Pith. sign in

REVIEW 4 major objections 5 minor 77 references

Using Diffusion Models to Estimate Uncertainties in Analytic Continuation

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A conditional diffusion model learns the full distribution of real-frequency spectra consistent with an imaginary-time correlation function, giving analytic continuation a principled uncertainty estimate.

desk verdict A genuinely new diffusion-based approach to analytic continuation with a useful UPV diagnostic, but the probabilistic uncertainty claim needs calibration and data-consistency checks before I'd trust it fully. read the letter →

arxiv 2608.13123 v1 pith:M5N37GND submitted 2026-08-13 physics.comp-ph cond-mat.stat-mech

Sagi Meir , Daniel Freedman , Barak Hirshberg This is my paper · ORCID
classification physics.comp-phcond-mat.stat-mech
keywords analyticcontinuationdiffusionmodelsuncertaintyquantificationimaginary-timecorrelationfunctionsinverseproblemspseudo-volumeliquidparahydrogengenerativemodeling
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 argues that the right way to treat the ill-posed inverse problem of analytic continuation is to model the full conditional distribution $p(C|G)$ of real-frequency spectra $C(\omega)$ given an imaginary-time correlation function $G(\tau)$, rather than to predict a single spectrum. Regression-based methods, including maximum entropy, collapse the solution space toward a conditional mean and hide the fact that many spectra fit the same data. The authors introduce a diffusion-model framework that samples many plausible spectra for one input, and define a new metric, the uncertainty pseudo-volume, that measures how spread out this solution set is after accounting for correlations between frequencies. Applied to synthetic spectra and to a path-integral simulation of liquid parahydrogen, the framework extracts a self-diffusion coefficient with an error bar and flags a high-frequency peak as unsupported by the data.

What carries the argument

The load-bearing mechanism is a conditional diffusion model with a linear interpolant forward process, $x_t=(1-t)x_0+t\epsilon$, in which a Diffusion Transformer trained to predict the clean spectrum $x_0$ from $(x_t,t,G)$ approximately minimizes the KL divergence to $p(C|G)$. At inference, an ensemble of reverse ODE trajectories, started from Gaussian noise and advanced with a first-order DDIM step followed by second-order DPM-Solver++ updates, produces samples of the posterior. A spectrum-specific principal component analysis then supplies a locally Gaussian model of the uncertainty, and the UPV metric $P_d=\prod_i(1+z_i/\lambda)^{-1}$ formed from the principal-axis percentiles measures the correlated spread that pointwise error bars miss.

What would settle it

Train the identical framework on a deliberately different spectral family, such as sharp narrow peaks confined to the upper half of the frequency domain, and then feed it iTCFs from that family to check whether the UPV and ensemble spread still track the known non-uniqueness; if the UPV stays small for a clearly ambiguous inversion, the metric is not measuring intrinsic hardness.

Watch

Extended reading notes

Core claim

The central claim is that a conditional diffusion model with a linear interpolant forward process can approximate the posterior distribution over power spectra for a given imaginary-time correlation function, and that the spread of samples from this learned distribution is a theoretically grounded uncertainty estimate. Pointwise standard deviations show where the data constrain the spectrum, while principal component analysis of the sampled ensemble reveals that the ambiguity is correlated across frequencies and concentrated in a few modes. The uncertainty pseudo-volume $P_d=\prod_i (1+z_i/\lambda)^{-1}$, built from the 5th and 95th percentiles of the principal-component coefficients, quantifies the intrinsic hardness of each inversion without access to the true spectrum. On liquid parahydrogen the model yields a self-diffusion coefficient $D=0.59\pm0.09$ Å$^2$/ps and a wide uncertainty band around a secondary peak near $\beta\hbar\omega\approx25$, which the authors flag as a likely spurious artifact.

Load-bearing premise

The uncertainty estimates are trustworthy only when the training spectra resemble the real system's spectrum, and the model was trained on procedural mixtures of one to four warped Gaussian bumps centered in the lower half of the frequency domain, so a real spectrum with very different structure could yield confident but wrong reconstructions.

Editorial extensions

If this is right

  • A single trained diffusion model yields distributional output, so uncertainty quantification does not require an ensemble of separately trained regression networks, which the paper finds converge to nearly identical spectra and underestimate ambiguity.
  • The UPV ranks inversions by difficulty without ground truth, allowing practitioners to flag imaginary-time traces whose reconstruction is dominated by the kernel's information loss.
  • Because the ambiguity is correlated and low-dimensional, with three to five principal components typically explaining over 90% of the variance, compact summaries of the solution space can replace full per-frequency error bars.
  • Quantities extracted from the spectrum inherit a principled error bar, as in the parahydrogen self-diffusion coefficient $D=0.59\pm0.09$ Å$^2$/ps.
  • The framework is formulated for any strongly smoothing inverse problem, so the same uncertainty-quantification logic transfers to other Laplace-type inversions beyond quantum correlation functions.

Reading between the lines

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

  • A testable extension is to use the UPV as a design objective, choosing experimental or simulation settings that reduce the pseudo-volume rather than merely reporting it after reconstruction.
  • Comparing the diffusion ensemble with solution sets produced by constrained stochastic analytic continuation on the same iTCFs would test whether the generative prior covers the full ambiguity or only a data-dependent subset.
  • Because the training distribution centers all spectral bumps in the lower half of the frequency domain, applying the framework to systems with dominant high-frequency structure would require retraining on a broader procedural family or adding an explicit distribution-shift diagnostic; the paper does not yet provide such a diagnostic.
  • The same correlated-uncertainty analysis could be applied to other inverse problems with smoothing kernels, such as NMR relaxometry or rheology, where the posterior is likewise low-dimensional despite high-dimensional data.
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

4 major / 5 minor

Summary. The paper introduces a diffusion-based generative model for analytic continuation, aiming to model the full conditional distribution p(C|G) of real-frequency spectra given an imaginary-time correlation function. The authors train a DiT-based denoiser on procedurally generated spectra built from 1-4 warped Gaussian bumps, condition on a four-channel representation of G(τ), and draw 1000 independent samples per input at inference. They summarize the resulting ensemble with a local PCA and propose a new metric, the uncertainty pseudo-volume (UPV), as a quantitative hardness measure for each inversion. The method is demonstrated on synthetic spectra and on a PIMD simulation of liquid parahydrogen, yielding a self-diffusion coefficient with an error bar and a high-frequency feature flagged as uncertain. The central claims are that the ensemble spread provides theoretically grounded uncertainty quantification and that the UPV diagnoses intrinsic inversion difficulty.

Significance. If validated, this would be a useful contribution to the analytic continuation literature, where joint-posterior uncertainty estimates remain rare. The synthetic experiments are carefully designed, the PCA parsimony analysis is a creative way to summarize correlated posterior structure, and the paper provides full implementation details, training and sampling pseudocode, and an explicit statement of a key limitation regarding training-family representativeness. However, the uncertainty statement is not yet anchored by a forward-consistency or calibration check, and the UPV scale is set relative to the compared spectra. These gaps are directly load-bearing for the paper's central claims, but they are addressable within the manuscript's scope. The method is a clear step beyond pointwise error bars, and the intended significance is evident, but the validation needs to be completed.

major comments (4)
  1. [III.B, IV.A (Eqs. 4-6)] The inference procedure never projects the generated spectrum through the forward kernel of Eq. (1), so nothing in the algorithm enforces that the sampled C(ω) reproduces the conditioning G(τ). The paper reports no residual or coverage test showing that the ensemble's forward transforms K C_j match G within the data noise level. Without such a check, the ensemble spread and the UPV in Section IV.C may conflate model approximation error with the intrinsic ambiguity of the inverse problem, which is exactly what the paper claims to measure. I request a forward-consistency experiment (e.g., the distribution of residuals ‖K C_j − G‖ across the ensemble, or the fraction of samples whose forward transform lies within a noise ball around G) and a corresponding calibration/coverage statistic.
  2. [IV.C (Eq. 11)] The UPV defined in Eq. (11) depends on the free scale parameter λ, which is set after the fact to half the maximal z_i across all compared spectra (Section IV.C). This makes the reported UPV values relative to the particular set of spectra included in the comparison. In particular, the claim in Section IV.D that P = 61 indicates "moderate" ambiguity relies on the synthetic examples in Figure 5 being used to fix λ; if the comparison set changes, P is not an absolute measure of inversion hardness. Please either fix λ in advance independent of the dataset, report P on an absolute scale, or demonstrate that the ranking of P across spectra is insensitive to the choice of λ.
  3. [IV.D, VI.B] The training distribution in Section VI.B consists of spectra built from 1 to 4 warped Gaussian bumps whose centers are uniformly sampled in the lower half of the frequency domain (ω ∈ [0, 25]). The authors explicitly state at the end of Section IV.D that the uncertainty estimate is valid only if the model was trained on spectra representative of the system at hand, but no diagnostic is provided to detect when this condition fails for a new iTCF. Given that the abstract and title promise uncertainty quantification for analytic continuation generally, the lack of a distribution-shift test or a characterization of when the method can be trusted is a load-bearing gap. At minimum, the claim of a theoretically grounded confidence estimate should be qualified to the training family, or the paper should add a shift-detection mechanism.
  4. [IV.A, IV.C] No calibration check is reported showing how often the true spectrum falls inside the diffusion credibility band. For example, one could compute the empirical coverage of the 5-95% ensemble interval, or the fraction of ground-truth spectra whose projection on the leading PCA components lies inside the UPV ellipsoid, across the synthetic test set. Without such a check, the probabilistic interpretation of the error bars in Figures 3 and 6 is asserted rather than demonstrated.
minor comments (5)
  1. [Figure 2] The axis labels in Figure 2 are garbled (e.g., "G( )" and non-rendered glyphs); the four channel names should be typeset properly so the figure is self-contained.
  2. [III.A] The sentence "predicting x0 is mathematically equivalent to predicting the score" could mislead readers because the score is defined for the marginal q_t(x_t) rather than the joint distribution; please spell out the relationship or cite the specific equivalence conditions from the referenced works.
  3. [IV.C, IV.D] The notation P_d is defined as a family of pseudo-volumes but is used interchangeably with P without the subscript in Figure 5 and Section IV.D; please make the notation consistent.
  4. [Abstract, IV.A] The abstract's phrase "concrete probabilistic basis" is stronger than what the training objective in Eq. (3) and the deterministic ODE sampler in Section III.B strictly justify; please soften the wording or state the approximation conditions explicitly.
  5. [S2] The sampling schedule is linearly spaced in t with t1 = 0.99, while training samples t from Beta(1, 2.5); the mismatch between the training-time distribution and the inference-time schedule is not discussed, and a brief comment on its effect would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the diffusion ensemble is trained on independently generated (C,G) pairs, and the UPV is a descriptive statistic of that ensemble, not a fitted prediction.

full rationale

The central claim, that the diffusion model samples p(C|G), is supported by the standard denoising objective (Eq. 3) and the conditional DiT parameterization; training pairs are generated on-the-fly by forward integration of Eq. (1), so the target conditional distribution is defined independently of the model. The UPV (Eq. 11) is a function of the PCA axis lengths z_i of the generated ensemble; it is not fitted to any ground-truth label, and the lambda normalization (half the value of the maximal z_i) only sets an overall scale for comparing inversions, so the relative hardness ordering is not forced by construction. The paper explicitly disclaims generalization beyond the training family (provided the model was trained on spectra representative of the system at hand, Section IVD), which is a stated limitation rather than a circular step. The one coauthor citation ([74], Belhasin et al., including D. Freedman) supplies a PCA-based correlated-uncertainty viewpoint, but the mathematical content used here (projection residual in Eqs. 7-9 and the volume-like metric in Eq. 11) is derived in the paper itself and is elementary linear algebra; the citation is not load-bearing. The skeptic's concern, absence of a forward-consistency projection or residual check, is a validity and accuracy risk about whether the learned ensemble covers only data-consistent spectra, not a reduction of the paper's outputs to its inputs by construction. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The central claim rests on one number fitted by hand (lambda in the UPV), several hyperparameters, and a strong domain assumption: that the procedural training distribution of spectra is representative of the systems to which the model is applied. The paper adds a method and a metric, rather than introducing new physical entities, and the falsifiability of the uncertainty claims is limited because the associated data and code are not provided.

free parameters (4)
  • lambda (UPV scale) = half the maximal z_i across compared spectra
    Equation 11 defines UPV with a positive constant lambda that must be chosen by hand; the paper sets it relative to the ensemble spread. Different lambda choices change the absolute UPV values, so the reported P=61 is not scale-free.
  • Beta distribution shape parameters (alpha=1, beta=2.5) = 1 and 2.5
    The training time distribution is chosen to emphasize low-noise regimes; the specific skew is a tuning choice that affects sample quality but is not claimed to be optimal.
  • Number of diffusion steps (40) and discretization = 40
    Inference uses 40 linearly spaced timesteps with a hybrid DDIM/DPM-Solver++ update. The paper does not report a convergence check over the number of sampling steps.
  • Synthetic spectrum generation parameters = 1 to 4 bumps, centers in lower half of frequency domain, widths 10-60%
    The training prior over spectra is defined by these hand-chosen procedural generation rules. The uncertainty estimate inherits this prior, and the paper itself states it is only valid for spectra representative of the system.
assumptions (5)
  • domain assumption The forward kernel in Equation 1, with beta hbar = 1 and the given symmetric kernel, is the correct mapping from power spectra to imaginary-time correlation functions.
    Stated in Section II and used throughout; standard in the analytic continuation literature.
  • ad hoc to paper Synthetic spectra built from 1 to 4 warped Gaussian bumps cover the relevant space of physical spectra for uncertainty quantification.
    The training distribution is a modeling choice that determines what the model considers plausible. The paper acknowledges this in Section IVD but provides no quantitative check of coverage for new systems.
  • domain assumption The conditional diffusion model trained with the MSE objective in Equation 3 learns the correct conditional distribution p(C|G).
    The paper cites standard diffusion theory connecting the denoising objective to the score, but the finite-capacity Transformer and finite training budget mean the learned distribution is only an approximation.
  • domain assumption Samples from the reverse ODE at 40 timesteps with classifier-free guidance disabled are distributed according to the learned conditional distribution.
    The paper uses a deterministic DDIM/DPM-Solver++ discretization and says pure conditional sampling is used, but no statistical test of the sample distribution is reported.
  • ad hoc to paper The locally Gaussian PCA model captures the full structure of the posterior distribution in the UPV calculation.
    The UPV and parsimony analysis assume a Gaussian or elliptically symmetric posterior. The paper shows this captures a few dominant modes but does not test for non-Gaussian structure.
invented entities (2)
  • Uncertainty pseudo-volume (UPV)
    purpose: A scalar metric that characterizes the magnitude and structure of inversion uncertainty by measuring the volume of the PCA ellipsoid of the diffusion ensemble.
    The UPV is defined in this paper and is not tied to any external observable. Its interpretation depends entirely on the internal normalization choice for lambda.
  • Four-channel conditioning representation
    purpose: A representation of the input imaginary-time correlation function that feeds four views of the same signal into the model to improve conditioning.
    This is a new architecture choice introduced in this paper. The authors provide qualitative motivation but no ablation study showing how much each channel contributes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Using Diffusion Models to Estimate Uncertainties in Analytic Continuation." pith.science (2026). https://pith.science/paper/M5N37GND

@misc{pith2026260813123,
  author       = {Pith},
  title        = {Pith review of: Using Diffusion Models to Estimate Uncertainties in Analytic Continuation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5N37GND}},
  note         = {Machine review of arXiv:2608.13123}
}
read the original abstract

Inverse problems are ubiquitous in physics, chemistry, and engineering, arising when reconstructing hidden quantities from indirect measurements. A key example is the analytic continuation of imaginary-time correlation functions (iTCFs) to the real-frequency domain. This process requires an inverse Laplace transform, which is inherently ill-posed and highly sensitive to small input variations. Recent neural network (NN)-based methods have shown promising results by learning mappings from imaginary-time to real-frequency spectra, often outperforming traditional techniques such as maximum entropy. However, because the problem is ill-posed, many spectra fit the same iTCF. Regression-based approaches output a single solution, which approximates an average over the true solution space, and therefore fail to capture the full distribution of plausible power spectra. To address this issue, we introduce a diffusion-based framework for analytic continuation that learns the distribution of spectra consistent with a given iTCF. It offers two key advantages. First, it quantifies uncertainty directly from the learned distribution. Second, by analyzing the spread and structure of this distribution, we can quantitatively assess the intrinsic hardness of each inversion problem. We measure this hardness with a new metric, the uncertainty pseudo-volume. Applying the framework to an iTCF from a path-integral molecular dynamics simulation of liquid parahydrogen, we obtain the self-diffusion coefficient with an error bar and flag a secondary high-frequency peak as a possible spurious artifact. In contrast to previous attempts at uncertainty quantification, our generative approach rests on a concrete probabilistic basis, providing a more theoretically grounded measure of confidence in the reconstructed power spectra.

Figures

Figures reproduced from arXiv: 2608.13123 by the authors.

Figure 1
Figure 1. Demonstration of the ill-posedness of equation (1). [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The four-channel representation of the iTCFs used for model conditioning. To provide a richer representation of the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Comparison of analytic continuation using regression and diffusion models across representative synthetic examples. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Step-by-step visualization of the PCA reconstruction for the correlated model ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Quantitative analysis of inversion ambiguity for the representative synthetic examples shown in figure 3. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Demonstration of analytic continuation of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

77 extracted references · 61 canonical work pages

  1. [1]

    Yaman, V

    F. Yaman, V. G. Yakhno, and R. Potthast, A survey on inverse problems for applied sciences, Mathematical Problems in Engineering2013, 1 (2013)

    work page 2013

  2. [2]

    ∂Pd ∂zi >0for alli= 1,...,d

    work page

  3. [3]

    hardness

    For anyz1,...,z d we have thatPd(z1,...,z i−1,0,z i+1,...,z d) =P d−1(z1,...,z i−1,zi+1,...,z d). Properties 1 and 2 are basic: if all axes have length zero, the pseudo-volume is correspondingly zero, and if any axis increases, then the pseudo-volume also increases. Property 3 is more subtle. It says that the pseudo-volume for a collection ofdaxes, one of...

    work page

  4. [4]

    Lesnic,Inverse problems with applications in science and engineering(Chapman and Hall/CRC, 2021)

    D. Lesnic,Inverse problems with applications in science and engineering(Chapman and Hall/CRC, 2021)

    work page 2021

  5. [5]

    Bingham, T

    D. Bingham, T. Butler, and D. Estep, Inverse problems for physics-based process models, Annual Review of Statistics and Its Application11, 461 (2024)

    work page 2024

  6. [6]

    R. N. Silver, D. S. Sivia, and J. E. Gubernatis, Maximum-entropy method for analytic continuation of quantum monte carlo data, Phys. Rev. B41, 2380 (1990)

    work page 1990

  7. [7]

    Rabani, D

    E. Rabani, D. R. Reichman, G. Krilov, and B. J. Berne, The calculation of transport properties in quantum liquids using the maximum entropy numerical analytic continuation method: Application to liquid para-hydrogen, Proceedings of the National Academy of Sciences99, 1129 (2002)

    work page 2002

  8. [8]

    Tripolt, P

    R.-A. Tripolt, P. Gubler, M. Ulybyshev, and L. Von Smekal, Numerical analytic continuation of euclidean data, Computer Physics Communications237, 129 (2019)

    work page 2019

Show all 77 references
  1. [9]

    Ying, Analytic continuation from limited noisy matsubara data, Journal of Computational Physics469, 111549 (2022)

    L. Ying, Analytic continuation from limited noisy matsubara data, Journal of Computational Physics469, 111549 (2022). 13

    2022

  2. [10]

    Shao and A

    H. Shao and A. W. Sandvik, Progress on stochastic analytic continuation of quantum monte carlo data, Physics Reports 1003, 1 (2023)

    2023

  3. [11]

    Gallicchio, S

    E. Gallicchio, S. A. Egorov, and B. J. Berne, On the application of numerical analytic continuation methods to the study of quantum mechanical vibrational relaxation processes, The Journal of Chemical Physics109, 7745 (1998)

    1998

  4. [12]

    Habershon, B

    S. Habershon, B. J. Braams, and D. E. Manolopoulos, Quantum mechanical correlation functions, maximum entropy analytic continuation, and ring polymer molecular dynamics, The Journal of Chemical Physics127, 174108 (2007)

    2007

  5. [13]

    Gunnarsson, M

    O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analytical continuation of imaginary axis data for optical conduc- tivity, Phys. Rev. B82, 165125 (2010)

    2010

  6. [14]

    Wlazłowski, P

    G. Wlazłowski, P. Magierski, and J. E. Drut, Shear viscosity of a unitary fermi gas, Phys. Rev. Lett.109, 020406 (2012)

    2012

  7. [15]

    Reymbaut, D

    A. Reymbaut, D. Bergeron, and A.-M. S. Tremblay, Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight, Phys. Rev. B92, 060509 (2015)

    2015

  8. [16]

    Tiihonen, I

    J. Tiihonen, I. Kylänpää, and T. T. Rantala, Computation of dynamic polarizabilities and van der waals coefficients from path-integral monte carlo, Journal of Chemical Theory and Computation14, 5750 (2018)

    2018

  9. [17]

    Hamann, T

    P. Hamann, T. Dornheim, J. Vorberger, Z. A. Moldabekov, and M. Bonitz, Dynamic properties of the warm dense electron gas based onab initiopath integral monte carlo simulations, Phys. Rev. B102, 125150 (2020)

    2020

  10. [18]

    J. G. McWhirter and E. R. Pike, On the numerical inversion of the laplace transform and similar fredholm integral equations of the first kind, Journal of Physics A: Mathematical and General11, 1729 (1978)

    1978

  11. [19]

    S. Shi, L. Wang, and K. Zhou, Rethinking the ill-posedness of the spectral function reconstruction—why is it fundamentally hard and how artificial neural networks can help, Computer Physics Communications282, 108547 (2023)

    2023

  12. [20]

    R. K. Bryan, Maximum entropy analysis of oversampled data problems, European Biophysics Journal18, 165 (1990)

    1990

  13. [21]

    J. E. Gubernatis, M. Jarrell, R. N. Silver, and D. S. Sivia, Quantum monte carlo simulations and maximum entropy: Dynamics from imaginary-time data, Phys. Rev. B44, 6011 (1991)

    1991

  14. [22]

    Jarrell and J

    M. Jarrell and J. E. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data, Physics Reports269, 133 (1996)

    1996

  15. [23]

    R. Levy, J. P. F. LeBlanc, and E. Gull, Implementation of the maximum entropy method for analytic continuation, Computer Physics Communications215, 149 (2017)

    2017

  16. [24]

    Bergeron and A.-M

    D. Bergeron and A.-M. S. Tremblay, Algorithms for optimized maximum entropy and diagnostic tools for analytic contin- uation, Phys. Rev. E94, 023303 (2016)

    2016

  17. [25]

    Rothkopf, Bryan’s maximum entropy method—diagnosis of a flawed argument and its remedy, Data5, 85 (2020)

    A. Rothkopf, Bryan’s maximum entropy method—diagnosis of a flawed argument and its remedy, Data5, 85 (2020)

    2020

  18. [26]

    Gunnarsson, M

    O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analytical continuation of imaginary axis data using maximum entropy, Phys. Rev. B81, 155107 (2010)

    2010

  19. [27]

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

    2022

  20. [28]

    A. W. Sandvik, Stochastic method for analytic continuation of quantum monte carlo data, Phys. Rev. B57, 10287 (1998)

    1998

  21. [29]

    A. W. Sandvik, Constrained sampling method for analytic continuation, Phys. Rev. E94, 063308 (2016)

    2016

  22. [30]

    K. S. D. Beach, R. J. Gooding, and F. Marsiglio, Reliable padé analytical continuation method based on a high-accuracy symbolic computation algorithm, Phys. Rev. B61, 5147 (2000)

    2000

  23. [31]

    Schött, I

    J. Schött, I. L. M. Locht, E. Lundin, O. Grånäs, O. Eriksson, and I. Di Marco, Analytic continuation by averaging padé approximants, Phys. Rev. B93, 075104 (2016)

    2016

  24. [32]

    Huang and C

    L. Huang and C. Yue, Barycentric rational function approximation made simple: A fast analytic continuation method for matsubara green’s functions, Phys. Rev. B111, 125139 (2025)

    2025

  25. [33]

    Fei, C.-N

    J. Fei, C.-N. Yeh, and E. Gull, Nevanlinna analytical continuation, Phys. Rev. Lett.126, 056402 (2021)

    2021

  26. [34]

    Fei, C.-N

    J. Fei, C.-N. Yeh, D. Zgid, and E. Gull, Analytical continuation of matrix-valued functions: Carathéodory formalism, Phys. Rev. B104, 165111 (2021)

    2021

  27. [35]

    Iskakov, A

    S. Iskakov, A. Hampel, N. Wentzell, and E. Gull, Triqs/nevanlinna: Implementation of the nevanlinna analytic continuation method for noise-free data, Computer Physics Communications304, 109299 (2024)

    2024

  28. [36]

    Goulko, A

    O. Goulko, A. S. Mishchenko, L. Pollet, N. Prokof’ev, and B. Svistunov, Numerical analytic continuation: Answers to well-posed questions, Phys. Rev. B95, 014102 (2017)

    2017

  29. [37]

    Rothkopf, Bayesian inference of real-time dynamics from lattice qcd, Frontiers in Physics10, 1028995 (2022)

    A. Rothkopf, Bayesian inference of real-time dynamics from lattice qcd, Frontiers in Physics10, 1028995 (2022)

    2022

  30. [38]

    Huang and S

    L. Huang and S. Liang, Reconstructing lattice qcd spectral functions with stochastic pole expansion and nevanlinna analytic continuation, Phys. Rev. D109, 054508 (2024)

    2024

  31. [39]

    Fournier, L

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

    2020

  32. [40]

    Yoon, J.-H

    H. Yoon, J.-H. Sim, and M. J. Han, Analytic continuation via domain knowledge free machine learning, Phys. Rev. B98, 245101 (2018)

    2018

  33. [41]

    X. Xie, F. Bao, T. Maier, and C. Webster, Analytic continuation of noisy data using adams bashforth residual neural network, Discrete and Continuous Dynamical Systems - Series S15, 877 (2022)

    2022

  34. [42]

    Kades, J

    L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, and F. P. Ziegler, Spectral reconstruction with deep neural networks, Phys. Rev. D102, 096001 (2020)

    2020

  35. [43]

    Zhang, M

    R. Zhang, M. E. Merkel, S. Beck, and C. Ederer, Training biases in machine learning for the analytic continuation of quantum many-body green’s functions, Phys. Rev. Res.4, 043082 (2022). 14

    2022

  36. [44]

    Z. Zhao, G. Wang, J. Xu, C. Wang, and Y. Yang, Analytic continuation by feature learning, Phys. Rev. B114, 065103 (2026)

    2026

  37. [45]

    Raghavan and A

    K. Raghavan and A. Lovato, Uncertainty-quantification-enabled inversion of nuclear responses, Phys. Rev. C110, 025504 (2024)

    2024

  38. [46]

    B. Chen, Y. Zhang, L. Wang, Z. Zhan, X. Guan, Z. Rong, Y. Cui, E. Lin, S. Cao, Y. Huang, Y. Yang, and Z. Chen, High- confidence reconstruction for Laplace inversion in NMR based on uncertainty-informed deep learning, Science Advances 11, eadw1379 (2025)

    2025

  39. [47]

    Schweighofer, L

    K. Schweighofer, L. Aichberger, M. Ielanskyi, G. Klambauer, and S. Hochreiter, Quantification of uncertainty with adver- sarial models, inAdvances in Neural Information Processing Systems, Vol. 36, edited by A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (...

    2023

  40. [48]

    Chen, H.-T

    S.-Y. Chen, H.-T. Ding, F.-Y. Liu, G. Papp, and C.-B. Yang, Machine learning spectral functions in lattice QCD (2021), arXiv:2110.13521 [hep-lat]

    2021 arXiv

  41. [49]

    J. Ho, A. Jain, and P. Abbeel, Denoising diffusion probabilistic models, inAdvances in Neural Information Processing Systems, Vol. 33, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin (Curran Associates, Inc., 2020) pp. 6840–6851

    2020

  42. [50]

    Kawar, M

    B. Kawar, M. Elad, S. Ermon, and J. Song, Denoising diffusion restoration models, inAdvances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc., 2022) pp. 23593–23606

    2022

  43. [51]

    Chung, J

    H. Chung, J. Kim, M. T. McCann, M. L. Klasky, and J. C. Ye, Diffusion posterior sampling for general noisy inverse problems, inInternational Conference on Learning Representations(2023) arXiv:2209.14687

    2023 arXiv

  44. [52]

    Daras, H

    G. Daras, H. Chung, C.-H. Lai, Y. Mitsufuji, J. C. Ye, P. Milanfar, A. G. Dimakis, and M. Delbracio, A survey on diffusion models for inverse problems (2024), arXiv:2410.00083 [cs.LG]

    2024 arXiv

  45. [53]

    B. T. Feng, J. Smith, M. Rubinstein, H. Chang, K. L. Bouman, and W. T. Freeman, Score-based diffusion models as principled priors for inverse imaging, inProceedings of the IEEE/CVF international conference on computer vision(2023) pp. 10486–10497

    2023

  46. [54]

    Z. Wu, Y. Sun, Y. Chen, B. Zhang, Y. Yue, and K. L. Bouman, Principled probabilistic imaging using diffusion models as plug-and-play priors, inAdvances in Neural Information Processing Systems, Vol. 37, edited by A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomc...

    2024

  47. [55]

    Rombach, A

    R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer, High-resolution image synthesis with latent diffusion models, in2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)(2022) pp. 10674–10685

    2022

  48. [56]

    Ramesh, P

    A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen, Hierarchical text-conditional image generation with clip latents (2022), arXiv:2204.06125 [cs.CV]

    2022 arXiv

  49. [57]

    X. Li, J. Thickstun, I. Gulrajani, P. S. Liang, and T. B. Hashimoto, Diffusion-lm improves controllable text generation, in Advances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associate...

    2022

  50. [58]

    S. Gong, M. Li, J. Feng, Z. Wu, and L. Kong, Diffuseq: Sequence to sequence text generation with diffusion models, in International Conference on Learning Representations(2023) arXiv:2210.08933

    2023 arXiv

  51. [59]

    J. L. Watson, D. Juergens, N. R. Bennett, B. L. Trippe, J. Yim, H. E. Eisenach, W. Ahern, A. J. Borst, R. J. Ragotte, L. F. Milles,et al., De novo design of protein structure and function with rfdiffusion, Nature620, 1089 (2023)

    2023

  52. [60]

    Corso, H

    G. Corso, H. Stärk, B. Jing, R. Barzilay, and T. Jaakkola, Diffdock: Diffusion steps, twists, and turns for molecular docking, inInternational Conference on Learning Representations(2023) arXiv:2210.01776

    2023 arXiv

  53. [61]

    Makri, Feynman path integration in quantum dynamics, Computer Physics Communications63, 389 (1991)

    N. Makri, Feynman path integration in quantum dynamics, Computer Physics Communications63, 389 (1991)

    1991

  54. [62]

    Krilov and B

    G. Krilov and B. J. Berne, Real time quantum correlation functions. ii. maximum entropy numerical analytic continuation of path integral monte carlo and centroid molecular dynamics data, The Journal of Chemical Physics111, 9147 (1999)

    1999

  55. [63]

    H. B. Meyer, Transport properties of the quark-gluon plasma: a lattice qcd perspective, The European Physical Journal A 47, 86 (2011)

    2011

  56. [64]

    Aarts and A

    G. Aarts and A. Nikolaev, Electrical conductivity of the quark-gluon plasma: perspective from lattice qcd, The European Physical Journal A57, 118 (2021)

    2021

  57. [65]

    J. Song, A. Vahdat, M. Mardani, and J. Kautz, Pseudoinverse-guided diffusion models for inverse problems, inInternational Conference on Learning Representations(2023)

    2023

  58. [66]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, inInternational Conference on Learning Representations(2021) arXiv:2011.13456

    2021 arXiv

  59. [67]

    Lipman, R

    Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le, Flow matching for generative modeling, inInternational Conference on Learning Representations(2023) arXiv:2210.02747

    2023 arXiv

  60. [68]

    Peebles and S

    W. Peebles and S. Xie, Scalable diffusion models with transformers, inProceedings of the IEEE/CVF International Con- ference on Computer Vision (ICCV)(2023) pp. 4172–4182, arXiv:2212.09748

    2023 arXiv

  61. [69]

    J. Chen, J. Yu, C. Ge, L. Yao, E. Xie, Y. Wu, Z. Wang, J. Kwok, P. Luo, H. Lu, and Z. Li, Pixart-α: Fast training of diffusion transformer for photorealistic text-to-image synthesis, inInternational Conference on Learning Representations (2024) arXiv:2310.00426

    2024 arXiv

  62. [70]

    J. Song, C. Meng, and S. Ermon, Denoising diffusion implicit models, inInternational Conference on Learning Represen- tations(2021). 15

    2021

  63. [71]

    Karras, M

    T. Karras, M. Aittala, T. Aila, and S. Laine, Elucidating the design space of diffusion-based generative models, inAdvances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc., ...

    2022

  64. [72]

    Kingma, T

    D. Kingma, T. Salimans, B. Poole, and J. Ho, Variational diffusion models, inAdvances in Neural Information Processing Systems, Vol. 34, edited by M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan (Curran Associates, Inc., 2021) pp. 21696–21707

    2021

  65. [73]

    C. Lu, Y. Zhou, F. Bao, J. Chen, C. Li, and J. Zhu, Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models, Machine Intelligence Research22, 730 (2025)

    2025

  66. [74]

    S.Fort, H.Hu,andB.Lakshminarayanan,Deepensembles: Alosslandscapeperspective(2020),arXiv:1912.02757[stat.ML]

    2020 arXiv

  67. [75]

    Ovadia, E

    Y. Ovadia, E. Fertig, J. Ren, Z. Nado, D. Sculley, S. Nowozin, J. Dillon, B. Lakshminarayanan, and J. Snoek, Can you trust your model’s uncertainty? evaluating predictive uncertainty under dataset shift, inAdvances in Neural Information Processing Systems, Vol. 32, edited by H...

    2019

  68. [76]

    Belhasin, Y

    O. Belhasin, Y. Romano, D. Freedman, E. Rivlin, and M. Elad, Principal uncertainty quantification with spatial correlation for image restoration problems, IEEE Transactions on Pattern Analysis and Machine Intelligence46, 3321 (2024)

    2024

  69. [77]

    Y. Li, K. Zhang, J. Cao, R. Timofte, M. Magno, L. Benini, and L. Van Gool, LocalViT: Analyzing locality in vision transformers, in2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(2023) pp. 9598– 9605, arXiv:2104.05707. 16 SUPPORTING INFORMATION S...

    2023 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.