Pith. sign in

REVIEW 4 major objections 7 minor 1 cited by

Application of normalizing flows to nuclear many-body perturbation theory

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Normalizing flows cut Monte Carlo integration error in nuclear many-body calculations by an order of magnitude, even for pole-divergent response integrals.

desk verdict A solid methods extension of normalizing-flow importance sampling to principal-value response integrals, but the order-of-magnitude advantage over VEGAS needs a specified baseline before it can be trusted. read the letter →

arxiv 2412.19777 v1 pith:T6HAASZB submitted 2024-12-27 nucl-th

Pengsheng Wen , Jeremy W. Holt , Albany Blackburn This is my paper · ORCID
classification nucl-th
keywords normalizingflowsimportancesamplingnuclearmatterequationofstatedensityresponsefunctionCauchyprincipalvalueMonteCarlointegrationquasi-Montemany-bodyperturbationtheory
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

Normalizing flows are trainable changes of variables that map a simple base distribution into a distribution shaped like the absolute value of an integrand, and the authors aim to show this makes high-dimensional Monte Carlo integrals in nuclear many-body perturbation theory much cheaper and more precise. If the claim holds, equation-of-state and density-response calculations that now limit astrophysical simulations could instead be tabulated accurately across the densities, temperatures, and energy-momentum transfers a supernova or neutron-star merger demands. The paper reports relative errors at least an order of magnitude smaller than the standard VEGAS importance-sampling algorithm for both a second-order grand-canonical-potential integral and a complex-valued response function whose real part needs a Cauchy principal value. It further reports that a trained flow transfers to neighboring phase-space and kinematic points, and even to a different nuclear force model, with little loss of precision.

What carries the argument

The engine is a normalizing flow built from alternating coupling layers of rational quadratic splines: each layer leaves some coordinates fixed while neural networks generate the spline parameters that transform the others, so the Jacobian stays triangular and the full map stays invertible. Training minimizes a divergence between the transformed density and the normalized absolute integrand $|\psi(\vec x)|/\tilde I$, and the same samples then estimate the integral by the importance weight $\psi(\vec x)/p(\vec x)$. For the response function, the pole line $k_{\rm pole}$ from Eq. (39) is handled either by reflection, folding one side of the integrand across the pole so the divergences cancel, or by subtraction, removing the singular piece and integrating the logarithm analytically; each resulting finite integral is assigned its own flow.

What would settle it

Run VEGAS on the same two integrals with the same total computational budget, scanning batch size and adaptation settings, and compare true errors against converged Gaussian-quadrature references; if a tuned VEGAS closes the factor-of-ten gap, the central performance claim collapses.

Watch

Extended reading notes

Core claim

The central claim is that normalizing flows provide a practical, high-accuracy Monte Carlo sampling tool for the integrands that appear in nuclear matter perturbation theory, including irregular integrands with a pole line. For the seven-dimensional second-order contribution $\Omega^{(2)}$ to the grand canonical potential and for the zeroth-order density-density response function $\chi(q,\omega)$, the authors compare the batch relative error of flow-based importance sampling with VEGAS and find the flow errors are at least an order of magnitude smaller after training. They show that both the reflection and subtraction strategies for Cauchy principal value integrals can be implemented with flows, with reflection giving a moderate efficiency gain. They also demonstrate model transfer: a flow trained at one density and temperature, or at one momentum and energy transfer $(q,\omega)$, can be reused at nearby values, and the flow adapts when the nuclear potential is replaced by a chiral pion-exchange interaction.

Load-bearing premise

The order-of-magnitude improvement over VEGAS assumes the VEGAS baseline was given a fair and reasonably optimized configuration, but the paper does not report VEGAS's batch size, adaptation schedule, or tuning effort.

Editorial extensions

If this is right

  • The second-order grand canonical potential and the zeroth-order density response function can be integrated with roughly an order of magnitude smaller batch uncertainty than VEGAS at the same sample count.
  • A flow trained at one density, temperature, and momentum/energy transfer can be transferred to nearby values with little retraining, which is what a full astrophysical tabulation requires.
  • Trained flows remain effective when the interaction is changed to a chiral pion-exchange potential, so uncertainty quantification across nuclear force models becomes more feasible.
  • Quasi-random sample generators stabilize training and improve true precision, especially at small batch sizes, and their pointwise variance estimates overstate the actual error.
  • Both reflection and subtraction pole treatments work with flows; reflection is modestly more efficient for the cases tested.

Reading between the lines

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

  • The economic case for the method is amortization: training is expensive, but if a single trained model covers a neighborhood of densities, temperatures, and kinematic points, the cost of filling an equation-of-state and response table should fall; a direct wall-clock comparison over a full table would test this.
  • The quasi-random generator results suggest that for low-discrepancy samples the per-batch variance estimate is not a reliable uncertainty reporter, so downstream users should quote ensemble spreads over repeated runs rather than the pointwise formula.
  • The reflection-versus-subtraction comparison hints that how a principal value pole is decomposed matters more than how the flow learns the remaining integrand; one could test this on other pole geometries, such as curved pole lines, to see whether the ordering persists.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper applies normalizing-flow importance sampling to two classes of integrals from nuclear many-body perturbation theory: the seven-dimensional second-order contribution to the finite-temperature grand canonical potential of symmetric nuclear matter (Eq. 32) and the two-dimensional zero-temperature density-density response function, whose real part requires principal-value integration (Eqs. 37-43). The authors compare eight loss functions and four random-number generators (pseudo-random, Sobol, Halton, Lattice), benchmark the EOS integral against converged Gaussian quadrature, and test transferring trained models across temperature, density, momentum/energy transfer, and one nuclear-potential change (scalar-boson exchange to chiral NLO pion exchange). They report that normalizing flows reach batch relative uncertainties at least an order of magnitude below VEGAS, that quasi-random generators improve stability and accuracy, and that point-estimate uncertainties for quasi-random sampling are conservative relative to the spread of independent estimates.

Significance. If the central performance claim survives scrutiny, the paper provides a useful demonstration that learned importance-sampling densities can handle both regular 7D MBPT integrands and singular principal-value integrands, with plausible transferability across thermodynamic and kinematic variables. The strengths are genuine: the EOS results are checked against converged Gaussian quadrature (Figs. 5, 7); the QRNG point-estimate-vs-true-error analysis (Figs. 4-5) is careful and correct in spirit; and treating the pole via both reflection and subtraction is a useful methodological comparison. The main weaknesses are that the VEGAS baseline is not specified, the response-function part lacks an analogous exact benchmark, the normalizing-flow architecture and interaction parameters are unreported, and the zero-fine-tuning transfer claim overstates the protocol actually used. No code is provided. As it stands, the order-of-magnitude claim is plausible but not yet reproducible; the practical 'efficient tabulations' promise is also asserted rather than costed, especially since the response-function examples are two-dimensional integrals where deterministic quadrature would be competitive.

major comments (4)
  1. [Section III B/III D; Figs. 3, 6, 11, 12] The central 'order of magnitude' claim over VEGAS is not reproducible as presented because the VEGAS configuration is never stated: the manuscript does not report VEGAS's samples per iteration, number of adaptation steps, damping parameter, grid binning, or whether it used the same batch sizes as the normalizing flows (5000 in Fig. 3, 1500 in Fig. 11). The claim in Section III B that 'VEGAS converges to its precision limit within about 100 iterations' is also inconsistent with the accumulation formula (Eqs. (33)-(34)), which for unbiased independent batches would give sigma_t proportional to 1/sqrt(N_iter); the observed plateau is better interpreted as a bias floor, and Fig. 7 confirms that VEGAS's reported uncertainties severely underpredict its true error at low temperature. Please report the VEGAS settings and present the comparison on the basis of true errors (against Gaussian quadrature or analytic results), not only the reported per-batch variance.
  2. [Section III B/III D; Section IV] The transferability claim is overstated relative to what is actually demonstrated. The conclusion states that a trained model 'can be transferred to nearby phase space points without additional fine training', but Section III B says 'After each new modification, we perform a 100-iteration update for the new models' (Fig. 6), and the Section III D protocol for Fig. 12 runs 100 iterations at the new kinematic point with continued Adam updates, since Section II d updates the neural networks at every iteration. These are warm-started retraining runs, not zero-update transfers. Please correct the wording, and ideally quantify the warm-start benefit by comparing against training from scratch at the target point with the same iteration budget.
  3. [Section III C/III D; Figs. 11-12] Unlike the EOS calculation, which is validated against converged Gaussian quadrature in Figs. 5 and 7, the response-function results in Figs. 11 and 12 are compared only against VEGAS. Since the conclusion claims an order-of-magnitude reduction of 'errors' for response-function calculations, and since Fig. 7 demonstrates that VEGAS-reported uncertainties can be unreliable, an external accuracy check is needed here. The reduced integral in Eq. (37) is only two-dimensional and the zero-temperature free-gas response has a known analytic (Lindhard) form, so a converged quadrature or analytic benchmark is straightforward to add and would make the response-function claim comparable in rigor to the EOS claim.
  4. [Section II; Section III A] The numerical setup is under-specified, which prevents reproduction and makes the comparison to VEGAS hard to interpret as a fair, tuned comparison. The text gives no values for the number of coupling layers, the number of spline bins K, the neural-network widths and depths, activation functions, or the learning-rate schedule (only the initial learning rate appears in figure captions). Likewise, the parameters of the scalar-boson interaction (g, m_phi in Eq. (29)) and the cutoff Lambda in Eq. (31) are never given, although all EOS results depend on them. These hyperparameters and coupling parameters, together with the training protocol, should be reported (or the code released) before the quantitative claims can be independently assessed.
minor comments (7)
  1. [Eq. (4)] Eq. (4) is garbled as printed: the sample-variance estimator should be sigma^2 approximately [1/(N(N-1))] times the sum over i of (psi(x_i)/p(x_i) - mean)^2; the current rendering has lost the 1/N normalization. Please correct.
  2. [Figure captions] Captions of Figures 3 and 4 state 'n = 0.16 fm^{-1}'; the density unit should be fm^{-3} (Figure 5 is correct).
  3. [Table I] Table I is difficult to parse and appears to contain typos: the 'rdkl' entry as printed is an unweighted average of log ratios rather than a KL-type divergence, and the 'var' entry as rendered reduces to N^{-1} sum_i (q_i/p_i)^2 - N^{-1} sum_i (q_i/p_i)^2, which vanishes identically; please provide the continuous definitions and the exact implemented estimators.
  4. [Figure 13] Figure 13 contains placeholder glyphs such as 'fm[box]1' and 'fm[box]2'; these appear to be unresolved LaTeX placeholders and should be replaced with proper units (fm^{-1}, fm^{-2}).
  5. [Section III D] Section III D, transfer paragraph: the sentence 'After the 100 iterations, new models are generated automatically' is ambiguous; please clarify whether the transferred model continues training at each new point or is re-initialized.
  6. [Section IV] Section IV: the 'efficient tabulations' motivation should be reconciled with the dimensionality of the examples: the response-function integrals in Eqs. (37) and (45) are two-dimensional after angular reduction, and for such low-dimensional integrals a cost comparison with deterministic quadrature (which the paper itself uses as the benchmark) would strengthen the practical-efficiency claim.
  7. [Section III B] Section III B, first paragraph: 'Since low temperatures lead to a steep integrands' should read 'a steep integrand' or 'steep integrands'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flow-based importance sampling is validated against independent Gaussian quadrature, and self-citations are contextual rather than load-bearing.

full rationale

The paper's central numerical claims are not derived from their own conclusions. The normalizing flow is trained to approximate the normalized absolute value of a fixed integrand (Eqs. 22-23), which is the standard importance-sampling target, and the resulting Monte Carlo estimates are benchmarked against well-converged Gaussian quadrature (Figure 5 and the Section III B text) and against direct VEGAS runs, so the reported integral values and error bars rest on an external reference rather than on the trained model itself. Citations to the authors' earlier PRL [47] provide motivation, a previously published integrand formula (Eq. 32), and earlier illustrative precision numbers; they are not used to establish the new transferability or order-of-magnitude claims, which are demonstrated in this paper in Figures 3-4, 6, 11-12 with per-point comparisons and chiral-potential tests. The possible concern that the VEGAS baseline may be undertuned is a comparison-fairness or correctness issue, not a circularity; no step in the manuscript defines a predicted quantity in terms of the fitted model parameters, and no uniqueness theorem is imported from prior work to force the method. Therefore no circular step is present.

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

No new physics entities are introduced. The method rests on standard importance sampling, normalizing flow theory, and standard many-body perturbation formulas. The main unstated burden is implementation detail: hyperparameters, architecture, and baseline configurations are not fully specified, which affects reproducibility more than circularity.

free parameters (4)
  • Normalizing flow hyperparameters = learning rate 1e-3/1e-4, batch sizes 5000/1500/500/1000, up to 35,000 iterations
    Algorithmic settings chosen by hand; they affect convergence and reported precision but are not physical parameters fitted to nuclear data.
  • Spline bins and flow layers = not stated
    Number of rational-quadratic spline bins K, coupling layers, masking scheme, and hidden-layer widths are not specified in the text; needed for exact reproduction.
  • VEGAS configuration = not stated
    VEGAS comparison is central to the claimed improvement, but its grid size, number of iterations, and adaptation settings are not given.
  • Toy model parameters = not stated in text except Eq. 31; g, m_phi unspecified
    The scalar boson exchange potential of Eq. (29) requires g, m_phi and cutoff Lambda values; these are inputs, not fitted to data, but are needed to reproduce Figure 3.
assumptions (5)
  • standard math Importance sampling estimator Eq. (2) is unbiased and Eq. (4) estimates the variance.
    Standard Monte Carlo theory.
  • standard math Rational quadratic spline transformations are invertible and have tractable Jacobians.
    Built into the normalizing flow construction; equations (18)-(21).
  • domain assumption The many-body perturbation expansion of the grand canonical potential, Eqs. (26) and (27), is valid for nuclear matter.
    Standard quantum many-body theory; references [67-70].
  • domain assumption The zeroth-order density response function, Eq. (36), can be decomposed using Sokhotski-Plemelj into principal value and delta function parts.
    Standard Lindhard function derivation; Eqs. (37)-(38).
  • domain assumption The trained flow distribution closely approximates the normalized absolute integrand.
    The whole method assumes the flow model class can express the target distribution sufficiently well after training; validated only on chosen examples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Application of normalizing flows to nuclear many-body perturbation theory." pith.science (2026). https://pith.science/paper/T6HAASZB

@misc{pith2026241219777,
  author       = {Pith},
  title        = {Pith review of: Application of normalizing flows to nuclear many-body perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6HAASZB}},
  note         = {Machine review of arXiv:2412.19777}
}
read the original abstract

Many-body perturbation theory provides a powerful framework to study the ground state and thermodynamic properties of nuclear matter as well as associated single-particle potentials and response functions within a systematic order-by-order expansion. However, computational challenges can emerge beyond the lowest orders of perturbation theory, especially when computing both single-particle potentials and response functions, which in general are complex-valued and require Cauchy principal value calculations of high-dimensional integrals. We demonstrate that normalizing flows are suitable for Monte Carlo importance sampling of both regular and irregular functions appearing in nuclear many-body calculations. Normalizing flows are a class of machine learning models that can be used to build and sample from complicated distributions through a bijective mapping from a simple base distribution. Furthermore, a well-trained model for a certain target integrand can be efficiently transferred to calculate related integrals with varying physical conditions. These features can enable more efficient tabulations of nuclear physics inputs to numerical simulations of supernovae and neutron star mergers across varying physical conditions and nuclear force models.

Figures

Figures reproduced from arXiv: 2412.19777 by the authors.

Figure 1
Figure 1. FIG. 1. Work flow for normalizing flows. A sample [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The diagrammatic 2nd-order normal contribution to [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Top panel) Batch relative error averaged over 1000 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Batch relative error for Ω [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Ratio of the actual error to the variance-estimated [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Zeroth-order static density-density response func [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The distribution of transformed samples at different training iterations within Region I. The right 3D figure shows [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The average relative pointwise uncertainties (top) [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Monte Carlo estimate for the real part of 0th-order response function Re [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. PMM-IMSRG emulator for the nuclear equation of state with quantified uncertainties

    nucl-th 2026-07 conditional novelty 6.0 of 10

    A parametric-matrix-model emulator reproduces IMSRG nuclear-matter energies with calibrated conformal-prediction error bars, enabling Bayesian fitting of three-nucleon couplings to saturation properties.

Reference graph

Works this paper leans on

74 extracted references · 61 canonical work pages · cited by 1 Pith paper

  1. [1]

    H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Nucl. Phys. A637, 435 (1998)

    work page 1998

  2. [2]

    Lattimer and D

    J. Lattimer and D. Swesty, Nucl. Phys. A535, 331 (1991)

    work page 1991

  3. [3]

    D. G. Yakovlev, O. Y. Gnedin, M. E. Gusakov, A. D. Kaminker, K. P. Levenfish, and A. Y. Potekhin, Nuclear Physics A Proceedings of the 22nd International Nuclear Physics Conference (Part 2), 752, 590 (2005)

    work page 2005

  4. [4]

    Thermal Effects in Dense Matter Beyond Mean Field Theory

    C. Constantinou, S. Lalit, and M. Prakash, Int. J. Mod. Phys. E 26, 1740005 (2017), arXiv:1608.04713 [nucl-th]

    work page Pith review arXiv 2017

  5. [5]

    E. F. Brown, A. Cumming, F. J. Fattoyev, C. J. Horowitz, D. Page, and S. Reddy, Phys. Rev. Lett. 120, 182701 (2018)

    2018

  6. [6]

    Sumiyoshi, Eur

    K. Sumiyoshi, Eur. Phys. J. A 57, 331 (2021)

    work page 2021

  7. [7]

    Annala, T

    E. Annala, T. Gorda, E. Katerini, A. Kurkela, J. N¨ attil¨ a, V. Paschalidis, and A. Vuorinen, Phys. Rev. X 12, 011058 (2022)

    work page 2022

  8. [8]

    Kumar et al

    R. Kumar et al. (MUSES), arXiv:2303.17021 (2023)

    arXiv 2023

Show all 74 references
  1. [9]

    X. Du, A. W. Steiner, and J. W. Holt, Phys. Rev. C 99, 025803 (2019)

    2019

  2. [10]

    Iwamoto and C

    N. Iwamoto and C. J. Pethick, Phys. Rev. D 25, 313 (1982)

    1982

  3. [11]

    R. F. Sawyer, Phys. Rev. C 40, 865 (1989)

    1989

  4. [12]

    nsam- ple

    At the beginning of the model transfer, the trained models are used to estimate the density response func- tion for 100 iterations with 1500 samples per iteration at the (q, ω) position where they are trained. After the 100 iterations, new models are generated automatically an...

  5. [13]

    Reddy, M

    S. Reddy, M. Prakash, and J. M. Lattimer, Phys. Rev. C 58, 013009 (1998)

    1998

  6. [14]

    Burrows and R

    A. Burrows and R. F. Sawyer, Phys. Rev. C 58, 554 (1998)

    1998

  7. [15]

    Lovato, O

    A. Lovato, O. Benhar, S. Gandolfi, and C. Losa, Phys. Rev. C 89, 025804 (2014)

    2014

  8. [16]

    Kotake, T

    K. Kotake, T. Takiwaki, T. Fischer, K. Nakamura, and G. Mart ´ ınez-Pinedo, ApJ853, 170 (2018)

    2018

  9. [17]

    C. Shen, U. Lombardo, N. Van Giai, and W. Zuo, Phys. Rev. C 68, 055802 (2003)

    2003

  10. [18]

    Pastore, M

    A. Pastore, M. Martini, D. Davesne, J. Navarro, S. Goriely, and N. Chamel, Phys. Rev. C 90, 025804 (2014)

    2014

  11. [19]

    C. J. Horowitz, M. A. P´ erez-Garc ´ ıa, D. K. Berry, and J. Piekarewicz, Phys. Rev. C 72, 035801 (2005)

    2005

  12. [20]

    E. Shin, E. Rrapaj, J. W. Holt, and S. K. Reddy, Phys. Rev. C 109, 015804 (2024)

    2024

  13. [21]

    L. F. Roberts, S. Reddy, and G. Shen, Phys. Rev. C 86, 065803 (2012)

    2012

  14. [22]

    Li, B.-J

    B.-A. Li, B.-J. Cai, L.-W. Chen, and J. Xu, Prog. Part. Nucl. Phys. 99, 29–119 (2018)

    2018

  15. [23]

    Rrapaj, J

    E. Rrapaj, J. W. Holt, A. Bartl, S. Reddy, and A. Schwenk, Phys. Rev. C 91, 035806 (2015)

    2015

  16. [24]

    Mart ´ ınez-Pinedo, T

    G. Mart ´ ınez-Pinedo, T. Fischer, A. Lohs, and L. Huther, Phys. Rev. Lett. 109, 251104 (2012)

    2012

  17. [25]

    Donati, P

    P. Donati, P. M. Pizzochero, P. F. Bortignon, and R. A. Broglia, Phys. Rev. Lett. 72, 2835 (1994)

    1994

  18. [26]

    Yasin, S

    H. Yasin, S. Sch¨ afer, A. Arcones, and A. Schwenk, Phys. Rev. Lett. 124, 092701 (2020)

    2020

  19. [27]

    Ishizuka, A

    C. Ishizuka, A. Ohnishi, K. Tsubakihara, K. Sumiyoshi, and S. Yamada, J. Phys. G: Nucl. Part. Phys.35, 085201 (2008)

    2008

  20. [28]

    Sotani and A

    H. Sotani and A. Dohi, Phys. Rev. D 105, 023007 (2022), 2201.00648

    2022 arXiv

  21. [29]

    A. W. Steiner, M. Hempel, and T. Fischer, ApJ 774, 17 (2013)

    2013

  22. [30]

    Oertel, C

    M. Oertel, C. Providˆ encia, F. Gulminelli, and A. R. Raduta, J. Phys. G: Nucl. Part. Phys. 42, 075202 (2015)

    2015

  23. [31]

    N. H. Tan, D. T. Loan, D. T. Khoa, and J. Margueron, Phys. Rev. C 93, 035806 (2016)

    2016

  24. [32]

    Hempel and J

    M. Hempel and J. Schaffner-Bielich, Nuclear Physics A 837, 210 (2010)

    2010

  25. [33]

    W. Zuo, A. Lejeune, U. Lombardo, and J. F. Mathiot, The European Physical Journal A - Hadrons and Nuclei 14, 469 (2002)

    2002

  26. [34]

    W. Zuo, A. Lejeune, U. Lombardo, and J. F. Mathiot, Nuclear Physics A 706, 418 (2002)

    2002

  27. [35]

    Wellenhofer, J

    C. Wellenhofer, J. W. Holt, and N. Kaiser, Phys. Rev. C 93, 055802 (2016)

    2016

  28. [36]

    Wen and J

    P. Wen and J. W. Holt, Phys. Rev. C103, 064002 (2021)

    2021

  29. [37]

    Gezerlis, I

    A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler, A. Nogga, and A. Schwenk, Phys. Rev. Lett. 111, 032501 (2013)

    2013

  30. [38]

    Akmal, V

    A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, 15 Phys. Rev. C 58, 1804 (1998)

    1998

  31. [39]

    I. Tews, S. Gandolfi, A. Gezerlis, and A. Schwenk, Phys. Rev. C 93, 024305 (2016)

    2016

  32. [40]

    Wlaz lowski, J

    G. Wlaz lowski, J. W. Holt, S. Moroz, A. Bulgac, and K. J. Roche, Phys. Rev. Lett. 113, 182503 (2014)

    2014

  33. [41]

    Alexandru, P

    A. Alexandru, P. Bedaque, E. Berkowitz, and N. C. Warrington, Phys. Rev. Lett. 126, 132701 (2021)

    2021

  34. [42]

    Buraczynski and A

    M. Buraczynski and A. Gezerlis, Phys. Rev. Lett. 116, 152501 (2016)

    2016

  35. [43]

    Holzmann, F

    M. Holzmann, F. Calcavecchia, D. M. Ceperley, and V. Olevano, Phys. Rev. Lett. 131, 186501 (2023)

    2023

  36. [44]

    Y.-Z. Ma, Z. Lin, B.-N. Lu, S. Elhatisari, D. Lee, N. Li, U.-G. Meißner, A. W. Steiner, and Q. Wang, Phys. Rev. Lett. 132, 232502 (2024)

    2024

  37. [45]

    Davesne, J

    D. Davesne, J. W. Holt, A. Pastore, and J. Navarro, Phys. Rev. C 91, 014323 (2015)

    2015

  38. [46]

    Roggero, F

    A. Roggero, F. Pederiva, and G. Orlandini, Phys. Rev. B 88, 094302 (2013)

    2013

  39. [47]

    Brady, P

    J. Brady, P. Wen, and J. W. Holt, Phys. Rev. Lett. 127, 062701 (2021)

    2021

  40. [48]

    G. Shen, S. Gandolfi, S. Reddy, and J. Carlson, Phys. Rev. C 87, 025802 (2013)

    2013

  41. [49]

    Tabak and C

    E. Tabak and C. Turner, Commun. Pure Appl. Math.66, 145 (2013)

    2013

  42. [50]

    E. G. Tabak and E. Vanden-Eijnden, Commun. Math. Sci. 8, 217 (2010)

    2010

  43. [51]

    Hahn, Comp

    T. Hahn, Comp. Phys. Comm. 168, 78 (2005)

    2005

  44. [52]

    Papamakarios, E

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mo- hamed, and B. Lakshminarayanan, arXiv:1912.02762 (2019)

    2019 arXiv

  45. [53]

    Drischler, K

    C. Drischler, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 122, 042501 (2019)

    2019

  46. [55]

    D. P. Kingma and P. Dhariwal, arXiv:1807.03039 [stat.ML] (2018), 10.48550/ARXIV.1807.03039

    2018 doi arXiv

  47. [56]

    E. G. Tabak and C. V. Turner, Communications on Pure and Applied Mathematics 66, 145 (2013)

    2013

  48. [57]

    Wirnsberger, A

    P. Wirnsberger, A. J. Ballard, G. Papamakarios, S. Aber- crombie, S. Racani` ere, A. Pritzel, D. Jimenez Rezende, and C. Blundell, J. Chem. Phys. 153, 144112 (2020)

    2020

  49. [58]

    G. Yang, X. Huang, Z. Hao, M.-Y. Liu, S. Belongie, and B. Hariharan, arXiv:1906.12320 [cs] (2019), 1906.12320

    2019 arXiv

  50. [59]

    C. Gao, S. H¨ oche, J. Isaacson, C. Krause, and H. Schulz, Phys. Rev. D 101, 076002 (2020)

    2020

  51. [60]

    Kanwar, M

    G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racani` ere, D. J. Rezende, and P. E. Shana- han, Phys. Rev. Lett. 125, 121601 (2020)

    2020

  52. [61]

    P. Wen, J. W. Holt, and M. Li, arXiv:2306.13007 (2023)

    2023 arXiv

  53. [62]

    Hajij, G

    M. Hajij, G. Zamzmi, R. Paul, and L. Thukar, in 2022 IEEE Healthcare Innovations and Point of Care Tech- nologies (HI-POCT) (2022) pp. 46–49

    2022

  54. [63]

    J. A. GREGORY and R. DELBOURGO, IMA Journal of Numerical Analysis 2, 123 (1982)

    1982

  55. [64]

    Durkan, A

    C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, arXiv:1906.04032 [cs, stat] (2019)

    2019 arXiv

  56. [65]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” (2014)

    2014

  57. [66]

    C. Gao, J. Isaacson, and C. Krause, Machine Learning: Science and Technology 1, 045023 (2020)

    2020

  58. [67]

    Kohn and J

    W. Kohn and J. M. Luttinger, Phys. Rev. 118, 41 (1960)

    1960

  59. [68]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, in Advances in Neural Information Pro-...

    2019

  60. [69]

    Wellenhofer, arXiv:1707.09222 [nucl-th] (2017)

    C. Wellenhofer, arXiv:1707.09222 [nucl-th] (2017)

    2017 arXiv

  61. [70]

    J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960)

    1960

  62. [71]

    J. W. Holt, M. Kawaguchi, and N. Kaiser, Frontiers in Physics 8, 100 (2020)

    2020

  63. [72]

    Wellenhofer, J

    C. Wellenhofer, J. W. Holt, N. Kaiser, and W. Weise, Phys. Rev. C 89, 064009 (2014)

    2014

  64. [73]

    G. P. Lepage, Journal of Computational Physics 439, 110386 (2021), 2009.05112

    2021 arXiv

  65. [74]

    S.-C. T. Choi, F. J. Hickernell, R. Jagadeeswaran, M. J. McCourt, and A. G. Sorokin, arXiv:2102.07833 [cs, math] (2021), 2102.07833

    2021 arXiv

  66. [100]

    maskings

    The precision of importance sampling integration is strongly dependent on how well the sampling distribution matches the normalized absolute value of the integrand. Normalizing flows map a simple base distribution (e.g., a uniform or Gaussian distribution) to a more complicate...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.