Pith. sign in

REVIEW 3 major objections 8 minor 32 references

Estimating theoretical uncertainties of the two-nucleon observables by using backpropagation

T0 review · 3 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read This paper claims that backpropagation can turn uncertainties in a two-nucleon potential into error bars on observables, and that those error bars match direct sampling.

desk verdict Genuinely new and mostly validated backpropagation approach to two-nucleon uncertainty propagation, but the Table 3 quadrature-node blowups require qualifying the abstract's 'very good agreement' claim. read the letter →

arxiv 2412.06304 v2 pith:UDL3GASG submitted 2024-12-09 nucl-th

K. Topolnicki , R. Skibiński , J. Golak This is my paper · ORCID
classification nucl-th PACS 03.65.Ge24.10.-i
keywords backpropagationautomaticdifferentiationtheoreticaluncertaintiesnucleon-nucleonscatteringdeuteronboundstateLippmann-Schwingerequationerrorpropagationnuclearpotential
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 backpropagation, the automatic-differentiation technique behind machine learning, can compute gradients of two-nucleon observables with respect to every value in the lattice that defines the nuclear potential, and that those gradients turn potential uncertainties into observable uncertainties through first-order error propagation. The authors implement the deuteron bound-state problem and the nucleon-nucleon scattering problem as differentiable tensor computations, then propagate uncorrelated Gaussian errors in the scalar potential functions to the binding energy and phase shifts. They validate the result by sampling many perturbed potentials and find good agreement with the backpropagated standard deviations. This makes uncertainty estimates that would otherwise require many repeated calculations available from a single backward pass, even when there are hundreds of thousands of input parameters.

What carries the argument

The carrying object is the differentiable implementation of the two-nucleon calculation. The potential is written as a linear combination of spin–momentum operators with scalar coefficient functions \(v_i(p',p,x)\) stored as a discrete array \(V_{ijkl}\); partial-wave matrix elements are built by tensor contractions with precomputed angular-integration arrays, and the resulting eigenproblem or linear system is solved with differentiable library routines. Because every operation is recorded in a computation graph, backpropagation computes the gradient of an observable with respect to all entries of \(V\) in one pass, and the uncorrelated-error formula \(\$\sigma$(O) = \sqrt{\sum_{ijkl} \$\sigma$(V_{ijkl})^2 (\partial O/\partial V_{ijkl})^2}\) turns those gradients into uncertainties.

What would settle it

Sweep the on-shell momentum \(p_0\) continuously between two adjacent quadrature nodes of the same grid and compare the backpropagated uncertainty with the Monte Carlo sampled uncertainty at each point; a sharp spike in the ratio at any node would confirm that the method's accuracy depends on grid placement rather than on the physics alone.

Watch

Extended reading notes

Core claim

The central claim is that error propagation from the scalar functions defining a two-nucleon potential to observables can be done through the calculation itself: once the eigenvalue equation for the deuteron and the Lippmann–Schwinger equation for scattering are written as differentiable tensor operations, the backward pass yields \(\partial O/\partial V_{ijkl}\) for each lattice value of each scalar function, and the variance formula with \(\$\sigma$(V_{ijkl}) = \$\alpha$(i)|V_{ijkl}|\) gives the observable uncertainty. Direct Monte Carlo sampling of the potential values confirms the result: for the \(^1S_0\) and \(^1P_1\) channels and the \(^3S_1\)-\(^3D_1\) and \(^3P_2\)-\(^3F_2\) coupled channels, the backpropagated and sampled standard deviations agree closely in most cases, and the sampled eigenvalue distributions are approximately Gaussian. The method is presented as especially efficient for many inputs and few outputs, and as a tool for identifying which parts of a potential drive the uncertainty of a given observable.

Load-bearing premise

The method assumes the momentum grid used in the scattering calculation keeps the on-shell momentum away from every quadrature node; when that fails, the backpropagated uncertainty becomes unreliable, as Table 3 shows at \(p_0 = 0.5\) and \(4.0\) fm\(^{-1}\).

Editorial extensions

If this is right

  • Deuteron binding-energy uncertainty scales roughly linearly with the assumed relative error \(\alpha\) for the values tested, so the backpropagated error bars can be rescaled when the input uncertainty changes.
  • For the uncoupled \(^1P_1\) channel at \(p_0 = 2\) fm\(^{-1}\), backpropagated and sampled phase-shift uncertainties agree across \(\alpha\) from 1% to 10% (e.g., 4.8 versus 4.6 degrees at \(\alpha = 0.1\)).
  • For the coupled \(^3S_1\)-\(^3D_1\) and \(^3P_2\)-\(^3F_2\) channels, backpropagation reproduces the sampled uncertainties for phase shifts and mixing parameters at \(p_0 = 2\) fm\(^{-1}\).
  • The gradient map (e.g., \(\partial\lambda/\partial v_1\) over momentum values) shows which regions of a scalar function most influence the binding energy, enabling targeted checks of potential models.
  • When the on-shell momentum sits close to a quadrature node, the backpropagated uncertainty blows up (3225.6 versus 0.4 degrees at \(p_0 = 0.5\) fm\(^{-1}\)), so the method doubles as a numerical-quality diagnostic.

Reading between the lines

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

  • The same gradient machinery could propagate correlated errors among lattice values, which the paper notes as future work, making realistic parameter covariances from chiral potentials accessible without large sample ensembles.
  • Because the method only requires the calculation to be an eigenproblem or a linear solve, it could be carried over to Faddeev three-nucleon equations and to nuclear structure calculations implemented in other languages, as the authors suggest.
  • Gradients from one backward pass could be reused to build linear emulators of observables around a reference potential, making uncertainty estimates for many energies or angles almost free once the reference calculation is done.
  • The near-node blow-up could be turned into a grid-adaptation criterion: choose quadrature points so that the backpropagated uncertainty for a test observable is stable under small grid shifts.
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

3 major / 8 minor

Summary. The manuscript presents a computational method for estimating uncertainties in the deuteron binding energy and in nucleon-nucleon phase shifts induced by uncertainties in the scalar functions that define a two-nucleon potential. The potential is represented on a discrete momentum-angle grid, and the deuteron eigenvalue problem and the Lippmann-Schwinger equation are implemented in PyTorch so that backpropagation can compute gradients of the selected eigenvalues and phase shifts with respect to every grid value of the potential. Assuming independent Gaussian errors with relative standard deviations alpha(i) on the scalar-function values, the authors propagate these errors through a first-order formula and compare the resulting uncertainties with direct Monte Carlo sampling of the same error model. Good agreement is reported for the deuteron eigenvalue and for most of the tested scattering channels, but Table 3 shows strong disagreements for p0 = 0.5 and 4.0 fm^-1, which the authors attribute to proximity to Gaussian quadrature nodes.

Significance. If the method is reliable, it is a useful addition to uncertainty quantification in few-nucleon calculations: it avoids repeated solves, works with hundreds of thousands of input parameters, and the code and plot data are publicly available. The approach is not circular, because the alpha(i) error model is an external input and the validation compares two ways of evaluating the same uncertainty model. However, the central validation claim is currently overstated: the discrepancies in Table 3 are not compatible with the abstract's unqualified 'very good agreement', and the deuteron binding-energy uncertainty is not validated by direct sampling of the fitted quantity. These issues are local and fixable, but they are load-bearing for the paper's main message.

major comments (3)
  1. [Abstract; §3.2, Table 3] The central validation claim is not supported by the full Table 3. At p0 = 0.5 fm^-1 the backpropagated uncertainty is 3225.6 degrees versus 0.4 degrees from sampling, and at p0 = 4.0 fm^-1 it is 217.8 degrees versus 26.0 degrees. The text's explanation (proximity to quadrature nodes 0.4999 and 4.0200 fm^-1, in the context of Eqs. (23)-(25)) is plausible, but the prescription that a large backpropagated uncertainty can be used as a signal is post hoc and provides no threshold or test. Because the paper presents this as a practical tool, please (i) qualify the abstract and the 'very good agreement' statement to exclude near-node momenta, (ii) provide an automatic diagnostic, for example min_k |p0 - p_k| or the condition number of the discretized A matrix in Eq. (25), and (iii) demonstrate that the diagnostic flags the failed rows and that refining or shifting the quadrature grid removes the discrepancy.
  2. [§2.2, Table 1 and Figs. 2, 4, 6] The sampling validation in the deuteron section is performed for the eigenvalue lambda at a single energy, not for the deuteron binding energy E_d reported in Table 1. The uncertainty sigma(E_d) is obtained from a weighted least-squares fit in which the errors of lambda(E_i) are treated as independent, although the lambda(E_i) values share the same random potential perturbation and are therefore correlated. The claimed validation by direct sampling therefore does not currently cover E_d. Please sample whole lambda(E) curves and compare the distribution of fitted E_d with the reported sigma(E_d), or restrict the validation claim to the single-energy eigenvalue uncertainties.
  3. [§3.2, Table 3] The p0 = 4.0 fm^-1 row shows that direct sampling is itself sensitive to the numerical grid, with sigma_samples = 26.0 degrees versus 1.1 and 1.6 degrees at the neighboring p0 = 3 and 5 fm^-1. The issue is therefore not only a backpropagation artifact but also a property of the discretized principal-value integral. The paper should include a grid-convergence study for at least one near-node energy and one safe energy, reporting how both sigma_backpropagation and sigma_samples change when the number of quadrature points or the quadrature rule is varied. Such a study would give users a practical way to decide when a computed uncertainty is trustworthy.
minor comments (8)
  1. [Abstract] The phrase 'thePythonpytorchlibrary' should read 'the Python pytorch library'.
  2. [Tables] Several table captions contain 'T able' instead of 'Table'; this should be corrected throughout.
  3. [§2.1] The text following Eq. (11) spells 'Clebsh-Gordan'; the standard spelling is 'Clebsch-Gordan'.
  4. [§3.2, Table 3] In Table 3, the units of p0 should appear in the column header (for example, 'p0 [fm^-1]') rather than only being explained in the text.
  5. [§3.1] The type of Gaussian quadrature and the momentum cutoff used for the scattering results in Tables 2-5 are not stated; Section 3.1 should report the same grid parameters that are given for the deuteron calculation.
  6. [Fig. 2 caption] The caption of Figure 2 states that 48 momentum points were used in the interval (0, 10) fm^-1, while Figure 1 uses (0, 20) fm^-1; please clarify whether these are different grids and why.
  7. [References] Reference [7] lists 'H. Krebs' twice in the author list; the correct list should be 'P. Reinert, H. Krebs, and E. Epelbaum'.
  8. [§2.1] Equation (17) is presented without a derivation or citation; a standard reference for first-order error propagation would help readers assess the approximation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uncertainty estimates are validated by direct Monte Carlo sampling of the same assumed input distribution, which is a self-consistency check rather than a construction that reduces to its inputs.

full rationale

The paper's derivation chain is self-contained. The input is a lattice of Bonn-B potential scalar-function values, with assumed uncorrelated Gaussian uncertainties sigma(V_ijkl) = alpha(i)|V_ijkl|, as stated in Eq. (16). Backpropagation through the PyTorch implementation yields gradients of the deuteron eigenvalue and scattering phase shifts with respect to these input values, and uncertainties are propagated by the first-order formula in Eq. (17). The validation consists of drawing new V' tensors from the same assumed normal distribution, recomputing the observables, and comparing the sample standard deviation with the backpropagated value. This is a check of the linear error-propagation approximation against direct Monte Carlo integration over the same assumed input distribution; no observable is used to infer its own uncertainty, no parameter is fitted to the quantity being predicted, and no external claim is derived from the chosen alpha values. The good agreement in Tables 2, 4, and 5 and in the deuteron figures reflects the accuracy of the linearization for small relative uncertainties, not an identity by construction. The two failures in Table 3 at p0 = 0.5 and 4.0 fm^-1 are numerical pathologies caused by the on-shell momentum lying close to Gaussian quadrature nodes; the authors explicitly attribute the discrepancy to this proximity and suggest using the large backpropagated value as a numerical warning. That is a robustness and correctness concern about the unqualified 'very good agreement' statement in the abstract, but it is not circular reasoning. The self-citation to Ref. [30] supplies the partial-wave decomposition formulas used in the implementation; it is a methodological reference, not a load-bearing uniqueness theorem, and the code is publicly available for independent verification. No circular step satisfying the required standard of exhibiting a specific equation that reduces to its own input is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard nuclear physics input (Bonn-B potential) and hand-chosen uncertainty and grid parameters. No new physical entities are introduced. The reported uncertainties are with respect to an assumed, uncorrelated error model, not a data-based determination.

free parameters (3)
  • alpha_i (relative standard deviation of scalar functions) = 0.01 to 0.1 (and 0.2, 0.4 in single-function tests)
    Chosen by hand to simulate theoretical uncertainty; the absolute error estimates scale with alpha. The authors state the values are arbitrary and carry no information about the quality of the potential.
  • Momentum and angle quadrature grids = 48 momentum points on (0,20) fm^-1 for bound state; 48 angle points; similar grids for scattering
    Numerical discretization parameters chosen by hand; Table 3 shows the results depend sensitively on the grid when p0 approaches a quadrature node.
  • Energy scan range for deuteron eigenvalue fit = E from -2.4 to -2.1 MeV approximately
    The deuteron binding energy is extracted from a least-squares line fit to eigenvalues over a chosen energy interval; the interval choice affects the inferred energy and uncertainty.
assumptions (5)
  • domain assumption The two-nucleon potential conserves total isospin and is expanded in six scalar functions times spin-momentum operators (Eqs. 3 and 4).
    Standard representation from prior work [30,31]; underlying the partial wave decomposition used throughout.
  • domain assumption The Bonn-B potential supplies the reference scalar function values V_ijkl.
    The potential is an input from [27]; the paper does not derive it and the uncertainty model is superimposed.
  • standard math Gaussian quadrature with 48 momentum and 48 angle points sufficiently approximates the integrals in (11), (14) and (23).
    Standard numerical integration assumption; Table 3 shows it can fail when the on-shell momentum is near a quadrature node.
  • domain assumption The scalar function uncertainties are uncorrelated, normally distributed, and small enough that first-order propagation (17) applies.
    Explicit assumption in the paper; the paper checks the linear approximation against sampling and notes correlations are left to future work.
  • domain assumption The selected eigenvalue closest to 1 in (2) is simple, so its derivative with respect to potential values is well-defined.
    Backpropagation through eigvals requires a differentiable eigenvalue; degeneracy would complicate the gradient.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Estimating theoretical uncertainties of the two-nucleon observables by using backpropagation." pith.science (2026). https://pith.science/paper/UDL3GASG

@misc{pith2026241206304,
  author       = {Pith},
  title        = {Pith review of: Estimating theoretical uncertainties of the two-nucleon observables by using backpropagation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UDL3GASG}},
  note         = {Machine review of arXiv:2412.06304}
}
read the original abstract

We present a novel approach to calculating theoretical uncertainties in few-nucleon calculations, making use of automatic differentiation via backpropagation, which is particularly efficient when there are many input variables but only a few outputs. The methods described in this paper constitute tools that can be used to investigate the properties of scalar functions used to define nuclear potentials and quantify their contribution to the uncertainty of few nucleon calculations. We demonstrate these methods in deuteron bound state and nucleon - nucleon scattering calculations. Backpropagation, implemented in the Python pytorch library, is used to calculate the gradients with respect to model parameters and propagate errors from these parameters to the deuteron binding energy and selected phase-shift parameters. The uncertainty values obtained using this approach are validated by directly sampling from the potential parameters. We find very good agreement between two ways of estimating that uncertainty.

Figures

Figures reproduced from arXiv: 2412.06304 by the authors.

Figure 1
Figure 1. Eigenvalues closest to 1 for different values of E in MeV from (2). The standard deviation of the scalar function values is taken to be σ(V1jkl) = 0.2|V1jkl|, other values have no uncertainty. Gaussian quadrature points and weights were used both for momenta and angles. For momenta in (6), 48 points were used in the interval (0 fm−1 , 20 fm−1 ). For x in (7), 48 points were used in the interval (−1, 1). All discrete… view at source ↗
Figure 2
Figure 2. Histogram of eigenvalues closest to 1 for single energy E = −2.14206 MeV from [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Similar to [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Similar to [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Upper plot: selected values of v1(0.686133, p, −0.767159) from (4) are plotted for differ￾ent values of p in fm−1 . Lower plot: the gradient of the eigenvalue λ from (2) that is closest to 1 (λ = 1.000212) for energy −2.22099 MeV with respect to the corresponding scala…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages

  1. [1]

    Error estimates of theoretical models: a guide,

    J. Dobaczewski, W. Nazarewicz, and P.-G. Reinhard, “Error estimates of theoretical models: a guide,” Jour- nal of Physics G: Nuclear and Particle Physics , vol. 41, p. 074001, may 2014

    work page 2014

  2. [2]

    Coarse- grained potential analysis of neutron-proton and proton- proton scattering below the pion production threshold,

    R. N. Pérez, J. E. Amaro, and E. R. Arriola, “Coarse- grained potential analysis of neutron-proton and proton- proton scattering below the pion production threshold,” Phys. Rev. C , vol. 88, p. 064002, Dec 2013

    work page 2013

  3. [3]

    Statistical error analysis for phenomenological nucleon- nucleon potentials,

    R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, “Statistical error analysis for phenomenological nucleon- nucleon potentials,” Phys. Rev. C , vol. 89, p. 064006, Jun 2014

    work page 2014

  4. [4]

    Mod- ern theory of nuclear forces,

    E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, “Mod- ern theory of nuclear forces,” Rev. Mod. Phys. , vol. 81, pp. 1773–1825, Dec 2009

    work page 2009

  5. [5]

    Chiral dynamics of few- and many-nucleon systems,

    E. Epelbaum and U.-G. Meißner, “Chiral dynamics of few- and many-nucleon systems,” Annual Review of Nuclear and Particle Science , vol. 62, no. 1, pp. 159–185, 2012

    work page 2012

  6. [6]

    Nuclear ef- fective field theory: Status and perspectives,

    H.-W. Hammer, S. König, and U. van Kolck, “Nuclear ef- fective field theory: Status and perspectives,” Rev. Mod. Phys., vol. 92, p. 025004, Jun 2020

    work page 2020

  7. [7]

    Semilo- cal momentum-space regularized chiral two-nucleon poten- tials up to fifth order.,

    P. Reinert, H. Krebs, E. Epelbaum, and H. Krebs, “Semilo- cal momentum-space regularized chiral two-nucleon poten- tials up to fifth order.,” Eur. Phys. J. A , vol. 54, p. 86, 2018

    work page 2018

  8. [8]

    Theoretical uncertainties of the elastic nucleon-deuteron scattering observables,

    R. Skibiński, Y. Volkotrub, J. Golak, K. Topolnicki, and H. Witała, “Theoretical uncertainties of the elastic nucleon-deuteron scattering observables,” Phys. Rev. C , vol. 98, p. 014001, Jul 2018

    work page 2018

Show all 32 references
  1. [9]

    Uncer- tainty of three-nucleon continuum observables arising from uncertainties of two-nucleon potential parameters,

    Y. Volkotrub, J. Golak, R. Skibiński, K. Topolnicki, H.Witała,E.Epelbaum,H.Krebs,andP.Reinert,“Uncer- tainty of three-nucleon continuum observables arising from uncertainties of two-nucleon potential parameters,”Jour- nal of Physics G: Nuclear and Particle Physics , vol. 47, p...

    2020

  2. [10]

    Posterior pre- dictive distributions of neutron-deuteron cross sections,

    S. B. S. Miller, A. Ekström, and C. Forssén, “Posterior pre- dictive distributions of neutron-deuteron cross sections,” Phys. Rev. C , vol. 107, p. 014002, Jan 2023

    2023

  3. [11]

    Significance of chiral three-nucleon force contact terms for understand- ing of elastic nucleon-deuteron scattering,

    H. Witała, J. Golak, and R. Skibiński, “Significance of chiral three-nucleon force contact terms for understand- ing of elastic nucleon-deuteron scattering,”Phys. Rev. C , vol. 105, p. 054004, May 2022

    2022

  4. [12]

    Colloquium: Eigenvector continuation and projection-based emulators,

    T. Duguet, A. Ekström, R. J. Furnstahl, S. König, and D. Lee, “Colloquium: Eigenvector continuation and projection-based emulators,” Rev. Mod. Phys. , vol. 96, p. 031002, Aug 2024

    2024

  5. [13]

    A non-hermitian quantum mechanics approach for extracting and emulating continuum physics based on bound-state-like calculations: technical details

    X. Zhang, “A non-hermitian quantum mechanics approach for extracting and emulating continuum physics based on bound-state-like calculations: technical details.”https:// arxiv.org/abs/2411.06712, 2024

    2024

  6. [14]

    Application of a momentum-space semi-locally regular- ized chiral potential to selected disintegration processes,

    V. Urbanevych, R. Skibiński, H. Witała, J. Golak, K. Topolnicki, A. Grassi, E. Epelbaum, and H. Krebs, “Application of a momentum-space semi-locally regular- ized chiral potential to selected disintegration processes,” Phys. Rev. C , vol. 103, p. 024003, Feb 2021

    2021

  7. [15]

    Improvedchi- ral nucleon-nucleon potential up to next-to-next-to-next- to-leading order.,

    E.Epelbaum,H.Krebs,andU.-G.Meißner,“Improvedchi- ral nucleon-nucleon potential up to next-to-next-to-next- to-leading order.,” Eur. Phys. J. A , vol. 51, p. 53, 2015

    2015

  8. [16]

    Precision nucleon-nucleon potential at fifth order in the chiral ex- pansion,

    E. Epelbaum, H. Krebs, and U.-G. Meißner, “Precision nucleon-nucleon potential at fifth order in the chiral ex- pansion,” Phys. Rev. Lett., vol. 115, p. 122301, Sep 2015

    2015

  9. [17]

    Towards high-order calculations of three-nucleon scattering in chi- ral effective field theory.,

    E. Epelbaum, J. Golak, K. Hebeler, H. Kamada, H. Krebs, U.-G. Meißner, A. Nogga, P. Reinert, R. Skibiński, K. Topolnicki, Volkotrub.Yu., and H. Witała, “Towards high-order calculations of three-nucleon scattering in chi- ral effective field theory.,”Eur. Phys. J. A , vol. 56, ...

    2020

  10. [18]

    Quantifying truncation errors in effective field theory,

    R.J.Furnstahl,N.Klco,D.R.Phillips,andS.Wesolowski, “Quantifying truncation errors in effective field theory,” Phys. Rev. C , vol. 92, no. 2, p. 024005, 2015

    2015

  11. [19]

    Quantifying Correlated Trun- cation Errors in Effective Field Theory,

    J. A. Melendez, R. J. Furnstahl, D. R. Phillips, M. T. Pra- tola, and S. Wesolowski, “Quantifying Correlated Trun- cation Errors in Effective Field Theory,” Phys. Rev. C , vol. 100, no. 4, p. 044001, 2019. K. Topolnicki et al.: Estimating theoretical uncertainties of the two-n...

    2019

  12. [20]

    Nuclear properties with semilocal momentum-space reg- ularized chiral interactions beyondn2LO,

    P. Maris, R. Roth, E. Epelbaum, R. J. Furnstahl, J. Go- lak, K. Hebeler, T. Hüther, H. Kamada, H. Krebs, H. Le, U.-G. Meißner, J. A. Melendez, A. Nogga, P. Reinert, R. Skibiński, J. P. Vary, H. Witała, and T. Wolfgruber, “Nuclear properties with semilocal momentum-space reg- u...

    2022

  13. [21]

    Assessing correlated trunca- tion errors in modern nucleon-nucleon potentials,

    P. J. Millican, R. J. Furnstahl, J. A. Melendez, D. R. Phillips, and M. T. Pratola, “Assessing correlated trunca- tion errors in modern nucleon-nucleon potentials,” Phys. Rev. C, vol. 110, p. 044002, Oct 2024

    2024

  14. [22]

    Uncertainty quantification of collective nuclear observables from the chiral potential parametrization,

    K. S. Becker, K. D. Launey, A. Ekström, T. Dytrych, D. Langr, G. H. Sargsyan, and J. P. Draayer, “Uncertainty quantification of collective nuclear observables from the chiral potential parametrization,”arXiv:2404.00063 [nucl- th], 2024

    2024 arXiv

  15. [23]

    Pytorch: An imperative style, high- performance deep learning library,

    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, “Pytorch: An imperative style, high- p...

    2019

  16. [24]

    Automatic differentiation in machine learning: a survey,

    A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” Journal of Machine Learning Research , vol. 18, no. 153, pp. 1–43, 2018

    2018

  17. [25]

    TensorFlow: Large-scale machine learning on heterogeneous systems,

    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. Leven- berg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, ...

    2015

  18. [26]

    Array programming with NumPy,

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay- lor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Shep- pa...

    2020

  19. [27]

    Machleidt, The Meson Theory of Nuclear Forces and Nuclear Structure, pp

    R. Machleidt, The Meson Theory of Nuclear Forces and Nuclear Structure, pp. 189–376. Boston, MA: Springer US, 1989

    1989

  20. [28]

    Uncertainty analysis and order-by-order op- timization of chiral nuclear interactions,

    B. D. Carlsson, A. Ekström, C. Forssén, D. F. Strömberg, G. R. Jansen, O. Lilja, M. Lindby, B. A. Mattsson, and K. A. Wendt, “Uncertainty analysis and order-by-order op- timization of chiral nuclear interactions,” Phys. Rev. X , vol. 6, p. 011019, Feb 2016

    2016

  21. [29]

    Angeles, The Application of Dual Algebra to Kinematic Analysis, p

    J. Angeles, The Application of Dual Algebra to Kinematic Analysis, p. 3–32. Springer Berlin Heidelberg, 1998

    1998

  22. [30]

    A new way to per- form partial-wave decompositions of few-nucleon forces,

    J. Golak, D. Rozpędzik, R. Skibiński, K. Topolnicki, H. Witała, W. Glöckle, A. Nogga, E. Epelbaum, H. Ka- mada, C. Elster, and I. Fachruddin, “A new way to per- form partial-wave decompositions of few-nucleon forces,” The European Physical Journal A,vol.43,p.241–250,Dec. 2009

    2009

  23. [31]

    Possible triple-scattering experiments,

    L. Wolfenstein, “Possible triple-scattering experiments,” Physical Review, vol. 96, p. 1654–1658, Dec. 1954

    1954

  24. [32]

    Mathematica, Version 14.1

    W. R. Inc., “Mathematica, Version 14.1.” Champaign, IL, 2024

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.