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REVIEW 5 major objections 5 minor 74 references

Machine Learning the 6d Supergravity Landscape

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Machine learning trained only on anomaly coefficients sorts 26 million 6d supergravity building blocks into likely landscape and likely swampland, and flags one model that resists combination.

desk verdict First ML pass over the 26M 6d Gram-matrix dataset, with honest caveats; the autoencoder results are the solid part, while the classifier precision claims rest on approximate labels and in-sample evaluation. read the letter →

arxiv 2505.16131 v2 pith:FFK3W4S5 submitted 2025-05-22 hep-th cs.LG

Nathan Brady , David Tennyson , Thomas Vandermeulen This is my paper
classification hep-thcs.LG
keywords 6dsupergravitystringlandscapeswamplandautoencoderanomalyinflowGrammatrixmachinelearningprobebranes
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

This paper asks whether machine learning can tell, from no physics input beyond a matrix of anomaly coefficients, which six-dimensional supergravity theories are likely to be part of the string landscape rather than the swampland. Using roughly 26 million building-block models tabulated in earlier work, the authors train an autoencoder on the Gram matrix of each model and show that a two-dimensional compressed representation clusters models by physical similarity. They also train two classifiers that predict whether a model passes the anomaly-inflow consistency conditions for probe strings, flagging about 214,000 models as likely consistent and about 1.9 million as likely inconsistent. If the scheme is right, a statistical map of a landscape too large for exhaustive analysis becomes available, and the clusters point to regions worth studying further.

What carries the argument

The Gram matrix $G$ of anomaly coefficients, the $(n+1)\times(n+1)$ matrix encoding the Green-Schwarz couplings $a$ and $b_i$, is the sole input to every network. Its upper triangle, flattened to a 136-dimensional vector, feeds a feed-forward autoencoder whose bottleneck is a two-dimensional latent layer, so each model becomes a point on a plane, with clustering done by the density-based hdbscan algorithm. For the supervised part, the machinery is a numerical labelling procedure: the paper relaxes the probe-string charge to the continuous space $\Lambda_S \otimes \mathbb{R}$, tests the inflow inequality at eight fixed values of $(q, k_0) = (Q\cdot Q, Q\cdot a)$, and labels models according to whether $\tilde{f}(k) = \sum_{i>0} k_i \dim G_i / (k_i + \check{h}_i)$ can be maximised or minimised below or above the corresponding central charge $c_l$. Classifiers with the same 136-dimensional input are then trained to reproduce these labels.

What would settle it

Run the exact integer-lattice anomaly-inflow test on a random sample of the 214,837 models the first classifier calls consistent: if the fraction that actually passes drops substantially below the claimed precision of roughly 78 percent, the labelling simplification is the culprit. A second check targets the outlier claim: attempt to combine the six-factor peculiar model with all 60 candidate partners at every admissible $n_T$; finding a single anomaly-free combination would undercut the claim that its presence in the landscape is extremely rare.

Watch

Extended reading notes

Core claim

The central claim is that anomaly coefficients, packaged as a Gram matrix, carry enough information for a neural network to recover physically meaningful structure. The autoencoder compresses the 136 entries of each model's Gram matrix into two latent coordinates with only a few percent reconstruction error, and the resulting points form clusters that track the number of gauge-group factors, clique structure, and tensor-multiplet data. The strongest evidence is the classifiers: trained on labels from a numerical check of the probe-brane unitarity inequality $c_l \geq \sum_i k_i \dim G_i / (k_i + \check{h}_i)$, the first classifier identifies 214,837 models predicted to pass the restricted anomaly-inflow condition (precision 0.78), and the second identifies 1,909,359 models predicted to violate it (precision 0.91). Projecting these predictions onto the autoencoder's latent layer shows predicted-consistent models clustering together, which the paper reads as the autoencoder having learned complex physical features from Gram matrices alone. The paper further reports that the hardest-to-reconstruct outlier, a six-factor model, resists combination into any anomaly-free theory: none of the 60 candidate partner models cancels its $\mathrm{tr}R^4$ anomaly, and the simplest successful combination requires 21 copies of an $E_6$ factor at $n_T = 62$.

Load-bearing premise

The entire prediction pipeline rests on the labelling shortcut: the test that generates the training data treats probe charges as continuous rather than discrete, checks only eight charge values near the origin, and drops the positive-tension condition, so any model the shortcut mislabels hands its error to every classifier prediction built on it.

Editorial extensions

If this is right

  • The 214,837 models flagged as consistent form a certified pool of building blocks, because trivially combining two of them again passes the restricted anomaly-inflow condition, so this set can generate a large number of candidate landscape theories.
  • The 1,909,359 models flagged as inconsistent are unlikely to appear in any consistent theory, since the inconsistency label is preserved under trivial combination with any other building block.
  • The latent-space clusters, in particular LC-14 with its high density of predicted-consistent models, identify regions of the landscape where targeted searches are most likely to succeed.
  • The peculiar six-factor model shows that being hard to combine is readable from the Gram matrix alone, making high reconstruction loss a cheap screening tool for models that resist anomaly cancellation.

Reading between the lines

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

  • The same Gram-matrix-only pipeline should transfer to other consistency questions in the 6d landscape, such as predicting which models admit F-theory realisations or satisfy stricter global-anomaly conditions, because the paper shows that physical labels and latent geometry correlate.
  • A cheap stress test of the whole scheme would be to enlarge the set of test points beyond the eight values of $(q,k_0)$ and re-measure the classifiers' precision; any sharp degradation would locate the boundary of validity of the continuous-relaxation labelling.
  • The reconstruction-loss ranking could be used prospectively: instead of enumerating models and then checking combinability, one could train the autoencoder once and use its loss outliers to shortlist candidates for exact analysis, turning anomaly detection into a search heuristic.
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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

5 major / 5 minor

Summary. This paper applies unsupervised and supervised machine learning to the 26,760,256 irreducible admissible 6d N=(1,0) supergravity building blocks tabulated by Hamada and Loges, using only the 136 upper-triangular Gram-matrix entries per model. An autoencoder with a two-dimensional latent space produces a clustered organisation of the data and identifies high-reconstruction-loss outliers; one six-gauge-factor outlier is shown, by direct calculation, to be extremely hard to embed in anomaly-free combinations (N=21 copies of E6 and nT=62 required). Two feed-forward classifiers are then trained on labels generated by a simplified anomaly-inflow check: one classifier targets models satisfying a restricted necessary condition (reported precision 0.78; 214,837 predicted positives in the full dataset), and the other targets models violating that condition in a relaxed real-lattice setting (reported precision 0.91; 1,909,359 predicted negatives). The classifier predictions are projected into the autoencoder latent space, where the predicted consistent models are reported to cluster. The paper releases its data, trained networks, and an interactive cluster website.

Significance. The core idea—using Gram matrices alone to build a statistical map of the 6d landscape/swampland—is genuinely interesting and fits a growing literature on ML for string-mathematics data. The autoencoder analysis is a useful organisational tool, and the peculiar-outlier argument is concrete and falsifiable: a model with reconstruction loss roughly forty times the average is shown by explicit anomaly-free combination counting to be extremely rare in the landscape. The supervised part is more fragile, but if the approximate labels are validated against the exact integer-lattice condition and the precisions are re-measured on held-out data, the counts 214,837 and 1,909,359 would become credible, falsifiable predictions. The public release of data, trained networks, and cluster outputs is a definite strength, as is the authors' explicit discussion of the limitations of their labelling procedure.

major comments (5)
  1. [4.1, Eqs. (4.22) and (4.25)] The labelling procedure replaces the charge lattice Λ_S by Λ_S⊗R (allowing k_i ∈ R), tests only the eight (q,k0) pairs of Eq. (4.6), and drops the positive-tension condition (2.41). The paper itself states that dataset-1 labels are necessary but not sufficient and that dataset-2 labels are neither necessary nor sufficient. Because these labels are the sole source of ground truth for both classifiers, the central quantitative claims in Section 4.3 (214,837 likely consistent and 1,909,359 likely inconsistent models) inherit an unvalidated simplification. I would need to see a comparison with the exact integer-lattice check on a random sample (for example 1,000 models per class), together with a sensitivity test to additional (q,k0) points, before accepting the headline counts.
  2. [4.3, Eqs. (4.36) and (4.40)] The reported precisions 0.78257 and 0.90933 are computed from confusion matrices on the full unbalanced training data after selecting the cut-offs p*, and the paper acknowledges that this is likely an over-estimate. Consequently, the propagated statements 'we might expect around 168,000' and 'around 1,736,000' lack a calibrated performance measure. A held-out test set, with p* fixed before evaluation and confusion matrices reported at the natural class prevalence, is necessary to support the precision claims made in the abstract.
  3. [4.3, Figures 9 and 11] The clustering of 'consistent' models is computed from classifier predictions, not from verified physical labels. Because the classifier and the autoencoder consume the same Gram-matrix input, the apparent clustering may encode the approximate labelling rule rather than the actual anomaly-inflow condition. The conclusion that consistent models cluster together should be re-derived using true labels for a validated subset, or explicitly qualified as clustering of predicted labels.
  4. [4.3, after Eq. (4.37)] The text says that 'trivially combining any two of these models will lead to another which will pass the anomaly inflow criteria.' This is stronger than what Section 4.1 establishes: the combination is guaranteed only when max f1 + max f2 ≤ c_l (displayed near the end of Section 4.1), which is not implied by each model individually being labelled 0. Please weaken the statement or prove the stronger property.
  5. [Appendix A] Models with λ+(G)=0 and λ−(G)<nT are assigned labels by fiat (1 in dataset 1 and 0 in dataset 2), despite Appendix A demonstrating that their inflow consistency is ambiguous. The size of this subclass should be reported, and the classifiers' sensitivity to these labels should be tested by removing them or by treating this subclass as a third class. Since these models are included in the training sets, arbitrary labels propagate into both classifiers.
minor comments (5)
  1. [3.2] The number of clusters is reported inconsistently: the text says re-clustering C-174 gives 100 additional clusters for a total of 275, while the caption of Figure 4b says 234 sub-clusters and Table 2 lists 22 LSCs. Please reconcile these numbers.
  2. [4.3, Eq. (4.33)] The Bayesian estimate treats the balanced validation accuracy 0.967 as the true-negative rate P(PN|TN); accuracy on a balanced set is not a class-conditional probability. Since the final precision is taken from a confusion matrix, this does not change the headline numbers, but the derivation should be corrected.
  3. [2.1] There are typographical errors such as 'Green-Schwaz-Sagnotti' in Section 2.1 and 'obatained' in Section 3.1; the manuscript would benefit from a careful proofread.
  4. [5] The summary refers to 'v.s. Figure 8' when discussing clustering of consistent models; the relevant figure is Figure 9.
  5. [4.1, Eq. (4.18)] The notation G^{-1} is used for the pseudo-inverse before its definition in Eq. (4.10) is connected to the displayed calculation; stating explicitly that G = DηD^T is substituted in the chain of equalities would make the argument easier to follow.

Circularity Check

2 steps flagged · score 4.0 of 10

Headline classifier precisions are computed on the same full training data used to tune the decision thresholds, so the reported 0.78/0.91 precision and the derived landscape/swampland counts are partly self-referential.

  1. fitted input called prediction [Section 4.3, first classifier, Eqs. (4.36)-(4.37) and accompanying footnote]
    "We find that, with value of p∗ = 2.95 × 10−6, applying the classifier to the full training dataset, we get the confusion matrix TP= 531, 531; TN= 3, 369; FP= 10, 280; FN= 936 which gives a 0-precision and 0-recall of 0-precision = 0.78257, 0-recall = 0.24683. ... This measure will likely be an over-estimation of the precision since the full unbalanced training dataset includes the data that the model was trained on."

    The decision threshold p∗ is chosen by inspecting the same full unbalanced training dataset on which the confusion matrix and precision are then reported. No held-out set is used for this headline statistic, so the 0.78257 precision is a fitted in-sample value rather than an out-of-sample prediction. The paper then uses this precision to convert the classifier output into the quantitative claim that about 168,000 of the 214,837 predicted models pass the anomaly inflow test, so the headline landscape count inherits the self-referential performance estimate.

  2. fitted input called prediction [Section 4.3, second classifier, Eqs. (4.40)-(4.41)]
    "We find that by raising the value of p∗ to 0.989, we get the confusion matrix TP= 8, 315; TN= 87, 572; FP= 829; FN= 14, 741 which gives a precision and recall of 1-precision = 0.90933, 1-recall = 0.36064. Applying this to the full dataset, we get a prediction of 1,909,359 models labeled 1 with around 1,736,000 models having a high likelihood of being inconsistent under anomaly-inflow."

    As with the first classifier, p∗ is tuned using the full training dataset and the confusion matrix is evaluated on that same dataset. The reported 0.90933 precision is therefore an in-sample fit, not an unbiased estimate of performance on unseen data. This precision is then used to convert the raw count of 1,909,359 into the swampland claim that about 1,736,000 models are likely inconsistent, making the headline estimate partly self-referential.

full rationale

The paper's main unsupervised result is not circular: the autoencoder is trained only on Gram matrices, and its identification of a peculiar model is subsequently checked against the external anomaly-combination criteria, with the full computation shown in the text. The supervised classifiers are also not definitionally circular in their training: the labels are generated from a stated numerical approximation to the anomaly-inflow conditions, and the paper explicitly warns that the simplified labels are necessary-but-not-sufficient for the first classifier and neither necessary nor sufficient for the second. That approximation is a correctness and validity risk, not a circularity. The genuine circularity is in the reported performance statistics: both headline precisions (0.78257 and 0.90933) are computed by applying the classifiers to the full unbalanced training data after choosing the cutoff p∗ on that same data, and the paper itself acknowledges the measure is an over-estimation. These in-sample precisions are then used to calibrate the predicted counts of 214,837 consistent and 1,909,359 inconsistent models, so the quantitative headline inherits a self-referential fit. There is no load-bearing self-citation chain or uniqueness argument imported from the authors' prior work; the cited dataset [18] and anomaly-inflow condition [11] are external sources. Overall, the central ML-feature-extraction claim retains independent content, but the classifier precision and resulting landscape/swampland counts are partially circular, giving a score of 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or mediators are introduced. The main free parameters are the ML thresholds, the hand-picked charge values, and the autoencoder architecture. The physical assumptions are standard for the 6d supergravity swampland program, with the continuous relaxation being the most consequential ad hoc simplification.

free parameters (4)
  • classifier-0 cutoff p* = 2.95e-6
    Chosen post hoc on the training data to raise 0-precision to 0.78; recall drops to 0.25. This is a hand-fit threshold, not a predicted quantity.
  • classifier-1 cutoff p* = 0.989
    Chosen post hoc on the training data to raise 1-precision to 0.91; recall is 0.36. This threshold is fit to achieve the reported precision.
  • set of (q,k0) points = (-1,1),(0,0),(0,2),(1,1),(1,3),(2,0),(2,2),(2,4)
    Hand-selected small central charges in eq (4.6); not derived from any principle. The labelling and all classifier predictions depend on this choice.
  • autoencoder latent dimension and architecture = 2 latent dims; encoder 512-256-64-16-2, decoder symmetric
    Chosen by comparing MSLE on a 500k sample across latent widths 2,4,6,8,10; width 2 was selected for plottability despite higher loss.
assumptions (5)
  • domain assumption The probe-brane unitarity condition from Kim-Shiu-Vafa, eq (2.44), is the correct consistency criterion.
    Used to label all data; if wrong, both classifiers are mislabeled. Introduced in Section 2.1 and applied throughout Section 4.
  • domain assumption The charge lattice and Gram matrix decomposition conditions from [18] are correct, and the 26,760,256 models are the relevant population of building blocks.
    Section 2.2; the finite dataset was obtained by imposing bounds on Delta(G_i) in [18], so completeness is bounded.
  • domain assumption A model labelled 0 by the continuous-relaxation test with k_i in R is consistent under probe-brane insertion for the eight chosen (q,k0) values.
    Section 4.1, eq (4.22); the relaxation from the integral lattice to the real vector space is the load-bearing simplification.
  • standard math Standard linear algebra for the pseudo-inverse and projector: q = k^T G^{-1} k, and P = D(D^T D)^{-1} D^T.
    Section 4.1, eqs (4.12)-(4.20); used to relate Q to the ki variables.
  • domain assumption Neural networks trained on balanced samples generalize to the full, imbalanced population.
    Section 4.2; the classifiers are evaluated on balanced validation sets, but applied to the full dataset with very different label frequencies.

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Cite this review

Pith. "Pith review of Machine Learning the 6d Supergravity Landscape." pith.science (2026). https://pith.science/paper/FFK3W4S5

@misc{pith2026250516131,
  author       = {Pith},
  title        = {Pith review of: Machine Learning the 6d Supergravity Landscape},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFK3W4S5}},
  note         = {Machine review of arXiv:2505.16131}
}
abstract

In this paper, we apply both supervised and unsupervised machine learning algorithms to the study of the string landscape and swampland in 6-dimensions. Our data are the (almost) anomaly-free 6-dimensional $\mathcal{N} = (1,0)$ supergravity models, characterised by the Gram matrix of anomaly coefficients. Our work demonstrates the ability of machine learning algorithms to efficiently learn highly complex features of the landscape and swampland. Employing an autoencoder for unsupervised learning, we provide an auto-classification of these models by compressing the Gram matrix data to 2-dimensions. Through compression, similar models cluster together, and we identify prominent features of these clusters. The autoencoder also identifies outlier models which are difficult to reconstruct. One of these outliers proves to be incredibly difficult to combine with other models such that the $\text{tr}R^{4}$ anomaly vanishes, making its presence in the landscape extremely rare. Further, we utilise supervised learning to build two classifiers predicting (1) model consistency under probe string insertion (precision: 0.78, predicting consistency for 214,837 models with reasonable certainty) and (2) inconsistency under anomaly inflow (precision: 0.91, predicting inconsistency for 1,909,359 models). Notably, projecting these predictions onto the autoencoder's 2-dimensional latent layer shows consistent models clustering together, further indicating that the autoencoder has learnt interesting and complex features of the set of models and potentially offers a novel approach to mapping the landscape and swampland of 6-dimensional supergravity theories.

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Works this paper leans on

74 extracted references · 24 canonical work pages

  1. [1]

    Vafa, The String landscape and the swampland , hep-th/0509212

    C. Vafa, The String landscape and the swampland , hep-th/0509212

    arXiv

  2. [2]

    van Beest, J

    M. van Beest, J. Calder´ on-Infante, D. Mirfendereski and I. Valenzuela, Lectures on the Swampland Program in String Compactifications , Phys. Rept. 989 (2022) 1 [ 2102.01111]

    arXiv 2022

  3. [3]

    Hamada and C

    Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the Swampland , JHEP 06 (2021) 178 [ 2104.05724]

    arXiv 2021

  4. [4]

    Montero and C

    M. Montero and C. Vafa, Cobordism Conjecture, Anomalies, and the String Lamppost Principle , JHEP 01 (2021) 063 [ 2008.11729]

    arXiv 2021

  5. [5]

    Bedroya, Y

    A. Bedroya, Y. Hamada, M. Montero and C. Vafa, Compactness of brane moduli and the String Lamppost Principle in d > 6, JHEP 02 (2022) 082 [ 2110.10157]

    arXiv 2022

  6. [6]

    Kumar, D

    V. Kumar, D. R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities , JHEP 11 (2010) 118 [ 1008.1062]

    arXiv 2010

  7. [7]

    Monnier and G

    S. Monnier and G. W. Moore, A Brief Summary Of Global Anomaly Cancellation In Six-Dimensional Supergravity, 1808.01335

    arXiv

  8. [8]

    Monnier and G

    S. Monnier and G. W. Moore, Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories, Commun. Math. Phys. 372 (2019) 963 [ 1808.01334]

    arXiv 2019

Show all 74 references
  1. [9]

    Davighi and N

    J. Davighi and N. Lohitsiri, Omega vs. pi, and 6d anomaly cancellation , JHEP 05 (2021) 267 [2012.11693]. – 44 –

    2021 arXiv

  2. [10]

    Monnier, G

    S. Monnier, G. W. Moore and D. S. Park, Quantization of anomaly coefficients in 6D N = (1, 0) supergravity, JHEP 02 (2018) 020 [ 1711.04777]

    2018 arXiv

  3. [11]

    H.-C. Kim, G. Shiu and C. Vafa, Branes and the Swampland , Phys. Rev. D 100 (2019) 066006 [1905.08261]

    2019 arXiv

  4. [12]

    Lee and T

    S.-J. Lee and T. Weigand, Swampland Bounds on the Abelian Gauge Sector , Phys. Rev. D 100 (2019) 026015 [ 1905.13213]

    2019 arXiv

  5. [13]

    Angelantonj, Q

    C. Angelantonj, Q. Bonnefoy, C. Condeescu and E. Dudas, String Defects, Supersymmetry and the Swampland , JHEP 11 (2020) 125 [ 2007.12722]

    2020 arXiv

  6. [14]

    Cheng, R

    P. Cheng, R. Minasian and S. Theisen, Anomalies as obstructions: from dimensional lifts to swampland, JHEP 01 (2022) 068 [ 2106.14912]

    2022 arXiv

  7. [15]

    Tarazi and C

    H.-C. Tarazi and C. Vafa, On The Finiteness of 6d Supergravity Landscape , 2106.10839

    arXiv

  8. [16]

    G. J. Loges, New infinite class of 6d, N = (1, 0) supergravities, Phys. Rev. D 109 (2024) 126006

    2024

  9. [17]

    S. D. Avramis and A. Kehagias, A Systematic search for anomaly-free supergravities in six dimensions, JHEP 10 (2005) 052 [ hep-th/0508172]

    2005 arXiv

  10. [18]

    Hamada and G

    Y. Hamada and G. J. Loges, Towards a complete classification of 6D supergravities , JHEP 02 (2024) 095 [ 2311.00868]

    2024 arXiv

  11. [19]

    Becker, A

    K. Becker, A. Kehagias, E. Sezgin, D. Tennyson and A. Violaris, New anomaly free supergravities in six dimensions , 2311.03337

    arXiv

  12. [20]

    LeCun, Y

    Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature 521 (2015) 436

    2015

  13. [21]

    Ruehle, Data science applications to string theory , Phys

    F. Ruehle, Data science applications to string theory , Phys. Rept. 839 (2020) 1

    2020

  14. [22]

    G. V. Cybenko, Approximation by superpositions of a sigmoidal function , Mathematics of Control, Signals and Systems 2 (1989) 303

    1989

  15. [23]

    M. R. Douglas, S. Lakshminarasimhan and Y. Qi, Numerical Calabi-Yau metrics from holomorphic networks , 2012.04797

    2012 arXiv

  16. [24]

    Halverson and F

    J. Halverson and F. Ruehle, Metric Flows with Neural Networks , 2310.19870

    arXiv

  17. [25]

    Jejjala, D

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

    2022 arXiv

  18. [26]

    Ashmore, Y.-H

    A. Ashmore, Y.-H. He and B. A. Ovrut, Machine Learning Calabi–Yau Metrics , Fortsch. Phys. 68 (2020) 2000068 [ 1910.08605]

    2020 arXiv

  19. [27]

    Ashmore, L

    A. Ashmore, L. Calmon, Y.-H. He and B. A. Ovrut, Calabi-Yau Metrics, Energy Functionals and Machine-Learning, International Journal of Data Science in the Mathematical Sciences 1 (2023) 49 [2112.10872]

    2023 arXiv

  20. [28]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Learning Size and Shape of Calabi-Yau Spaces , 2111.01436

    arXiv

  21. [29]

    A. Apte, A. Ashmore, C. Cordova and T.-C. Huang, Deep learning lattice gauge theories , 2405.14830

    arXiv

  22. [30]

    Ashmore, R

    A. Ashmore, R. Deen, Y.-H. He and B. A. Ovrut, Machine learning line bundle connections , Phys. Lett. B 827 (2022) 136972 [ 2110.12483]. – 45 –

    2022 arXiv

  23. [31]

    Cipriani, A

    A. Cipriani, A. De Santis, G. Di Russo, A. Grillo and L. Tabarroni, Hamiltonian Neural Networks approach to fuzzball geodesics , 2502.20881

    arXiv

  24. [32]

    He, Deep-Learning the Landscape, 1706.02714

    Y.-H. He, Deep-Learning the Landscape, 1706.02714

    arXiv

  25. [33]

    Bull, Y.-H

    K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Getting CICY High , Phys. Lett. B 795 (2019) 700 [1903.03113]

    2019 arXiv

  26. [34]

    Bull, Y.-H

    K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Machine Learning CICY Threefolds , Phys. Lett. B 785 (2018) 65 [ 1806.03121]

    2018 arXiv

  27. [35]

    He and S.-J

    Y.-H. He and S.-J. Lee, Distinguishing elliptic fibrations with AI , Phys. Lett. B 798 (2019) 134889 [1904.08530]

    2019 arXiv

  28. [36]

    Krefl and R.-K

    D. Krefl and R.-K. Seong, Machine Learning of Calabi-Yau Volumes , Phys. Rev. D 96 (2017) 066014 [1706.03346]

    2017 arXiv

  29. [37]

    Wang and Z

    Y.-N. Wang and Z. Zhang, Learning non-Higgsable gauge groups in 4D F-theory , JHEP 08 (2018) 009 [1804.07296]

    2018 arXiv

  30. [38]

    Carifio, J

    J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson, Machine Learning in the String Landscape, JHEP 09 (2017) 157 [ 1707.00655]

    2017 arXiv

  31. [39]

    Deen, Y.-H

    R. Deen, Y.-H. He, S.-J. Lee and A. Lukas, Machine learning string standard models , Phys. Rev. D 105 (2022) 046001 [ 2003.13339]

    2022 arXiv

  32. [40]

    G. E. Hinton and R. R. Salakhutdinov, Reducing the dimensionality of data with neural networks , Science 313 (2006) 504 [ https://www.science.org/doi/pdf/10.1126/science.1127647]

    2006 doi

  33. [41]

    Klaewer and L

    D. Klaewer and L. Schlechter, Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties, Phys. Lett. B 789 (2019) 438 [ 1809.02547]

    2019 arXiv

  34. [42]

    M¨ utter, E

    A. M¨ utter, E. Parr and P. K. S. Vaudrevange,Deep learning in the heterotic orbifold landscape , Nucl. Phys. B 940 (2019) 113 [ 1811.05993]

    2019 arXiv

  35. [43]

    K´ antor, V

    G. K´ antor, V. Niarchos and C. Papageorgakis,Solving Conformal Field Theories with Artificial Intelligence, Phys. Rev. Lett. 128 (2022) 041601 [ 2108.08859]

    2022 arXiv

  36. [44]

    K´ antor, V

    G. K´ antor, V. Niarchos and C. Papageorgakis,Conformal bootstrap with reinforcement learning, Phys. Rev. D 105 (2022) 025018 [ 2108.09330]

    2022 arXiv

  37. [45]

    K´ antor, V

    G. K´ antor, V. Niarchos, C. Papageorgakis and P. Richmond, 6D (2,0) bootstrap with the soft-actor-critic algorithm , Phys. Rev. D 107 (2023) 025005 [ 2209.02801]

    2023 arXiv

  38. [46]

    Halverson, B

    J. Halverson, B. Nelson and F. Ruehle, Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning, JHEP 06 (2019) 003 [ 1903.11616]

    2019 arXiv

  39. [47]

    Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape , JHEP 08 (2017) 038 [ 1706.07024]

    F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape , JHEP 08 (2017) 038 [ 1706.07024]

    2017 arXiv

  40. [48]

    Damian, L

    C. Damian, L. R. Diaz-Barron, O. Loaiza-Brito and M. Sabido, Slow-Roll Inflation in Non-geometric Flux Compactification, JHEP 06 (2013) 109 [ 1302.0529]

    2013 arXiv

  41. [49]

    Damian and O

    C. Damian and O. Loaiza-Brito, More stable de Sitter vacua from S-dual nongeometric fluxes , Phys. Rev. D 88 (2013) 046008 [ 1304.0792]

    2013 arXiv

  42. [50]

    Cabo Bizet, C

    N. Cabo Bizet, C. Damian, O. Loaiza-Brito, D. K. M. Pe˜ na and J. A. Monta˜ nez Barrera,Testing Swampland Conjectures with Machine Learning , Eur. Phys. J. C 80 (2020) 766 [ 2006.07290]. – 46 –

    2020 arXiv

  43. [51]

    A. Cole, A. Schachner and G. Shiu, Searching the Landscape of Flux Vacua with Genetic Algorithms, JHEP 11 (2019) 045 [ 1907.10072]

    2019 arXiv

  44. [52]

    Abel and J

    S. Abel and J. Rizos, Genetic Algorithms and the Search for Viable String Vacua , JHEP 08 (2014) 010 [ 1404.7359]

    2014 arXiv

  45. [53]

    I. Bena, J. Bl ˚ ab¨ ack, M. Gra˜ na and S. L¨ ust,Algorithmically Solving the Tadpole Problem , Adv. Appl. Clifford Algebras 32 (2022) 7 [ 2103.03250]

    2022 arXiv

  46. [54]

    AbdusSalam, S

    S. AbdusSalam, S. Abel, M. Cicoli, F. Quevedo and P. Shukla, A systematic approach to K¨ ahler moduli stabilisation, JHEP 08 (2020) 047 [ 2005.11329]

    2020 arXiv

  47. [55]

    Bershadsky and C

    M. Bershadsky and C. Vafa, Global anomalies and geometric engineering of critical theories in six-dimensions, hep-th/9703167

    arXiv

  48. [56]

    Machine learning the 6d sugra landscape

    N. Brady, D. Tennyson and T. Vandermeulen, “Machine learning the 6d sugra landscape.” GitHub Page, 2025. https://nait400.github.io/ML-6d-sugra-landscape/

    2025

  49. [57]

    Machine learning the 6d sugra landscape

    N. Brady, D. Tennyson and T. Vandermeulen, “Machine learning the 6d sugra landscape.” GitHub Repository, 2025. https://github.com/nait400/ML-6d-sugra-landscape

    2025

  50. [58]

    6d-sugra-classify

    G. Loges, “6d-sugra-classify.” https://github.com/gloges/6d-sugra-classify

  51. [59]

    6d-sugra-data

    G. Loges, “6d-sugra-data.” https://github.com/gloges/6d-sugra-data

  52. [60]

    Becker, E

    K. Becker, E. Sezgin and D. Tennyson, Global and inflow anomalies in 6d gauged supergravity , in preparation

  53. [61]

    Sagnotti, A Note on the Green-Schwarz mechanism in open string theories , Phys

    A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories , Phys. Lett. B 294 (1992) 196 [ hep-th/9210127]

    1992 arXiv

  54. [62]

    Witten, An SU(2) Anomaly , Phys

    E. Witten, An SU(2) Anomaly , Phys. Lett. B 117 (1982) 324

    1982

  55. [63]

    Suzuki and Y

    R. Suzuki and Y. Tachikawa, More anomaly-free models of six-dimensional gauged supergravity , J. Math. Phys. 47 (2006) 062302 [ hep-th/0512019]

    2006 arXiv

  56. [64]

    Deser, A

    S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, P-brane dyons and electric magnetic duality, Nucl. Phys. B 520 (1998) 179 [ hep-th/9712189]

    1998 arXiv

  57. [65]

    Seiberg and W

    N. Seiberg and W. Taylor, Charge Lattices and Consistency of 6D Supergravity , JHEP 06 (2011) 001 [1103.0019]

    2011 arXiv

  58. [66]

    V. V. Nikulin, Integral symmetric bilinear forms and some of their applications , Mathematics of the USSR-Izvestiya 14 (1980) 103

    1980

  59. [67]

    Polchinski, Monopoles, duality, and string theory , Int

    J. Polchinski, Monopoles, duality, and string theory , Int. J. Mod. Phys. A 19S1 (2004) 145 [hep-th/0304042]

    2004 arXiv

  60. [68]

    Haghighat, S

    B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, F-Theory, Spinning Black Holes and Multi-string Branches, JHEP 01 (2016) 009 [ 1509.00455]

    2016 arXiv

  61. [69]

    Hamada and G

    Y. Hamada and G. J. Loges, Enumerating 6D supergravities with T ≤ 1, 2404.08845

    arXiv

  62. [70]

    Escalante-Notario, I

    E. Escalante-Notario, I. Portillo-Castillo and S. Ramos-Sanchez, An autoencoder for heterotic orbifolds with arbitrary geometry , 2024

    2024

  63. [71]

    Otsuka and K

    H. Otsuka and K. Takemoto, Deep learning and k-means clustering in heterotic string vacua with line bundles , Journal of High Energy Physics 2020 (2020) . – 47 –

    2020

  64. [72]

    Ishiguro, S

    K. Ishiguro, S. Nishimura and H. Otsuka, Autoencoder-driven clustering of intersecting d-brane models via tadpole charge , 2023

    2023

  65. [73]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel et al., Scikit-learn: Machine learning in Python , Journal of Machine Learning Research 12 (2011) 2825

    2011

  66. [74]

    Cs294a lecture notes

    A. Ng, “Cs294a lecture notes.” https://web.stanford.edu/class/cs294a/sparseAutoencoder.pdf. – 48 – Local # of Elements k-Clique Types Group Sub-type Lie-types Cluster # : {A, B, C, D, E, F, G } (A,B,C,D,E6,7,8,F4,G2) LC-01 14,241,455 Given in Table 2 LC-02 2,437,669 5, 6*, 7, ...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.