Pith. sign in

REVIEW 2 major objections 5 minor 177 references

Pre-Strings Lectures on Artificial Intelligence

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A field theory can be defined by a neural-network architecture and a density on its parameters, recovering free fields, Liouville, string amplitudes, BKT, and D=26.

desk verdict Solid lecture notes that package the 2025–26 NN-FT papers into a usable three-day course; the value is synthesis and pedagogy, not a new theorem. read the letter →

arxiv 2607.02905 v1 pith:X5WCTNN4 submitted 2026-07-03 hep-th

James Halverson This is my paper · ORCID
classification hep-th
keywords neuralnetworkfieldtheoryNNGPLiouvillebosonicstringcriticaldimensionBKTtransitionWardidentitiesCalabi-Yaumetrics
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

These lecture notes argue that the data of a neural network—its architecture together with a probability density on its parameters—already define a field theory, because correlators can be computed by integrating over parameters rather than over field configurations. Infinite-width networks become free fields by the central limit theorem; interactions appear when that theorem is broken by finite width or by deforming the density. With this definition the notes reconstruct Liouville structure constants, the Virasoro–Shapiro and Veneziano amplitudes, the BKT transition via discrete topological latents, Ward identities and anomalies on parameter space, and a mode-counting derivation of the bosonic-string critical dimension D=26. They also survey how the same tools reverse: physics-informed networks for Calabi–Yau metrics, reinforcement learning over string vacua and knots, and interpretable models aimed at conjecture generation. A sympathetic reader cares because the construction supplies both a new language for field theory and concrete computational routes to quantities that usually require path integrals or lattice methods.

What carries the argument

Neural Network Field Theory (NN-FT): the data (ϕ_θ, P(θ)) whose parameter-space partition function Z[J] = ∫ dθ P(θ) exp(∫ J ϕ_θ) defines all correlators. Free limits follow from the CLT; interactions from finite N or density deformations; symmetries and anomalies from flows on parameter space that leave or break that integral.

What would settle it

Construct an explicit finite-width NN-FT whose four-point and higher correlators can be computed exactly, check whether the Osterwalder–Schrader reflection-positivity inequalities hold after analytic continuation, and compare the resulting Lorentzian spectrum or S-matrix against a known continuum QFT or string amplitude.

Watch

Extended reading notes

Core claim

The central claim is that a neural-network field theory is the pair (architecture, parameter density), with the partition function an integral over parameters. From this definition one engineers free theories by the CLT and spectrum shaping, interactions by 1/N corrections or independence-breaking deformations, and quantum theories by clearing Osterwalder–Schrader conditions. Concrete ensembles then recover Liouville DOZZ constants, string amplitudes, BKT phenomenology, Ward identities, and a Jacobian mode-count that yields D=26.

Load-bearing premise

That the Euclidean correlators of these engineered ensembles continue to unitary Lorentzian quantum field theories, which requires reflection positivity and cluster decomposition that are verified only in free and quantum-mechanics cases so far.

Editorial extensions

If this is right

  • Any Euclidean QFT in the constructive sense admits an NN-FT description with countably many parameters (universality).
  • Liouville three-point structure constants can be obtained by Monte Carlo sampling of a conditioned zero-mode density plus free spherical-harmonic modes.
  • Virasoro–Shapiro and Veneziano amplitudes arise as exact finite-dimensional integrals over network parameters once free bosons and ghosts are realized as infinite-width ensembles.
  • Topological sectors enter as discrete latent variables, allowing vortex unbinding and the BKT jump to be read off from network correlators.
  • The critical dimension D=26 follows from a regularized mismatch between bosonic and Grassmann Jacobians under a Weyl flow on parameter space.

Reading between the lines

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

  • If reflection positivity can be engineered systematically, NN-FT becomes a constructive route to interacting QFTs that bypasses the need to write an action first.
  • The same parameter-space Ward identity machinery that produced D=26 should apply to other anomaly coefficients (central charges, chiral anomalies) once the relevant flows are identified.
  • Agentic verification loops wrapped around NN-FT correlator codes could turn the Monte-Carlo Liouville and BKT checks into automated, continuously refined numerical proofs.
  • Architecture-robust free-boson constructions suggest that finite-N non-Gaussian corrections to string amplitudes are themselves architecture-dependent observables worth classifying.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. These lecture notes present a three-day course on AI for string theorists. Day 1 develops neural-network essentials (expressivity, statistics via NNGP and 1/N non-Gaussianities, dynamics via NTK and µP) with a field-theoretic vocabulary. Day 2 defines Neural Network Field Theory (NN-FT) as the data of an architecture ϕ_θ together with a density P(θ), with correlators computed from the parameter-space partition function; free theories arise by CLT plus spectrum shaping, interactions by CLT violation, and recent constructions recover Liouville DOZZ constants, Virasoro-Shapiro/Veneziano amplitudes, BKT phenomenology with vortices, Ward identities/anomalies, and a mode-counting derivation of D=26. Day 3 surveys applied ML for strings (agents, PINNs for Calabi-Yau metrics, RL on vacua and unknotting, conjecture generation). The notes synthesize and pedagogically reorganize results largely already posted as research papers.

Significance. If the NN-FT definition and constructions hold as stated, the notes supply a coherent, field-theoretic language for ensembles of networks and a concrete engineering toolkit (CLT free theories, independence-breaking interactions, absorption for symmetries, discrete latents for topology) that recovers standard results of 2d CFT, string theory, and topological phase transitions. Strengths include analytic recovery of string amplitudes from Gaussian parameter integrals, percent-level numerical DOZZ agreement with reported error bars, an explicit zeta-regularized Jacobian count yielding D=26, and open acknowledgment that OS axioms beyond free/QM cases remain open. As lecture notes they are valuable for training and for consolidating a rapidly growing literature; they do not claim a single new theorem beyond the cited papers.

major comments (2)
  1. §3.1.3 (and the OS checklist): the claim that NN ensembles define quantum field theories rests on Osterwalder–Schrader reconstruction. The notes correctly state that reflection positivity and cluster decomposition are verified only for free cases and quantum mechanics, and remain open for the interacting constructions (Liouville, strings, BKT). This is load-bearing for any Lorentzian/unitary reading of Day 2. The Euclidean constructions and amplitude recoveries stand independently, but the manuscript should either (i) restrict the “QFT” language more carefully to Euclidean correlators throughout §§3.2.2–3.2.5, or (ii) supply at least a sketch of how RP/cluster are expected to hold (or fail) for the deformed densities used in Liouville and ϕ^4.
  2. §3.2.3, Eqs. (156)–(165) and the D=26 count in §3.2.5, Eqs. (189)–(194): the free-boson and ghost architectures are engineered so that the two-point functions match the standard worldsheet propagators (spectrum shaping + Grassmann Gaussians). Once that is fixed, Virasoro-Shapiro/Veneziano and the critical-dimension count follow by standard Gaussian integrals and zeta regularization of mode Jacobians. This is design, not circular derivation of the target numbers, but the notes should state more explicitly that the physical content is the choice of architecture/density that realizes the free theory, after which the amplitudes and anomaly are recovered rather than independently predicted. A short paragraph distinguishing “engineering the free theory” from “deriving the amplitude” would prevent misreading.
minor comments (5)
  1. Introduction and §5: the notes are dated July 2026 and cite contemporaneous results (IMO gold, Erdős counterexamples, agentic systems). For archival publication, a brief note on the snapshot nature of the “current AI landscape” paragraphs would help future readers.
  2. Fig. 3 (Liouville DOZZ): the caption reports L=30, 10×50k runs, and error bars smaller than markers. Adding a short statement of the Monte-Carlo estimator variance or a reference to the companion paper’s numerical appendix would strengthen reproducibility claims.
  3. §2.4.3 (µP scaling): the one-parameter family and the unique solution with η∼O_N(1) are dense. A small table of (a_ℓ,b_ℓ,c,d) for the standard NTK vs µP regimes would aid readers who skip the index algebra.
  4. §4.1 (agents): the GPD pipeline and “order-of-magnitude drop in implementation barrier” are useful but anecdotal. A single concrete before/after example (e.g., time to reproduce a Day-2 baseline) would make the claim more falsifiable.
  5. Typos and polish: the disclaimer already notes residual typos from rapid posting; a pass for consistency of notation (e.g., θ vs θ(t), G^{(n)} vs G_c^{(4)}) and for missing cross-references between Day-1 NTK and Day-3 metric flows would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: free theories are engineered by design (CLT + spectrum shaping), after which amplitudes, DOZZ match, BKT exponents, and the D=26 mode count are independent analytic or numerical checks against external benchmarks.

full rationale

The manuscript is lecture notes whose core definition (NN-FT = architecture ϕ_θ plus density P(θ), Def. 3.1) is stipulative, not a derivation of a prior claim. Free theories are obtained by the CLT plus explicit spectrum-shaping of G^{(2)} (Scalar-net, free-boson random features, ghosts); this is engineering, openly stated as a recipe, not a prediction of the free propagator. Once the free ensemble is fixed, the Liouville Monte-Carlo match to DOZZ, the analytic recovery of Virasoro-Shapiro/Veneziano from parameter-space Gaussian integrals, the BKT power-law/essential-singularity/stiffness-jump phenomenology, and the zeta-regularized Jacobian count that yields D=26 are all independent of the free-theory input and are checked against external formulas or known results. The universality theorem invokes the Borel isomorphism theorem (external mathematics) for existence of a (possibly non-constructive) architecture. Self-citations to the author's recent papers supply details of the constructions but are not used as unverified uniqueness theorems that force the target numbers; the sketches given in the notes are self-contained enough to exhibit the reductions. The only openly acknowledged gap (OS reflection positivity beyond free/QM cases) is orthogonal to circularity. Score 1 only for the ordinary presence of author-overlapping citations that are not load-bearing for the claimed recoveries.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

NN-FT rests on the definition that a field theory is an architecture plus a parameter density, the CLT for free limits, the Osterwalder–Schrader reconstruction for quantumness, and standard QFT/ML background (Ward identities via change of variables, zeta regularization, calibrated geometry for CY volumes). Free parameters appear as architectural scales (width N, weight variances, cutoffs) that are fixed to match known free propagators or physical constants such as α′. No new particles or forces are postulated; the main invented entity is the NN-FT definition itself.

free parameters (4)
  • output-weight variance σ_a (or equivalent)
    Sets the string tension α′ via α′ = σ_a^{2}/(Λ^{2}−ϵ^{2}) in the free-boson architecture; chosen to match the desired physical value.
  • width N and UV/IR cutoffs Λ, ϵ
    Control the continuum and infinite-width limits; regulators that must be sent to infinity after correlators are computed.
  • Liouville zero-mode conditioning parameters (b, μ, α_i)
    Physical Liouville parameters; sampled or fixed to compare against DOZZ, not fitted to produce the match.
  • vortex fugacity y and stiffness K_0 (or b)
    Control the BKT temperature; scanned to locate the transition rather than fitted post hoc to force η=1/4.
assumptions (5)
  • standard math Central Limit Theorem in function space: sum of N i.i.d. neurons becomes a Gaussian process as N→∞
    Used throughout Day 1–2 to obtain free NN-FTs (NNGP correspondence).
  • domain assumption Osterwalder–Schrader reconstruction: Euclidean correlators obeying the OS axioms define a unitary Lorentzian QFT
    Invoked in §3.1.3 to claim that certain NN ensembles are quantum; reflection positivity remains open beyond QM.
  • standard math Borel isomorphism theorem: any two standard Borel spaces are isomorphic, so every probability measure on S′(R^d) is the push-forward of a parameter measure
    Underpins the universality theorem (Theorem 3.3).
  • domain assumption Yau’s theorem: a compact Kähler manifold with c1=0 admits a unique Ricci-flat metric in each Kähler class
    Background for the PINN/Calabi–Yau section; existence is assumed, the network approximates the metric.
  • ad hoc to paper Absorption mechanism: a symmetry of the ensemble exists when a field transformation can be absorbed into a reparameterization that leaves dθ P(θ) invariant
    Core technical device of Day 1–2 for engineering global symmetries and later Ward identities; introduced in the author’s prior work and used as definitional here.
invented entities (1)
  • Neural Network Field Theory (NN-FT) independent evidence
    purpose: Defines a field theory by the pair (architecture ϕ_θ, density P(θ)) with partition function an integral over parameters rather than fields.
    The central conceptual object of Day 2; independent evidence is the recovery of known correlators and amplitudes, but the definition itself is a modeling choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Pre-Strings Lectures on Artificial Intelligence." pith.science (2026). https://pith.science/paper/X5WCTNN4

@misc{pith2026260702905,
  author       = {Pith},
  title        = {Pith review of: Pre-Strings Lectures on Artificial Intelligence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5WCTNN4}},
  note         = {Machine review of arXiv:2607.02905}
}
read the original abstract

These notes are based on six lectures given over three days at the Pre-Strings 2026 school in Shanghai. Day 1 develops neural network essentials, organized around the expressivity, statistics, and dynamics of neural networks, presented with a field-theoretic lens. Day 2 develops a neural network approach to field theory (NN-FT), in which a field theory is defined by a network architecture and a density on its parameters, and surveys recent results. Examples include a universality theorem, a neural network realization of Liouville theory, famous string amplitudes, topological sectors and the Kosterlitz-Thouless transition, Ward identities and anomalies, and a new derivation of the critical dimension of the bosonic string. Day 3 turns the lens around and covers applied AI for string theory: agentic workflows that are changing how the other techniques are implemented, physics-informed neural networks and Calabi-Yau metrics, reinforcement learning and search in the string landscape and in knot theory, and interpretable supervised learning with an eye towards conjecture generation.

Figures

Figures reproduced from arXiv: 2607.02905 by the authors.

Figure 1
Figure 1. The mechanism behind the Universal Approximation The [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. MLPs versus KANs: functional forms, the mathematical [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The Liouville structure constant C(α1, α2, α3) from NN correlators (points; α1 = α2 fixed and α3 varied, for several b) versus the exact DOZZ formula (solid curves), which lies within all error bars. NN-FT run: L = 30 spherical harmonics, 10 experiments of 50,000 runs, sphere pixelated at 100 × 200 points; the error bars (variance across experiments) are smaller than the markers [67]. the free-boson logarithm for 1/… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Vortex density ρv versus the coupling b (the dashed line marks bc = 1 2 ): consistent with zero below the transition, rising steeply above it as vortex–antivortex pairs unbind. From [69]. the integral and its measure are invariant, and read off a conservation law when …
Figure 5
Figure 5. Figure 5: The agent loop. A reasoning model repeatedly thinks, acts [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]
Figure 6
Figure 6. Figure 6: Hierarchy of neural-network metric flows [152]. The gen [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 7
Figure 7. Figure 7: The Artin generator σ1 ∈ B2: two strands with one cross￾ing. Closing the braid (joining each strand’s top to its own bottom) gives a single loop, the unknot. where destabilization, free cancellation, and the braid relation are the braid-word avatars of Reidemeister mov…
Figure 8
Figure 8. Figure 8: The braid relation σiσi+1σi = σi+1σiσi+1 (here i=1 on three strands), the Reidemeister-III move in braid-word form, and the move the trained agent relies on most. The remaining move types act on the word the same algebraic way: far commutation σiσj = σjσi , free cancel…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

177 extracted references · 13 canonical work pages

  1. [1]

    Halverson,TASI lectures on physics for machine learning,

    J. Halverson,TASI lectures on physics for machine learning,

  2. [2]

    https://arxiv.org/abs/2408.00082

    arXiv

  3. [3]

    Isaacson,The Innovators: How a Group of Hackers, Geniuses, and Geeks Created the Digital Revolution

    W. Isaacson,The Innovators: How a Group of Hackers, Geniuses, and Geeks Created the Digital Revolution. Simon & Schuster, New York, first simon & schuster hardcover edition ed., 2014

    2014

  4. [4]

    Jaech, A

    OpenAI, :, A. Jaech, A. Kalai, A. Lerer, A. Richardson et al., OpenAI o1 system card, 2026. https://arxiv.org/abs/2412.16720

    arXiv 2026

  5. [5]

    DeepSeek-AI, D. Guo, D. Yang, H. Zhang, J. Song, P. Wang et al.,DeepSeek-R1: Incentivizing reasoning capability in LLMs via reinforcement learning, 2026. https://doi.org/10.1038/s41586-025-09422-z, https://arxiv.org/abs/2501.12948

    arXiv doi:10.1038/s41586-025-09422-z 2026

  6. [6]

    https://deepmind.google/discover/blog/advanced-version-of- gemini-with-deep-think-officially-achieves-gold-medal- standard-at-the-international-mathematical-olympiad/

    Google DeepMind,Advanced version of Gemini with Deep Think officially achieves gold-medal standard at the international mathematical olympiad, July, 2025. https://deepmind.google/discover/blog/advanced-version-of- gemini-with-deep-think-officially-achieves-gold-medal- standard-at-the-international-mathematical-olympiad/

    2025

  7. [7]

    OpenAI model earns gold-medal score at international math olympiad and advances path to artificial general intelligence

    D. E. B´ echard, “OpenAI model earns gold-medal score at international math olympiad and advances path to artificial general intelligence.” Scientific American, 2025. https://www.scientificamerican.com/article/openai-model- earns-gold-medal-score-at-international-math-olympiad-and/

    2025

  8. [8]

    The Erd˝ os problems project

    T. F. Bloom, “The Erd˝ os problems project.” https://www.erdosproblems.com

Show all 177 references
  1. [9]

    https://openai.com/index/model- disproves-discrete-geometry-conjecture/

    OpenAI,An OpenAI model has disproved a central conjecture in discrete geometry, 2026. https://openai.com/index/model- disproves-discrete-geometry-conjecture/. Pre-Strings Lectures on Artificial Intelligence 43

    2026

  2. [10]

    N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar et al.,Remarks on the disproof of the unit distance conjecture,2605.20695

    arXiv

  3. [11]

    S. Yao, J. Zhao, D. Yu, N. Du, I. Shafran, K. Narasimhan et al.,React: Synergizing reasoning and acting in language models, 2023. https://arxiv.org/abs/2210.03629

    2023 arXiv

  4. [12]

    https://claude.com/claude-code

    Anthropic,Claude Code, 2025. https://claude.com/claude-code

    2025

  5. [13]

    Romera-Paredes, M

    B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M. P. Kumar, E. Dupont et al.,Mathematical discoveries from program search with large language models,Nature625 (2024) 468

    2024

  6. [14]

    Novikov, N

    A. Novikov, N. V˜ u, M. Eisenberger, E. Dupont, P.-S. Huang, A. Z. Wagner et al.,Alphaevolve: A coding agent for scientific and algorithmic discovery, 2025. https://arxiv.org/abs/2506.13131. [14]Pre-strings school 2026, 2026. https://www.pre-strings2026.com/

    2025 arXiv

  7. [15]

    A Stochastic Approximation Method

    H. Robbins and S. Monro, “A Stochastic Approximation Method.” Annals of Mathematical Statistics, 1951. 10.1214/aoms/1177729586, https://doi.org/10.1214/aoms/1177729586

    1951 doi

  8. [16]

    Rosenblatt,The perceptron: a probabilistic model for information storage and organization in the brain., Psychological review65 6(1958) 386

    F. Rosenblatt,The perceptron: a probabilistic model for information storage and organization in the brain., Psychological review65 6(1958) 386

    1958

  9. [17]

    D. P. Kingma and J. Ba,Adam: A method for stochastic optimization, 2017. https://arxiv.org/abs/1412.6980

    2017 arXiv

  10. [18]

    G. B. D. Luca and E. Silverstein,Born-infeld (BI) for AI: Energy-conserving descent (ECD) for optimization, 2022. https://arxiv.org/abs/2201.11137

    2022 arXiv

  11. [19]

    G. B. D. Luca, A. Gatti and E. Silverstein,Improving energy conserving descent for machine learning: Theory and practice,

  12. [20]

    https://arxiv.org/abs/2306.00352

    arXiv

  13. [21]

    Approximation by Superpositions of a Sigmoidal Function

    G. Cybenko, “Approximation by Superpositions of a Sigmoidal Function.” Mathematics of Control, Signals and Systems, 1989. 10.1007/BF02551274, https://doi.org/10.1007/bf02551274

    1989 doi

  14. [22]

    Hornik, M

    K. Hornik, M. Stinchcombe and H. White,Approximation capabilities of multilayer feedforward networks,Neural Networks4(1991) 251

    1991

  15. [23]

    A. N. Kolmogorov,On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition,Doklady Akademii Nauk SSSR114(1957) 953

    1957

  16. [24]

    V. I. Arnold,On functions of three variables,Doklady Akademii Nauk SSSR114(1957) 679

    1957

  17. [25]

    Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljaˇ ci´ c et al.,Kan: Kolmogorov-arnold networks, 2025. https://arxiv.org/abs/2404.19756

    2025 arXiv

  18. [26]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez et al.,Attention is all you need, 2023. https://arxiv.org/abs/1706.03762

    2023 arXiv

  19. [27]

    T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal et al.,Language models are few-shot learners,

  20. [28]

    https://arxiv.org/abs/2005.14165

    2005 arXiv

  21. [29]

    C. Yun, S. Bhojanapalli, A. S. Rawat, S. J. Reddi and S. Kumar,Are transformers universal approximators of sequence-to-sequence functions?, 2020. https://arxiv.org/abs/1912.10077. Pre-Strings Lectures on Artificial Intelligence 44

    2020 arXiv

  22. [30]

    J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia et al.,Chain-of-thought prompting elicits reasoning in large language models, 2023. https://arxiv.org/abs/2201.11903

    2023 arXiv

  23. [31]

    Merrill and A

    W. Merrill and A. Sabharwal,The expressive power of transformers with chain of thought, 2024. https://arxiv.org/abs/2310.07923

    2024 arXiv

  24. [32]

    G. Feng, B. Zhang, Y. Gu, H. Ye, D. He and L. Wang, Towards revealing the mystery behind chain of thought: A theoretical perspective, 2023. https://arxiv.org/abs/2305.15408

    2023 arXiv

  25. [33]

    R. M. Neal,Bayesian learning for neural networks,Lecture Notes in Statistics118(1996)

    1996

  26. [34]

    C. K. Williams,Computing with infinite networks,Advances in neural information processing systems(1996)

    1996

  27. [35]

    Krizhevsky, I

    A. Krizhevsky, I. Sutskever and G. E. Hinton,Imagenet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems (F. Pereira, C. Burges, L. Bottou and K. Weinberger, eds.), vol. 25, Curran Associates, Inc., 2012, https://procee...

    2012

  28. [36]

    Yang,Tensor programs i: Wide feedforward or recurrent neural networks of any architecture are gaussian processes,

    G. Yang,Tensor programs i: Wide feedforward or recurrent neural networks of any architecture are gaussian processes,

  29. [37]

    https://arxiv.org/abs/1910.12478

    1910 arXiv

  30. [38]

    Demirtas, J

    M. Demirtas, J. Halverson, A. Maiti, M. D. Schwartz and K. Stoner,Neural network field theories: Non-gaussianity, actions, and locality, 2023. https://arxiv.org/abs/2307.03223

    2023 arXiv

  31. [39]

    T. S. Cohen and M. Welling,Group equivariant convolutional networks, 2016. https://arxiv.org/abs/1602.07576

    2016 arXiv

  32. [40]

    Winkels and T

    M. Winkels and T. S. Cohen,3D G-CNNs for pulmonary nodule detection, 2018. https://arxiv.org/abs/1804.04656

    2018 arXiv

  33. [41]

    Kaplan, S

    J. Kaplan, S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child et al.,Scaling laws for neural language models, 2020. https://arxiv.org/abs/2001.08361

    2020 arXiv

  34. [42]

    Batzner, A

    S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth et al.,E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, 2021. https://doi.org/10.1038/s41467-022-29939-5, https://arxiv.org/abs/2101.03164

    2021 doi arXiv

  35. [43]

    N. Frey, R. Soklaski, S. Axelrod, S. Samsi, R. Gomez-Bombarelli, C. Coley et al.,Neural scaling of deep chemical models,ChemRxiv(2022)

    2022

  36. [44]

    Boyda, G

    D. Boyda, G. Kanwar, S. Racani` ere, D. J. Rezende, M. S. Albergo, K. Cranmer et al.,Sampling usingsu(n)gauge equivariant flows, 2020. https://doi.org/10.1103/PhysRevD.103.074504, https://arxiv.org/abs/2008.05456

    2020 doi arXiv

  37. [45]

    Maiti, K

    A. Maiti, K. Stoner and J. Halverson,Symmetry-via-duality: Invariant neural network densities from parameter-space correlators, 2021. https://arxiv.org/abs/2106.00694

    2021 arXiv

  38. [46]

    Halverson, A

    J. Halverson, A. Maiti and K. Stoner,Neural networks and quantum field theory, 2021. https://doi.org/10.1088/2632-2153/abeca3, https://arxiv.org/abs/2008.08601

    2021 doi arXiv

  39. [47]

    Halverson,Building quantum field theories out of neurons,

    J. Halverson,Building quantum field theories out of neurons,

  40. [48]

    https://arxiv.org/abs/2112.04527

    arXiv

  41. [49]

    Jacot, F

    A. Jacot, F. Gabriel and C. Hongler,Neural tangent kernel: Convergence and generalization in neural networks, 2020. https://arxiv.org/abs/1806.07572. Pre-Strings Lectures on Artificial Intelligence 45

    2020 arXiv

  42. [50]

    J. Lee, L. Xiao, S. S. Schoenholz, Y. Bahri, R. Novak, J. Sohl-Dickstein et al.,Wide neural networks of any depth evolve as linear models under gradient descent, 2019. https://doi.org/10.1088/1742-5468/abc62b, https://arxiv.org/abs/1902.06720

    2019 doi arXiv

  43. [51]

    Pehlevan and B

    C. Pehlevan and B. Bordelon,Lecture notes on infinite-width limits of neural networks, August, 2024. https://mlschool.princeton.edu/events/2023/pehlevan

    2024

  44. [52]

    Yang and E

    G. Yang and E. J. Hu,Feature learning in infinite-width neural networks, 2022. https://arxiv.org/abs/2011.14522

    2022 arXiv

  45. [53]

    Bordelon and C

    B. Bordelon and C. Pehlevan,Self-consistent dynamical field theory of kernel evolution in wide neural networks, 2022. https://arxiv.org/abs/2205.09653

    2022 arXiv

  46. [54]

    D. A. Roberts, S. Yaida and B. Hanin,The principles of deep learning theory, 2021. https://doi.org/10.1017/9781009023405, https://arxiv.org/abs/2106.10165

    2021 doi arXiv

  47. [55]

    Yaida,Meta-principled family of hyperparameter scaling strategies, 2022

    S. Yaida,Meta-principled family of hyperparameter scaling strategies, 2022. https://arxiv.org/abs/2210.04909

    2022 arXiv

  48. [56]

    S. S. Schoenholz, J. Gilmer, S. Ganguli and J. Sohl-Dickstein, Deep information propagation, 2017. https://arxiv.org/abs/1611.01232

    2017 arXiv

  49. [57]

    Frank, J

    S. Frank, J. Halverson, A. Maiti and F. Ruehle,Fermions and supersymmetry in neural network field theories, 2025. https://arxiv.org/abs/2511.16741

    2025

  50. [58]

    Osterwalder and R

    K. Osterwalder and R. Schrader,Axioms for Euclidean Green’s Functions,Commun. Math. Phys.31(1973) 83

    1973

  51. [59]

    Simmons-Duffin,TASI lectures on the conformal bootstrap,

    D. Simmons-Duffin,TASI lectures on the conformal bootstrap,

  52. [60]

    https://arxiv.org/abs/1602.07982

    arXiv

  53. [61]

    Ferko and J

    C. Ferko and J. Halverson,Quantum mechanics and neural networks, 2025. https://arxiv.org/abs/2504.05462

    2025 arXiv

  54. [62]

    Halverson, J

    J. Halverson, J. Naskar and J. Tian,Conformal fields from neural networks, 2025. https://doi.org/10.1007/JHEP10(2025)039, https://arxiv.org/abs/2409.12222

    2025 doi

  55. [63]

    Robinson,Virasoro symmetry in neural network field theories, 2026

    B. Robinson,Virasoro symmetry in neural network field theories, 2026. https://arxiv.org/abs/2512.24420

    2026

  56. [64]

    Capuozzo, B

    P. Capuozzo, B. Robinson and B. Suzzoni,Conformal defects in neural network field theories, 2026. https://doi.org/10.1007/JHEP05(2026)124, https://arxiv.org/abs/2512.07946

    2026 doi arXiv

  57. [65]

    Ferko, S

    C. Ferko, S. Frank, J. Halverson and V. Jejjala,Anomalies in neural network field theory, 2026. https://arxiv.org/abs/2605.12488

    2026 arXiv

  58. [66]

    David, A

    F. David, A. Kupiainen, R. Rhodes and V. Vargas,Liouville quantum gravity on the riemann sphere, 2015. https://arxiv.org/abs/1410.7318

    2015 arXiv

  59. [67]

    Probabilistic paths to Quantum Field Theory

    Simons Collaboration on Probabilistic Paths to Quantum Field Theory, “Probabilistic paths to Quantum Field Theory.”https://probabilistic-qft.org/, 2026

    2026

  60. [68]

    Jiang, B

    M. Jiang, B. P. Czajka, E. B. James, A. J. Hardaway, C. D. Achammer, Q. Su et al.,Neural network approach to dirac quantum field theory,Phys. Rev. A112(2025) 062223

    2025

  61. [69]

    J. N. Howard, M. S. Klinger, A. Maiti and A. G. Stapleton, Bayesian rg flow in neural network field theories, 2025. https://doi.org/10.21468/SciPostPhysCore.8.1.027, https://arxiv.org/abs/2405.17538

    2025 doi arXiv

  62. [70]

    D. S. Ageev and Y. A. Ageeva,Neural network quantum field theory from transformer architectures, 2026. https://arxiv.org/abs/2602.10209. Pre-Strings Lectures on Artificial Intelligence 46

    2026

  63. [71]

    Zhang,Optimal architecture and fundamental bounds in neural network field theory, 2026

    Z. Zhang,Optimal architecture and fundamental bounds in neural network field theory, 2026. https://arxiv.org/abs/2604.27050

    2026 arXiv

  64. [72]

    Ferko, J

    C. Ferko, J. Halverson and A. Mutchler,Universality of neural network field theory, 2026. https://arxiv.org/abs/2601.14453

    2026

  65. [73]

    Frank and J

    S. Frank and J. Halverson,String theory from infinite width neural networks, 2026. https://arxiv.org/abs/2601.06249

    2026

  66. [74]

    Ferko, J

    C. Ferko, J. Halverson, V. Jejjala and B. Robinson, Topological effects in neural network field theory, 2026. https://arxiv.org/abs/2604.02313

    2026 arXiv

  67. [75]

    Dorn and H

    H. Dorn and H. J. Otto,Two and three-point functions in liouville theory, 1994. https://doi.org/10.1016/0550-3213(94)00352-1, https://arxiv.org/abs/hep-th/9403141

    1994 doi arXiv

  68. [76]

    A. B. Zamolodchikov and A. B. Zamolodchikov,Structure constants and conformal bootstrap in liouville field theory,

  69. [77]

    https://doi.org/10.1016/0550-3213(96)00351-3, https://arxiv.org/abs/hep-th/9506136

    doi arXiv

  70. [78]

    Chatterjee and E

    S. Chatterjee and E. Witten,Liouville theory: An introduction to rigorous approaches, 2024. https://arxiv.org/abs/2404.02001

    2024 arXiv

  71. [79]

    Song and S

    Y. Song and S. Ermon,Generative modeling by estimating gradients of the data distribution, 2020. https://arxiv.org/abs/1907.05600

    2020 arXiv

  72. [80]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon and B. Poole,Score-based generative modeling through stochastic differential equations, 2021. https://arxiv.org/abs/2011.13456

    2021 arXiv

  73. [81]

    Sohl-Dickstein, E

    J. Sohl-Dickstein, E. A. Weiss, N. Maheswaranathan and S. Ganguli,Deep unsupervised learning using nonequilibrium thermodynamics, 2015. https://arxiv.org/abs/1503.03585

    2015 arXiv

  74. [82]

    E. T. Newman and R. Penrose,Note on the Bondi-Metzner-Sachs group,J. Math. Phys.7(1966) 863

    1966

  75. [83]

    J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich and E. C. G. Sudarshan,Spin-sspherical harmonics andð,J. Math. Phys.8(1967) 2155

    1967

  76. [84]

    He,Deep-learning the landscape, 2018

    Y.-H. He,Deep-learning the landscape, 2018. https://arxiv.org/abs/1706.02714

    2018 arXiv

  77. [85]

    Krefl and R.-K

    D. Krefl and R.-K. Seong,Machine learning of calabi-yau volumes, 2017. https://doi.org/10.1103/PhysRevD.96.066014, https://arxiv.org/abs/1706.03346

    2017 doi arXiv

  78. [86]

    Ruehle,Evolving neural networks with genetic algorithms to study the string landscape, 2017

    F. Ruehle,Evolving neural networks with genetic algorithms to study the string landscape, 2017. https://doi.org/10.1007/JHEP08(2017)038, https://arxiv.org/abs/1706.07024

    2017 doi arXiv

  79. [87]

    Carifio, J

    J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson, Machine learning in the string landscape, 2017. https://doi.org/10.1007/JHEP09(2017)157, https://arxiv.org/abs/1707.00655

    2017 doi arXiv

  80. [88]

    Ruehle,Data science applications to string theory,Physics Reports839(2020) 1

    F. Ruehle,Data science applications to string theory,Physics Reports839(2020) 1

    2020

  81. [89]

    He,The calabi-yau landscape: from geometry, to physics, to machine-learning, 2020

    Y.-H. He,The calabi-yau landscape: from geometry, to physics, to machine-learning, 2020. https://arxiv.org/abs/1812.02893

    2020 arXiv

  82. [90]

    Gukov, J

    S. Gukov, J. Halverson and F. Ruehle,Rigor with machine learning from field theory to the poincar´ e conjecture, 2024. https://arxiv.org/abs/2402.13321

    2024 arXiv

  83. [91]

    Bull, Y.-H

    K. Bull, Y.-H. He, V. Jejjala and C. Mishra,Machine learning cicy threefolds, 2018. https://doi.org/10.1016/j.physletb.2018.08.008, https://arxiv.org/abs/1806.03121. Pre-Strings Lectures on Artificial Intelligence 47

    2018 doi arXiv

  84. [92]

    Erbin and R

    H. Erbin and R. Finotello,Machine learning for complete intersection calabi-yau manifolds: a methodological study,

  85. [93]

    https://doi.org/10.1103/PhysRevD.103.126014, https://arxiv.org/abs/2007.15706

    2007 doi arXiv

  86. [94]

    Klaewer and L

    D. Klaewer and L. Schlechter,Machine learning line bundle cohomologies of hypersurfaces in toric varieties, 2018. https://doi.org/10.1016/j.physletb.2019.01.002, https://arxiv.org/abs/1809.02547

    2018 doi arXiv

  87. [95]

    C. R. Brodie, A. Constantin, R. Deen and A. Lukas,Machine learning line bundle cohomology, 2019. https://doi.org/10.1002/prop.201900087, https://arxiv.org/abs/1906.08730

    2019 doi arXiv

  88. [96]

    Deen, Y.-H

    R. Deen, Y.-H. He, S.-J. Lee and A. Lukas,Machine learning string standard models, 2020. https://arxiv.org/abs/2003.13339

    2020 arXiv

  89. [97]

    Hashimoto, S

    K. Hashimoto, S. Sugishita, A. Tanaka and A. Tomiya,Deep learning and ads/cft, 2018. https://doi.org/10.1103/PhysRevD.98.046019, https://arxiv.org/abs/1802.08313

    2018 doi arXiv

  90. [98]

    Choi and R.-K

    E. Choi and R.-K. Seong,Machine learning regularization for the minimum volume formula of toric calabi-yau 3-folds,

  91. [99]

    https://doi.org/10.1103/PhysRevD.109.046015, https://arxiv.org/abs/2310.19276

    doi arXiv

  92. [100]

    Gukov and R.-K

    S. Gukov and R.-K. Seong,Learning bps spectra and the gap conjecture, 2024. https://doi.org/10.1103/PhysRevD.110.046016, https://arxiv.org/abs/2405.09993

    2024 doi arXiv

  93. [101]

    Halverson, B

    J. Halverson, B. Nelson and F. Ruehle,Branes with brains: Exploring string vacua with deep reinforcement learning,

  94. [102]

    https://doi.org/10.1007/JHEP06(2019)003, https://arxiv.org/abs/1903.11616

    2019 doi arXiv

  95. [103]

    A. Cole, A. Schachner and G. Shiu,Searching the landscape of flux vacua with genetic algorithms, 2019. https://doi.org/10.1007/JHEP11(2019)045, https://arxiv.org/abs/1907.10072

    2019 doi arXiv

  96. [104]

    Krippendorf, R

    S. Krippendorf, R. Kroepsch and M. Syvaeri,Revealing systematics in phenomenologically viable flux vacua with reinforcement learning, 2021. https://arxiv.org/abs/2107.04039

    2021 arXiv

  97. [105]

    A. Cole, S. Krippendorf, A. Schachner and G. Shiu,Probing the structure of string theory vacua with genetic algorithms and reinforcement learning, 2021. https://arxiv.org/abs/2111.11466

    2021 arXiv

  98. [106]

    Abel and J

    S. Abel and J. Rizos,Genetic algorithms and the search for viable string vacua, 2014. https://doi.org/10.1007/JHEP08(2014)010, https://arxiv.org/abs/1404.7359

    2014 doi arXiv

  99. [107]

    Constantin, T

    A. Constantin, T. R. Harvey and A. Lukas,Heterotic string model building with monad bundles and reinforcement learning, 2021. https://doi.org/10.1002/prop.202100186, https://arxiv.org/abs/2108.07316

    2021 doi arXiv

  100. [108]

    S. Abel, A. Constantin, T. R. Harvey and A. Lukas,Evolving heterotic gauge backgrounds: Genetic algorithms versus reinforcement learning, 2021. https://doi.org/10.1002/prop.202200034, https://arxiv.org/abs/2110.14029

    2021 doi arXiv

  101. [109]

    S. Abel, A. Constantin, T. R. Harvey and A. Lukas,String model building, reinforcement learning and genetic algorithms,

  102. [110]

    https://arxiv.org/abs/2111.07333

    arXiv

  103. [111]

    Berglund, Y.-H

    P. Berglund, Y.-H. He, E. Heyes, E. Hirst, V. Jejjala and A. Lukas,New calabi-yau manifolds from genetic algorithms,

  104. [112]

    Pre-Strings Lectures on Artificial Intelligence 48

    https://doi.org/10.1016/j.physletb.2024.138504, https://arxiv.org/abs/2306.06159. Pre-Strings Lectures on Artificial Intelligence 48

    2024 doi

  105. [113]

    T. He, J. Lee and H. Ooguri,Exploring the holographic entropy cone via reinforcement learning, 2026. https://doi.org/10.1007/JHEP06(2026)267, https://arxiv.org/abs/2601.19979

    2026 doi arXiv

  106. [114]

    T. R. Harvey and A. Lukas,Particle physics model building with reinforcement learning, 2021. https://arxiv.org/abs/2103.04759

    2021 arXiv

  107. [115]

    M. C. N. Cheng, V. Anagiannis, M. Weiler, P. de Haan, T. S. Cohen and M. Welling,Covariance in physics and convolutional neural networks, 2019. https://arxiv.org/abs/1906.02481

    2019 arXiv

  108. [116]

    Krippendorf and M

    S. Krippendorf and M. Syvaeri,Detecting symmetries with neural networks, 2020. https://arxiv.org/abs/2003.13679

    2020 arXiv

  109. [117]

    Halverson and C

    J. Halverson and C. Long,Statistical predictions in string theory and deep generative models, 2020. https://doi.org/10.1002/prop.202000005, https://arxiv.org/abs/2001.00555

    2020 doi arXiv

  110. [118]

    Ashmore, Y.-H

    A. Ashmore, Y.-H. He and B. Ovrut,Machine learning calabi-yau metrics, 2020. https://doi.org/10.1002/prop.202000068, https://arxiv.org/abs/1910.08605

    2020 doi arXiv

  111. [119]

    M. R. Douglas, S. Lakshminarasimhan and Y. Qi,Numerical calabi-yau metrics from holomorphic networks, 2021. https://arxiv.org/abs/2012.04797

    2021 arXiv

  112. [120]

    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, 2021. https://arxiv.org/abs/2012.04656

    2021 arXiv

  113. [121]

    Jejjala, D

    V. Jejjala, D. K. M. Pena and C. Mishra,Neural network approximations for calabi-yau metrics, 2021. https://arxiv.org/abs/2012.15821

    2021 arXiv

  114. [122]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider,Learning size and shape of calabi-yau spaces, 2021. https://arxiv.org/abs/2111.01436

    2021 arXiv

  115. [123]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider,Numerical metrics for complete intersection and kreuzer-skarke calabi-yau manifolds, 2022. https://arxiv.org/abs/2205.13408

    2022 arXiv

  116. [124]

    Gerdes and S

    M. Gerdes and S. Krippendorf,Cyjax: A package for calabi-yau metrics with jax, 2023. https://doi.org/10.1088/2632-2153/acdc84, https://arxiv.org/abs/2211.12520

    2023 doi arXiv

  117. [125]

    Jejjala, A

    V. Jejjala, A. Kar and O. Parrikar,Deep learning the hyperbolic volume of a knot, 2019. https://doi.org/10.1016/j.physletb.2019.135033, https://arxiv.org/abs/1902.05547

    2019 doi

  118. [126]

    Craven, M

    J. Craven, M. Hughes, V. Jejjala and A. Kar,Learning knot invariants across dimensions, 2022. https://doi.org/10.21468/SciPostPhys.14.2.021, https://arxiv.org/abs/2112.00016

    2022 doi arXiv

  119. [127]

    Gukov, J

    S. Gukov, J. Halverson, F. Ruehle and P. Su lkowski,Learning to unknot, 2020. https://arxiv.org/abs/2010.16263

    2020 arXiv

  120. [128]

    Davies, P

    A. Davies, P. Veliˇ ckovi´ c, L. Buesing, S. Blackwell, D. Zheng, N. Tomaˇ sev et al.,Advancing mathematics by guiding human intuition with AI,Nature600(2021) 70

    2021

  121. [129]

    Gukov, J

    S. Gukov, J. Halverson, C. Manolescu and F. Ruehle, Searching for ribbons with machine learning, 2025. https://arxiv.org/abs/2304.09304

    2025 arXiv

  122. [130]

    D. A. Boiko, R. MacKnight, B. Kline and G. Gomes, Autonomous chemical research with large language models, Nature624(2023) 570. Pre-Strings Lectures on Artificial Intelligence 49

    2023

  123. [131]

    C. Lu, C. Lu, R. T. Lange, J. Foerster, J. Clune and D. Ha, The AI scientist: Towards fully automated open-ended scientific discovery, 2024. https://arxiv.org/abs/2408.06292

    2024 arXiv

  124. [132]

    Ghafarollahi and M

    A. Ghafarollahi and M. J. Buehler,Sciagents: Automating scientific discovery through multi-agent intelligent graph reasoning, 2024. https://arxiv.org/abs/2409.05556

    2024 arXiv

  125. [133]

    https://github.com/psi-oss/get-physics-done

    Physical Superintelligence PBC,Get Physics Done (GPD): the first open-source agentic AI physicist, 2026. https://github.com/psi-oss/get-physics-done

    2026

  126. [134]

    Amodei, C

    D. Amodei, C. Olah, J. Steinhardt, P. Christiano, J. Schulman and D. Man´ e,Concrete problems in AI safety,

  127. [135]

    https://arxiv.org/abs/1606.06565

    arXiv

  128. [136]

    Skalse, N

    J. Skalse, N. H. R. Howe, D. Krasheninnikov and D. Krueger, Defining and characterizing reward hacking, 2025. https://arxiv.org/abs/2209.13085

    2025 arXiv

  129. [137]

    Raissi, P

    M. Raissi, P. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,Journal of Computational Physics378(2018)

    2018

  130. [138]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang and L. Yang,Physics-informed machine learning, Nature Reviews Physics3(2021)

    2021

  131. [139]

    Platt,Non-uniqueness and symmetries for the nirenberg problem using computer assistance, 2026

    D. Platt,Non-uniqueness and symmetries for the nirenberg problem using computer assistance, 2026. https://arxiv.org/abs/2603.29544

    2026

  132. [140]

    Y. Wang, M. Bennani, J. Martens, S. Racani` ere, S. Blackwell, A. Matthews et al.,Discovery of unstable singularities, 2025. https://arxiv.org/abs/2509.14185

    2025

  133. [141]

    Carleo and M

    G. Carleo and M. Troyer,Solving the quantum many-body problem with artificial neural networks, 2016. https://doi.org/10.1126/science.aag2302, https://arxiv.org/abs/1606.02318

    2016 doi arXiv

  134. [142]

    D. Pfau, J. S. Spencer, A. G. de G. Matthews and W. M. C. Foulkes,Ab-initio solution of the many-electron schr¨ odinger equation with deep neural networks, 2021. https://doi.org/10.1103/PhysRevResearch.2.033429, https://arxiv.org/abs/1909.02487

    2021 doi arXiv

  135. [143]

    Hermann, Z

    J. Hermann, Z. Sch¨ atzle and F. No´ e,Deep neural network solution of the electronic schr¨ odinger equation, 2020. https://doi.org/10.1038/s41557-020-0544-y, https://arxiv.org/abs/1909.08423

    2020 doi arXiv

  136. [144]

    Hibat-Allah, M

    M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko and J. Carrasquilla,Recurrent neural network wave functions,

  137. [145]

    https://doi.org/10.1103/PhysRevResearch.2.023358, https://arxiv.org/abs/2002.02973

    2002 doi arXiv

  138. [146]

    S.-T. Yau,On the ricci curvature of a compact k¨ ahler manifold and the complex monge-amp´ ere equation, i, Communications on Pure and Applied Mathematics31(1978) 339 [https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160310304]

    1978 doi

  139. [147]

    Kreuzer and H

    M. Kreuzer and H. Skarke,Complete classification of reflexive polyhedra in four dimensions, 2000. https://arxiv.org/abs/hep-th/0002240

    2000 arXiv

  140. [148]

    Demirtas, L

    M. Demirtas, L. McAllister and A. Rios-Tascon,Bounding the kreuzer-skarke landscape, 2020. https://arxiv.org/abs/2008.01730

    2020 arXiv

  141. [149]

    Chandra, A

    A. Chandra, A. Constantin, K. Fraser-Taliente, T. R. Harvey and A. Lukas,Enumerating calabi-yau manifolds: Placing bounds on the number of diffeomorphism classes in the kreuzer-skarke list, 2023. https://arxiv.org/abs/2310.05909. Pre-Strings Lectures on Artificial Intelligence 50

    2023 arXiv

  142. [150]

    Halverson, C

    J. Halverson, C. Long and B. Sung,On algorithmic universality in f-theory compactifications, 2017. https://doi.org/10.1103/PhysRevD.96.126006, https://arxiv.org/abs/1706.02299

    2017 doi arXiv

  143. [151]

    Taylor and Y.-N

    W. Taylor and Y.-N. Wang,Scanning the skeleton of the 4d f-theory landscape, 2017. https://doi.org/10.1007/JHEP01(2018)111, https://arxiv.org/abs/1710.11235

    2017 doi arXiv

  144. [152]

    Demirtas, C

    M. Demirtas, C. Long, L. McAllister and M. Stillman,The kreuzer-skarke axiverse, 2018. https://arxiv.org/abs/1808.01282

    2018 arXiv

  145. [153]

    S. V. P. Fallon, J. Halverson, L. McAllister and Y. Zhu, F-theory axiverse, 2025. https://arxiv.org/abs/2511.20458

    2025

  146. [154]

    Harvey and H

    R. Harvey and H. B. Lawson,Calibrated geometries,Acta Mathematica148(1982) 47

    1982

  147. [155]

    Halverson and D

    J. Halverson and D. R. Morrison,On gauge enhancement and singular limits ing 2 compactifications of m-theory, 2016. https://doi.org/10.1007/JHEP04(2016)100, https://arxiv.org/abs/1507.05965

    2016 doi arXiv

  148. [156]

    S. K. Donaldson,Some numerical results in complex differential geometry, 2005. https://arxiv.org/abs/math/0512625

    2005 arXiv

  149. [157]

    M. R. Douglas, R. L. Karp, S. Lukic and R. Reinbacher, Numerical calabi-yau metrics, 2006. https://doi.org/10.1063/1.2888403, https://arxiv.org/abs/hep-th/0612075

    2006 doi arXiv

  150. [158]

    Headrick and T

    M. Headrick and T. Wiseman,Numerical ricci-flat metrics on k3, 2006. https://doi.org/10.1088/0264-9381/22/23/002, https://arxiv.org/abs/hep-th/0506129

    2006 doi arXiv

  151. [159]

    Headrick and A

    M. Headrick and A. Nassar,Energy functionals for calabi-yau metrics, 2010. https://arxiv.org/abs/0908.2635

    2010 arXiv

  152. [160]

    Butbaia, D

    G. Butbaia, D. M. Pe˜ na, J. Tan, P. Berglund, T. H¨ ubsch, V. Jejjala et al.,Physical yukawa couplings in heterotic string compactifications, 2024. https://doi.org/10.4310/ATMP.241119041341, https://arxiv.org/abs/2401.15078

    2024 doi arXiv

  153. [161]

    Berglund, G

    P. Berglund, G. Butbaia, T. H¨ ubsch, V. Jejjala, C. Mishra, D. M. Pe˜ na et al.,cymyc – calabi-yau metrics, yukawas, and curvature, 2024. https://doi.org/10.1007/JHEP03(2025)028, https://arxiv.org/abs/2410.19728

    2024 doi arXiv

  154. [162]

    Mishra and J

    C. Mishra and J. Tan,Hermitian yang–mills connections on general vector bundles: geometry and physical yukawa couplings, 2025. https://arxiv.org/abs/2512.10907

    2025

  155. [163]

    Ashmore,Eigenvalues and eigenforms on calabi-yau threefolds, 2023

    A. Ashmore,Eigenvalues and eigenforms on calabi-yau threefolds, 2023. https://doi.org/10.1016/j.geomphys.2023.105028, https://arxiv.org/abs/2011.13929

    2023 doi

  156. [164]

    Larfors, A

    M. Larfors, A. Lukas, F. Ruehle and R. Schneider,cymetric,

  157. [165]

    https://github.com/pythoncymetric/cymetric

  158. [166]

    Halverson and F

    J. Halverson and F. Ruehle,Metric flows with neural networks, 2024. https://doi.org/10.1088/2632-2153/ad8533, https://arxiv.org/abs/2310.19870

    2024 doi arXiv

  159. [167]

    R. S. Sutton and A. G. Barto,Reinforcement learning: an introduction, Adaptive computation and machine learning series. The MIT Press, Cambridge, Massachusetts, second edition ed., 2018

    2018

  160. [168]

    V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare et al.,Human-level control through deep reinforcement learning,Nature518(2015) 529. Pre-Strings Lectures on Artificial Intelligence 51

    2015

  161. [169]

    Schulman, S

    J. Schulman, S. Levine, P. Moritz, M. I. Jordan and P. Abbeel,Trust region policy optimization, 2017. https://arxiv.org/abs/1502.05477

    2017 arXiv

  162. [170]

    V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. P. Lillicrap, T. Harley et al.,Asynchronous methods for deep reinforcement learning, 2016. https://arxiv.org/abs/1602.01783

    2016 arXiv

  163. [171]

    Silver, J

    D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez et al.,Mastering the game of go without human knowledge,Nature550(2017) 354

    2017

  164. [172]

    Silver, T

    D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez et al.,Mastering chess and shogi by self-play with a general reinforcement learning algorithm, 2017. https://arxiv.org/abs/1712.01815

    2017 arXiv

  165. [173]

    Blumenhagen, M

    R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu, Toward realistic intersecting d-brane models, 2005. https://doi.org/10.1146/annurev.nucl.55.090704.151541, https://arxiv.org/abs/hep-th/0502005

    2005 doi

  166. [174]

    Manolescu and L

    C. Manolescu and L. Piccirillo,From zero surgeries to candidates for exotic definite four-manifolds, 2023. https://arxiv.org/abs/2102.04391

    2023 arXiv

  167. [175]

    M. Bies, M. Cvetic, R. Donagi, L. Lin, M. Liu and F. Ruehle, Machine learning and algebraic approaches towards complete matter spectra in 4d f-theory, 2020. https://doi.org/10.1007/JHEP01(2021)196, https://arxiv.org/abs/2007.00009

    2020 doi arXiv

  168. [176]

    Davies, A

    A. Davies, A. Juh´ asz, M. Lackenby and N. Tomasev,The signature and cusp geometry of hyperbolic knots, 2022. https://doi.org/10.2140/gt.2024.28.2313, https://arxiv.org/abs/2111.15323

    2022 doi arXiv

  169. [177]

    Ginsparg,Applied conformal field theory, 1988

    P. Ginsparg,Applied conformal field theory, 1988. https://arxiv.org/abs/hep-th/9108028

    1988 arXiv

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.