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REVIEW 2 major objections 6 minor 59 references

DeepONet ensembles can reconstruct lattice spectral densities with far smaller total uncertainty than the leading linear method on the same correlators.

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2026-07-12 03:20 UTC pith:TS6H32FO

load-bearing objection First solid DeepONet pipeline for lattice spectral reconstruction; precision win over HLT is real on the O(3) data but not yet shown to be prior-independent. the 2 major comments →

arxiv 2607.03311 v1 pith:TS6H32FO submitted 2026-07-03 hep-lat

Operator Learning in Lattice QCD: Spectral Reconstruction

Alessandro De Santis This is my paper
classification hep-lat
keywords spectral reconstructionoperator learningDeepONetlattice QCDEuclidean correlatorssmeared spectral densityO(3) sigma modelHansen-Lupo-Tantalo
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Lattice QCD computes Euclidean correlators; many physical rates are spectral densities obtained by inverting a Laplace transform. That inversion is ill-posed once noise is present. This paper reframes the map from correlator to smeared spectral density as an operator-learning problem and approximates it with DeepONet-style networks. An ensemble of networks, trained on mock correlators drawn from a Gaussian-process prior that encodes the two-particle structure of the O(3) model, supplies both a central prediction and a systematic uncertainty for the network approximation itself. After continuum, infinite-volume and zero-smearing limits the reconstructed inclusive spectral density agrees with the known analytic answer to within 1.1 standard deviations and shows a large reduction in total error relative to the Hansen–Lupo–Tantalo result obtained from identical correlators. The author notes that how much of the gain survives without the physics-informed prior has yet to be quantified, yet the architecture itself is reusable for any fixed-time-grid lattice observable that can be cast as an operator.

Core claim

An ensemble of DeepONets trained on Gaussian-process mock data can approximate the operator that maps a noisy Euclidean correlator to its Gaussian-smeared spectral density continuously in energy and smearing width; after all continuum and volume limits the reconstruction of the O(3) inclusive rate matches the analytic density and carries substantially smaller total uncertainty than the same data analysed with the Hansen–Lupo–Tantalo algorithm.

What carries the argument

DeepONet estimator: the product of a Branch net (latent encoding of the correlator vector) and a Trunk net (latent encoding of continuous (E,σ)), whose ensemble-weighted average and spread furnish both the reconstruction and its systematic error.

Load-bearing premise

That the large error reduction versus the linear method is not mainly an artifact of training on a Gaussian process whose mean is already the exact two-particle contribution of the target model.

What would settle it

Repeat the entire reconstruction pipeline on identical O(3) correlators after replacing the physics-informed GP mean by a flat or zero mean and increasing the prior variance; if the uncertainty advantage over HLT disappears while consistency with the analytic density is lost, the central claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper reformulates spectral reconstruction of smeared densities from Euclidean correlators as an operator-learning problem and approximates the map with DeepONet architectures (Branch net on the correlator vector, Trunk net on continuous (E, σ)). An ensemble of 15 networks of varying capacity and training-set size is trained on noisy mock correlators drawn from a Gaussian-process prior whose mean is the exact two-particle contribution of the O(3) model and whose variance is suppressed below the four-particle threshold; the ensemble supplies both the central value and a systematic uncertainty for the operator approximation. After validation on 1000 unseen noisy mocks (coverage consistent with or more conservative than Gaussian), the method is applied to the same Monte-Carlo correlators used in the HLT benchmark of Ref. [4]. Infinite-volume, continuum and σ→0 limits are performed with explicit systematic budgets, yielding a spectral density consistent with the known analytic result (largest deviation 1.1σ) and with total uncertainties up to an order of magnitude smaller than HLT on the same data. The authors note that the contribution of the physics-informed prior to this precision gain remains unquantified.

Significance. If the operator-learning formulation and ensemble systematics prove robust beyond the present prior, the work supplies a flexible, continuous-parameter alternative to HLT that can be reused across lattice spacings and smearing kernels without retraining, and that can be exported to other lattice operations. The first application of DeepONets in lattice QCD, the thorough mock-data closure tests, the explicit multi-source error budget (statistical, network, FVE, continuum, σ→0 via BMA), and the direct analytic comparison on a controlled integrable model are genuine strengths. The demonstrated reduction relative to HLT on identical correlators is striking, even though its origin is not yet disentangled from the GP mean. The method therefore has clear potential for high-impact observables such as the R-ratio once the prior dependence is quantified.

major comments (2)
  1. Abstract, §I, §V.D (Fig. 15) and §VI: the headline claim of a “significant reduction in the total uncertainty” relative to HLT is obtained exclusively with networks trained on a GP whose mean is exactly the analytic two-particle density ρ₂^{O(3)}(E) (Eq. 23) and whose local variance is deliberately suppressed below the four-particle threshold (Eqs. 24–25, Fig. 2). The paper itself states that “the extent to which this improvement persists in the absence of prior physical knowledge remains to be quantified.” Without at least one controlled ablation (e.g., a flat or zero mean, or a substantially larger σ_GP as sketched in App. C) the comparison cannot be read as a pure test of the DeepONet architecture versus HLT. Either such a test or a more prominent, quantitative caveat in the abstract and conclusions is required before the precision gain can be advertised as a general feature of the me
  2. §III.B and App. D: the out-of-distribution checks enlarge only σ_GP while retaining the same two-particle mean. They therefore do not address the load-bearing question of how much of the final precision and the recovery of multi-particle content is inherited from the mean function itself. A short additional paragraph quantifying the sensitivity of the continuum-extrapolated, σ→0 result to the choice of mean (or to the correlation-length range) would strengthen the central claim that the networks are learning the operator rather than merely interpolating the prior.
minor comments (6)
  1. §II.C and Fig. 1: the DeepONet schematic is clear, but the text never states whether a bias term is included in the final linear combination (Eq. 13). A one-sentence clarification would help reproducibility.
  2. Table II: the validation-loss values and weights are given to only one significant figure after the decimal; reporting one more digit would make the relative ranking of the networks more transparent.
  3. Fig. 9: the interpolation of the A2 correlator is shown only for a short time window; a supplementary panel covering the full range used by the Branch net (up to 540 a m) would reassure the reader that the cubic-spline procedure does not introduce visible artifacts at large t.
  4. §V.B: the continuum ansatz (Eq. 51) saturates the three available lattice spacings. While the subsequent erf-based systematic is conservative, a brief remark on the expected size of residual a^6 or logarithmic terms (citing Balog et al.) would be useful.
  5. App. B: the correction of two typographical errors in the published formulae of Ref. [4] is valuable; it would be helpful to state explicitly that the numerical values in Table III were recomputed with the corrected expressions.
  6. Throughout: the noise-level parameter λ is introduced in §III.C but the two ensembles (λ=1 and λ=3.5) are sometimes referred to only by the ensemble labels A/B; a consistent notation would reduce cognitive load.

Circularity Check

2 steps flagged

Training GP mean is exactly the target two-particle spectral density, so the headline precision gain vs HLT is not demonstrated to be prior-free.

specific steps
  1. fitted input called prediction [Section III.B, Eqs. (23)–(25) and Fig. 2; abstract, §I, §VI, App. C]
    "we choose as mean function the two-particle contribution to the O(3) inclusive rate, which can be written in closed form, µ(E)=ρ_O(3)_2(E)=… The covariance kernel is instead … with ξ(E)=σ_GP/(1+e^{-(E-4m)/ε}). … While the extent to which this improvement persists in the absence of prior physical knowledge remains to be quantified"

    The supervised training distribution is constructed so that its mean is exactly the dominant analytic component of the target spectral density that is later reconstructed; variance is deliberately kept small below the four-particle threshold. The networks therefore receive the leading piece of the answer as a fitted/prior input. The subsequent claim of a large precision gain versus HLT (same correlators) is obtained only under this prior; without an ablation that removes or flattens the mean, the reduction cannot be attributed solely to the DeepONet operator approximation.

  2. other [Appendix D and Fig. 15]
    "we generated out-of-distribution spectral functions by enlarging σ_GP … by a factor of five … while still using the two-particle contribution … as the mean function. … at high energies, our reconstructed spectral density differs from the GP mean by more than one standard deviation"

    Even the out-of-distribution checks retain the same exact two-particle mean and only enlarge the variance; they therefore do not remove the load-bearing prior that supplies the dominant piece of the target. The high-energy deviation from the GP mean shows some generalization, but does not eliminate the circularity of the precision comparison.

full rationale

The paper is largely self-contained: reconstruction is performed on independent Monte-Carlo correlators (ensembles A1–B2), multi-particle content above the four-particle threshold is recovered, the final continuum+volume+σ→0 result lies outside the one-σ GP band at high energy, and the analytic O(3) spectral density (including n=4,6) is an external benchmark. The ensemble uncertainty procedure is validated on held-out mock data. The only partial circularity is that the supervised training distribution is deliberately centered on the exact dominant piece of the target (ρ₂^{O(3)}), with variance suppressed below 4m; the claimed factor-of-ten uncertainty reduction relative to HLT is therefore obtained only under this physics-informed prior, and the paper itself repeatedly flags that the improvement without prior knowledge remains unquantified. This is a fitted/prior-informed input used for a closely related prediction, not a pure self-definitional loop or load-bearing self-citation of an unverified uniqueness theorem. Score 4 reflects partial rather than fatal circularity.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 2 invented entities

The central numerical claim rests on (i) the DeepONet universal-approximation property, (ii) a carefully engineered but non-unique GP prior whose mean is the exact two-particle spectral density, (iii) a large set of architectural and training hyperparameters, and (iv) the assumption that the ensemble spread faithfully quantifies operator-approximation error. No new physical entities are postulated; the free parameters are algorithmic.

free parameters (6)
  • GP correlation-length range ℓ
    Sampled uniformly in [15m,100m]; controls smoothness of training spectral functions and therefore the inductive bias of every network.
  • GP local variance σ_GP and transition scale ε
    Fixed by hand to 0.04 and 0.5m; set the amplitude of allowed deviations from the two-particle mean above the four-particle threshold.
  • Ensemble weight temperature α
    Set to 10^5; converts validation losses into normalized network weights that enter the final central value and systematic error.
  • Noise levels λ=1 and λ=3.5
    Chosen to match the noise-to-signal ratios of the A and B ensembles; two separate network ensembles are trained.
  • Latent dimension p, layer counts, neuron counts
    Fifteen architectures with p∈{16,32,64,128}, 1–3 layers, 64–256 neurons; selected by validation loss rather than by a first-principles criterion.
  • Domains Ω_t, Ω_E, Ω_σ and prediction grids
    Fixed a priori in physical units (t up to 540 a m, E∈[4m,40m], σ∈[0.5m,7m]); determine the operator that is actually learned.
axioms (4)
  • standard math DeepONet universal approximation theorem for continuous operators (Chen & Chen 1995 / Lu et al. 2021)
    Invoked in Section II.C and Appendix A to justify that a neural network can approximate the map from correlator to smeared spectral density.
  • domain assumption Finite-volume spectral densities admit a well-defined infinite-volume limit only after smearing with a Schwartz kernel
    Standard lattice-QCD lore used throughout Sections II.A and V; underpins the whole reconstruction programme.
  • ad hoc to paper The two-particle contribution ρ₂^{O(3)}(E) is an adequate mean function for the GP that generates the training set
    Chosen in Section III.B; the paper itself flags that the quantitative impact of this choice is unquantified.
  • ad hoc to paper Ensemble spread of independently trained networks of varying capacity supplies a reliable estimate of operator-approximation systematic error
    Core methodological claim of Section IV.B; validated only inside the same GP family used for training.
invented entities (2)
  • Physics-informed GP training distribution for spectral reconstruction no independent evidence
    purpose: Restricts the function space seen by the networks while still allowing multi-particle deviations above 4m
    Novel construction combining the exact two-particle formula with a non-stationary covariance; no independent experimental handle outside the O(3) benchmark.
  • Weighted neural-network ensemble for operator systematic uncertainty no independent evidence
    purpose: Convert validation losses into a data-driven systematic error bar on the reconstructed spectral density
    Reinterpretation of the ensemble idea of Ref. [25]; the weighting formula (Eq. 40) is specific to this work.

pith-pipeline@v1.1.0-grok45 · 42777 in / 3175 out tokens · 29166 ms · 2026-07-12T03:20:37.789903+00:00 · methodology

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read the original abstract

In this work, we propose a novel supervised machine-learning-based strategy for extracting smeared spectral functions from Euclidean correlation functions. The strategy revisits the numerically ill-posed spectral reconstruction problem within the framework of Operator Learning through the use of DeepONet-like architectures. To illustrate the method, we construct an ensemble of neural networks trained on mock data generated from a specific class of functions. This ensemble is then employed to estimate the systematic uncertainty associated with the fact that a neural network provides only an approximation to the target operator. The procedure is fully validated on previously unseen noisy mock data. To demonstrate the potential of the method for phenomenological applications, we reconstruct the inclusive rate above the four-particle threshold and up to high energies in the $1+1$-dimensional O(3) non-linear $\sigma$ model, starting from correlation functions computed in Monte Carlo simulations. The final result is consistent with the known analytic spectral density and, compared with the state-of-the-art Hansen-Lupo-Tantalo algorithm, exhibits a significant reduction in the total uncertainty. While the extent to which this improvement persists in the absence of prior physical knowledge remains to be quantified, the proposed strategy can be naturally extended to more phenomenologically relevant observables and, more generally, to other operations commonly encountered in lattice QCD.

Figures

Figures reproduced from arXiv: 2607.03311 by Alessandro De Santis.

Figure 1
Figure 1. Figure 1: FIG. 1. DeepONet architecture employed in this work and inspired by Ref.[37]. The architecture consists of two separate neural [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Space of functions that are generated from a GP to [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Branch (red points) and Trunk (black points) latent representations for ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Global bias for the neural network with index [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 2
Figure 2. Figure 2: To avoid introducing additional notation, it [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Average and one standard deviation of the ratio [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Validation of the uncertainty estimate provided [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Examples of reconstructed smeared spectral func [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of the Euclidean correlation function [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Pull variables for the finite-spatial (top panel) and [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Examples of the reconstructed smeared spectral [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Examples of continuum extrapolations for ( [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Examples of [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Relative error budget (multiplied by 100) of our [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Our final result (red circles) for the spectral density of the 1+1-dimensional O(3) non-linear [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: The coefficients cn are then generated randomly. The de￾gree of model independence is controlled by the dimen￾sion of the basis, NB. Small values of NB only allow the generation of very smooth spectral functions, whereas in the limit NB → ∞ any sufficiently regular function can be represented. By generating several training sets with pro￾gressively increasing values of NB, it is therefore possible to quan… view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Examples of reconstructions for spectral func [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗

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Reference graph

Works this paper leans on

59 extracted references · 29 linked inside Pith

  1. [1]

    Hansen, A

    M. Hansen, A. Lupo, and N. Tantalo, Extraction of spec- tral densities from lattice correlators, Phys. Rev. D99, 094508 (2019), arXiv:1903.06476 [hep-lat]

    Pith/arXiv arXiv 2019

  2. [2]

    Lupo and N

    A. Lupo and N. Tantalo, Extraction of spectral densi- ties from lattice correlators: decoupling signal from noise (2026), arXiv:2605.14652 [hep-lat]

    Pith/arXiv arXiv 2026

  3. [3]

    Backus and F

    G. Backus and F. Gilbert, The resolving power of gross earth data, Geophysical Journal International16, 169 (1968)

    1968

  4. [4]

    Bulava, M

    J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella, and N. Tantalo, Inclusive rates from smeared spectral den- sities in the two-dimensional O(3) non-linearσ-model, 29 JHEP07, 034, arXiv:2111.12774 [hep-lat]

    Pith/arXiv arXiv

  5. [5]

    Alexandrouet al.(Extended Twisted Mass Collabo- ration (ETMC)), Probing the Energy-Smeared R Ratio Using Lattice QCD, Phys

    C. Alexandrouet al.(Extended Twisted Mass Collabo- ration (ETMC)), Probing the Energy-Smeared R Ratio Using Lattice QCD, Phys. Rev. Lett.130, 241901 (2023), arXiv:2212.08467 [hep-lat]

    Pith/arXiv arXiv 2023

  6. [6]

    Evangelista, R

    A. Evangelista, R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, and V. Lubicz (Extended Twisted Mass), Inclusive hadronic decay rate of theτlep- ton from lattice QCD, Phys. Rev. D108, 074513 (2023), arXiv:2308.03125 [hep-lat]

    Pith/arXiv arXiv 2023

  7. [7]

    Alexandrouet al.(Extended Twisted Mass), Inclusive Hadronic Decay Rate of theτLepton from Lattice QCD: The ¯usFlavor Channel and the Cabibbo Angle, Phys

    C. Alexandrouet al.(Extended Twisted Mass), Inclusive Hadronic Decay Rate of theτLepton from Lattice QCD: The ¯usFlavor Channel and the Cabibbo Angle, Phys. Rev. Lett.132, 261901 (2024), arXiv:2403.05404 [hep- lat]

    Pith/arXiv arXiv 2024

  8. [8]

    De Santiset al., Inclusive semileptonic decays of the Ds meson: A first-principles lattice QCD calculation, Phys

    A. De Santiset al., Inclusive semileptonic decays of the Ds meson: A first-principles lattice QCD calculation, Phys. Rev. D112, 054503 (2025), arXiv:2504.06063 [hep- lat]

    arXiv 2025

  9. [9]

    De Santiset al., Inclusive Semileptonic Decays of the Ds Meson: Lattice QCD Confronts Experiments, Phys

    A. De Santiset al., Inclusive Semileptonic Decays of the Ds Meson: Lattice QCD Confronts Experiments, Phys. Rev. Lett.135, 121901 (2025), arXiv:2504.06064 [hep- lat]

    arXiv 2025

  10. [10]

    De Santiset al., Inclusive ¯Bs 7→X ¯scℓ¯νdecays from lattice QCD: computational strategy and a first physical result (2026), arXiv:2607.01116 [hep-lat]

    A. De Santiset al., Inclusive ¯Bs 7→X ¯scℓ¯νdecays from lattice QCD: computational strategy and a first physical result (2026), arXiv:2607.01116 [hep-lat]

    Pith/arXiv arXiv 2026

  11. [11]

    Bonanno, F

    C. Bonanno, F. D’Angelo, M. D’Elia, L. Maio, and M. Naviglio, Sphaleron rate from a modified backus- gilbert inversion method, Phys. Rev. D108, 074515 (2023)

    2023

  12. [12]

    Bonanno, F

    C. Bonanno, F. D’Angelo, M. D’Elia, L. Maio, and M. Naviglio, Sphaleron rate ofN f = 2 + 1 qcd, Phys. Rev. Lett.132, 051903 (2024)

    2024

  13. [13]

    J. C. A. Barata and K. Fredenhagen, Particle scattering in Euclidean lattice field theories, Commun. Math. Phys. 138, 507 (1991)

    1991

  14. [14]

    Bailas, S

    G. Bailas, S. Hashimoto, and T. Ishikawa, Recon- struction of smeared spectral function from Euclidean correlation functions, PTEP2020, 043B07 (2020), arXiv:2001.11779 [hep-lat]

    Pith/arXiv arXiv 2020

  15. [15]

    Asakawa, T

    M. Asakawa, T. Hatsuda, and Y. Nakahara, Maxi- mum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys.46, 459 (2001), arXiv:hep- lat/0011040

    arXiv 2001

  16. [17]

    Burnier and A

    Y. Burnier and A. Rothkopf, Bayesian Approach to Spec- tral Function Reconstruction for Euclidean Quantum Field Theories, Phys. Rev. Lett.111, 182003 (2013), arXiv:1307.6106 [hep-lat]

    Pith/arXiv arXiv 2013

  17. [18]

    Del Debbio, A

    L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Bayesian solution to the inverse problem and its rela- tion to Backus–Gilbert methods, Eur. Phys. J. C85, 185 (2025), arXiv:2409.04413 [hep-lat]

    Pith/arXiv arXiv 2025

  18. [19]

    Tsuji and S

    R. Tsuji and S. Hashimoto, Spectral reconstruction from Euclidean lattice correlators through singular value de- composition (2026), arXiv:2605.15674 [hep-lat]

    Pith/arXiv arXiv 2026

  19. [20]

    Abbott, S

    R. Abbott, S. Fields, W. I. Jay, P. Oare, and M. Sac- cardi, The Causal Bootstrap: Bounding Smeared Spec- tral Functions from Non-Perturbative Euclidean Data (2026), arXiv:2605.20509 [hep-lat]

    Pith/arXiv arXiv 2026

  20. [21]

    Bruno, L

    M. Bruno, L. Giusti, and M. Saccardi, Spectral densities from Euclidean lattice correlators via the Mellin transform, Phys. Rev. D111, 094515 (2025), arXiv:2407.04141 [hep-lat]

    Pith/arXiv arXiv 2025

  21. [22]

    Bergamaschi, W

    T. Bergamaschi, W. I. Jay, and P. R. Oare, Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D108, 074516 (2023), arXiv:2305.16190 [hep- lat]

    Pith/arXiv arXiv 2023

  22. [23]

    W. I. Jay and M. Saccardi, Kernel transformations and bounds for smeared spectral functions (2026), arXiv:2606.19503 [hep-lat]

    Pith/arXiv arXiv 2026

  23. [24]

    Giusti, M

    L. Giusti, M. Saccardi, and D. Toniolo, Spectral densities from Euclidean correlators via integral transforms: the- oretical framework (2026), arXiv:2606.28167 [hep-lat]

    Pith/arXiv arXiv 2026

  24. [25]

    Buzzicotti, A

    M. Buzzicotti, A. De Santis, and N. Tantalo, Teaching to extract spectral densities from lattice correlators to a broad audience of learning-machines, Eur. Phys. J. C84, 32 (2024), arXiv:2307.00808 [hep-lat]

    Pith/arXiv arXiv 2024

  25. [26]

    Del Debbio, Some Inverse Problems in Particle Physics (2026) arXiv:2606.08316 [hep-lat]

    L. Del Debbio, Some Inverse Problems in Particle Physics (2026) arXiv:2606.08316 [hep-lat]

    Pith/arXiv arXiv 2026

  26. [27]

    Kades, J

    L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, and F. P. G. Ziegler, Spectral Reconstruction with Deep Neural Networks, Phys. Rev. D102, 096001 (2020), arXiv:1905.04305 [physics.comp-ph]

    Pith/arXiv arXiv 2020

  27. [28]

    Karpie, K

    J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks, JHEP04, 057, arXiv:1901.05408 [hep-lat]

    Pith/arXiv arXiv 1901

  28. [29]

    L. Wang, S. Shi, and K. Zhou, Automatic differen- tiation approach for reconstructing spectral functions with neural networks, in35th Conference on Neural In- formation Processing Systems(2021) arXiv:2112.06206 [physics.comp-ph]

    Pith/arXiv arXiv 2021

  29. [30]

    Chen, H.-T

    S.-Y. Chen, H.-T. Ding, F.-Y. Liu, G. Papp, and C.- B. Yang, Machine learning Hadron Spectral Functions in Lattice QCD, PoSLATTICE2021, 148 (2022), arXiv:2112.00460 [hep-lat]

    Pith/arXiv arXiv 2022

  30. [31]

    L. Wang, S. Shi, and K. Zhou, Reconstructing spec- tral functions via automatic differentiation, Phys. Rev. D106, L051502 (2022)

    2022

  31. [32]

    S. Y. Chen, H. T. Ding, F. Y. Liu, G. Papp, and C. B. Yang, Machine learning spectral functions in lattice qcd (2022), arXiv:2110.13521 [hep-lat]

    Pith/arXiv arXiv 2022

  32. [33]

    M. Zhou, F. Gao, J. Chao, Y.-X. Liu, and H. Song, Appli- cation of radial basis functions neutral networks in spec- tral functions, Phys. Rev. D104, 076011 (2021)

    2021

  33. [34]

    Lechien and D

    T. Lechien and D. Dudal, Neural network approach to reconstructing spectral functions and complex poles of confined particles, SciPost Phys.13, 097 (2022), arXiv:2203.03293 [hep-lat]

    Pith/arXiv arXiv 2022

  34. [35]

    Fournier, L

    R. Fournier, L. Wang, O. V. Yazyev, and Q. Wu, Artifi- cial neural network approach to the analytic continuation problem, Phys. Rev. Lett.124, 056401 (2020)

    2020

  35. [36]

    Andratschke, B

    C. Andratschke, B. B. Brandt, E. Garnacho-Velasco, L. Pannullo, S. Singh, and A. D. M. Valois, Spectral reconstruction techniques, their shortcomings and rele- vance to the electric conductivity coefficient, in42nd In- ternational Symposium on Lattice Field Theory(2026) arXiv:2603.19156 [hep-lat]

    arXiv 2026

  36. [37]

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature machine intelligence3, 218 (2021). 30

    2021

  37. [38]

    Patella and N

    A. Patella and N. Tantalo, Scattering amplitudes from Euclidean correlators: Haag-Ruelle theory and approxi- mation formulae, JHEP01, 091, arXiv:2407.02069 [hep- lat]

    Pith/arXiv arXiv

  38. [39]

    Bulava and M

    J. Bulava and M. T. Hansen, Scattering amplitudes from finite-volume spectral functions, Phys. Rev. D100, 034521 (2019)

    2019

  39. [40]

    M. T. Hansen, H. B. Meyer, and D. Robaina, From deep inelastic scattering to heavy-flavor semileptonic decays: Total rates into multihadron final states from lattice qcd, Phys. Rev. D96, 094513 (2017)

    2017

  40. [41]

    Chen and H

    T. Chen and H. Chen, Universal approximation to non- linear operators by neural networks with arbitrary acti- vation functions and its application to dynamical sys- tems, IEEE transactions on neural networks6, 911 (1995)

    1995

  41. [42]

    Zaheer, S

    M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. R. Salakhutdinov, and A. J. Smola, Deep sets, Advances in neural information processing systems30(2017)

    2017

  42. [43]

    Cipriani, A

    A. Cipriani, A. De Santis, G. Di Russo, A. Grillo, and L. Tabarroni, Hamiltonian neural network approach to fuzzball geodesics, Phys. Rev. D112, 026018 (2025), arXiv:2502.20881 [hep-th]

    Pith/arXiv arXiv 2025

  43. [44]

    Margari, S

    F. Margari, S. Bacchio, A. De Santis, A. Evangelista, R. Frezzotti, G. Gagliardi, M. Garofalo, F. Sanfilippo, and N. Tantalo (Extended Twisted Mass), The smeared R-ratio in isoQCD from first-principles lattice simula- tions (2026), arXiv:2603.19070 [hep-lat]

    arXiv 2026

  44. [45]

    A. P. Valentine and M. Sambridge, Gaussian process models—i. a framework for probabilistic continuous in- verse theory, Geophysical Journal International220, 1632 (2020)

    2020

  45. [46]

    C. K. Williams and C. E. Rasmussen,Gaussian processes for machine learning, Vol. 2 (MIT press Cambridge, MA, 2006)

    2006

  46. [47]

    Luscher and U

    M. Luscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation, Nucl. Phys. B339, 222 (1990)

    1990

  47. [48]

    Lellouch and M

    L. Lellouch and M. Luscher, Weak transition matrix ele- ments from finite volume correlation functions, Commun. Math. Phys.219, 31 (2001), arXiv:hep-lat/0003023

    Pith/arXiv arXiv 2001

  48. [49]

    Tancik, P

    M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich- Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Bar- ron, and R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains, Ad- vances in neural information processing systems33, 7537 (2020)

    2020

  49. [50]

    Hendrycks and K

    D. Hendrycks and K. Gimpel, Gaussian error linear units (gelus) (2023), arXiv:1606.08415 [cs.LG]

    Pith/arXiv arXiv 2023

  50. [51]

    Google, Google colaboratory,https://colab.google (2026)

    2026

  51. [52]

    Abadi, A

    M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Is- ard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Lev- enberg, D. Man´ e, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Van...

    2015

  52. [53]

    Cholletet al., Keras,https://keras.io(2015)

    F. Cholletet al., Keras,https://keras.io(2015)

    2015

  53. [54]

    D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)

    Pith/arXiv arXiv 2014

  54. [55]

    F. A. Bresciani, M. Bruno, and M. T. Hansen, Finite- volume effects on smeared spectral densities (2026), arXiv:2606.14349 [hep-lat]

    arXiv 2026

  55. [56]

    Balog, F

    J. Balog, F. Niedermayer, and P. Weisz, Logarithmic cor- rections to o (a2) lattice artifacts, Physics Letters B676, 188 (2009)

    2009

  56. [57]

    Balog, F

    J. Balog, F. Niedermayer, and P. Weisz, The puzzle of apparent linear lattice artifacts in the 2d non-linearσ- model and symanzik’s solution, Nuclear physics B824, 563 (2010)

    2010

  57. [58]

    Akaike, A new look at the statistical model identifi- cation, IEEE transactions on automatic control19, 716 (1974)

    H. Akaike, A new look at the statistical model identifi- cation, IEEE transactions on automatic control19, 716 (1974)

    1974

  58. [59]

    D. A. Nix and A. S. Weigend, Estimating the mean and variance of the target probability distribution, inPro- ceedings of 1994 ieee international conference on neural networks (ICNN’94), Vol. 1 (IEEE, 1994) pp. 55–60

    1994

  59. [60]

    Balog and M

    J. Balog and M. Niedermaier, Off-shell dynamics of the O(3) NLS model beyond Monte Carlo and perturba- tion theory, Nucl. Phys. B500, 421 (1997), arXiv:hep- th/9612039

    arXiv 1997