejjala, D. K. Mayorga Pena, and C. Mishra, “Neural network approximations for Calabi-Yau metrics,” JHEP 08 (2022) 105, arXiv:2012.15821 [hep-th]

    2022 arXiv

  24. [32]

    Numerical Calabi-Yau metrics from holomorphic networks,

    M. R. Douglas, S. Lakshminarasimhan, and Y. Qi, “Numerical Calabi-Yau metrics from holomorphic networks,” arXiv:2012.04797 [hep-th]

    2012 arXiv

  25. [33]

    Learning Size and Shape of Calabi-Yau Spaces,

    M. Larfors, A. Lukas, F. Ruehle, and R. Schneider, “Learning Size and Shape of Calabi-Yau Spaces,” arXiv:2111.01436 [hep-th]

    arXiv

  26. [34]

    Machine-learned Calabi–Yau metrics and curvature,

    P. Berglund, G. Butbaia, T. H¨ uubsch, V. Jejjala, D. Mayorga Pe˜ na, C. Mishra, and J. Tan, “Machine-learned Calabi–Yau metrics and curvature,” Adv. Theor. Math. Phys. 27 no. 4, (2023) 1107–1158, arXiv:2211.09801 [hep-th]

    2023 arXiv

  27. [35]

    Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties,

    D. Klaewer and L. Schlechter, “Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties,” Phys. Lett. B789 (2019) 438–443, arXiv:1809.02547 [hep-th]

    2019 arXiv

  28. [36]

    Machine Learning Line Bundle Cohomology,

    C. R. Brodie, A. Constantin, R. Deen, and A. Lukas, “Machine Learning Line Bundle Cohomology,” Fortsch. Phys.68 no. 1, (2020) 1900087, arXiv:1906.08730 [hep-th]

    2020 arXiv

  29. [37]

    A new construction of Calabi–Yau manifolds: Generalized CICYs,

    L. B. Anderson, F. Apruzzi, X. Gao, J. Gray, and S.-J. Lee, “A new construction of Calabi–Yau manifolds: Generalized CICYs,” Nucl. Phys. B906 (2016) 441–496, arXiv:1507.03235 [hep-th]

    2016 arXiv

  30. [38]

    Machine learning on generalized complete intersection Calabi-Yau manifolds,

    W. Cui, X. Gao, and J. Wang, “Machine learning on generalized complete intersection Calabi-Yau manifolds,” Phys. Rev. D107 no. 8, (2023) 086004, arXiv:2209.10157 [hep-th]

    2023 arXiv

  31. [39]

    Machine learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds,

    D. Aggarwal, Y.-H. He, E. Heyes, E. Hirst, H. N. S. Earp, and T. S. R. Silva, “Machine learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds,” Phys. Lett. B 850 (2024) 138517, arXiv:2310.03064 [math.DG]. 24

    2024 arXiv

  32. [40]

    Constructing and Machine Learning Calabi-Yau Five-Folds,

    R. Alawadhi, D. Angella, A. Leonardo, and T. S. Gherardini, “Constructing and Machine Learning Calabi-Yau Five-Folds,” Fortsch. Phys.72 no. 2, (2024) 2300262, arXiv:2310.15966 [hep-th]

    2024 arXiv

  33. [41]

    Calabi-Yau four-, five-, sixfolds as Pwn hypersurfaces: Machine learning, approximation, and generation,

    E. Hirst and T. S. Gherardini, “Calabi-Yau four-, five-, sixfolds as Pwn hypersurfaces: Machine learning, approximation, and generation,” Phys. Rev. D109 no. 10, (2024) 106006, arXiv:2311.17146 [hep-th]

    2024 arXiv

  34. [42]

    Machine Learning Calabi-Yau Three-Folds, Four-Folds, and Five-Folds,

    K. M. Keita and Y. H. Dicko, “Machine Learning Calabi-Yau Three-Folds, Four-Folds, and Five-Folds,” arXiv:2503.00139 [hep-th]

  35. [43]

    Attention is all you need,

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” arXiv:1706.03762 [cs.CL]

    arXiv

  36. [44]

    Gemini: A family of highly capable multimodal models,

    G. Team, “Gemini: A family of highly capable multimodal models,” 2025. https://arxiv.org/abs/2312.11805

    2025 arXiv

  37. [45]

    Team, “Gemini,” 2025

    G. Team, “Gemini,” 2025. https://gemini.google.com

    2025

  38. [46]

    Claude sonnet 4,

    Anthropic, “Claude sonnet 4,” 2025. https://www.anthropic.com/news/claude-sonnet-4

    2025

  39. [47]

    Deepseek-v3 technical report,

    DeepSeek-AI, “Deepseek-v3 technical report,” 2025. https://arxiv.org/abs/2412.19437

    2025 arXiv

  40. [48]

    Deepseek,

    DeepSeek-AI, “Deepseek,” 2025. https://www.deepseek.com/

    2025

  41. [49]

    Chatgpt: Applications, opportunities, and threats,

    A. Bahrini, M. Khamoshifar, H. Abbasimehr, R. J. Riggs, M. Esmaeili, R. M. Majdabadkohne, and M. Pasehvar, “Chatgpt: Applications, opportunities, and threats,”

  42. [50]

    Chatgpt,

    OpenAI, “Chatgpt,” 2025. https://chat.openai.com/

    2025

  43. [51]

    A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States,

    R. Orus, “A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States,” Annals Phys. 349 (2014) 117–158, arXiv:1306.2164 [cond-mat.str-el]

    2014 arXiv

  44. [52]

    Tensor networks for quantum machine learning,

    H.-M. Rieser, F. K¨ oster, and A. P. Raulf, “Tensor networks for quantum machine learning,” Proc. Roy. Soc. Lond. A479 no. 2275, (2023) 20230218, arXiv:2303.11735 [quant-ph]. 25

    2023 arXiv

  45. [2023]

    https://arxiv.org/abs/2304.09103

    arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.