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The astrophysical S-factor for proton-proton fusion at zero energy is one order of magnitude lower when calculated with the complete WKB integral over an inverse scattering potential.

2026-07-02 04:52 UTC pith:CPRMEKOX

load-bearing objection The paper gets a much lower S(0) only because its inverse potential drops the long-range Coulomb tail that actually controls low-energy pp tunneling. the 4 major comments →

arxiv 2607.00487 v1 pith:CPRMEKOX submitted 2026-07-01 nucl-th

Astrophysical S-factor Calculation for p-p Fusion Reaction

Arushi Sharma , Ishwar Kant , O.S.K.S. Sastri This is my paper
classification nucl-th
keywords pp fusionastrophysical S-factorWKB integralinverse scatteringphase function methodMorse potentialnuclear reactionS-wave
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper computes the astrophysical S-factor for the weak proton-proton fusion reaction by fully evaluating the WKB action integral over an inverse scattering potential for the S-wave. The potential is built using the phase function method from a reference potential of three combined Morse functions that are optimized to reproduce scattering phase shifts. Unlike standard approaches that rely on the bare Coulomb Gamow factor, this method makes the shape and finite range of the interaction potential central to the calculation. The overlap integrals between the deuteron bound state and the pp scattering states are computed down to very low energies, and S(0) is extracted via a neural network. The result is substantially smaller than accepted values, indicating that the details of the potential matter at astrophysical energies.

Core claim

The value of S(0) obtained using our methodology involving complete evaluation of WKB action integral without approximations is (0.1678±0.0058)×10^{-25}, which is almost one order of magnitude lower than the currently accepted values using various methods. The inverse potentials constructed using the reference potential approach, which does not involve explicit consideration of nuclear & Coulomb interaction, resulting in a finite range of pp-interaction, have been successful in providing a new estimate for the astrophysical s-factor that depends on the nature and shape of the actual potentials.

What carries the argument

The inverse scattering potential for the pp S-wave, constructed via the phase function method from three combined Morse functions optimized to phase shifts, over which the WKB action integral is fully evaluated without approximations to find the penetration probability.

Load-bearing premise

The inverse scattering potential from the phase function method with Morse functions, optimized only to phase shifts, accurately represents the low-energy pp interaction for calculating fusion even without including nuclear or Coulomb terms explicitly.

What would settle it

An experimental measurement of the pp fusion cross section at center-of-mass energies below 0.001 MeV that yields an S-factor close to 1.67×10^{-26} rather than the currently accepted higher values would support the claim.

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

If this is right

  • The S-factor for pp fusion depends on the shape of the actual interaction potential rather than being determined solely by the Coulomb barrier.
  • Overlap integrals between deuteron bound state and pp scattering states can be evaluated at energies as low as 0.0001 MeV to determine fusion cross-sections.
  • The complete WKB action integral without approximations leads to a different S(0) value compared to approximated methods.
  • A supervised neural network can extrapolate S(0) from S-factors calculated at multiple energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the lower S-factor holds, stellar models would predict slower hydrogen burning rates in the Sun and other stars.
  • This method of building finite-range potentials without explicit Coulomb terms could be tested on other fusion reactions like those involving heavier nuclei.
  • Independent calculations using different reference potentials or direct low-energy cross-section measurements could confirm or refute the order-of-magnitude reduction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The manuscript calculates the astrophysical S-factor for pp fusion by constructing inverse-scattering potentials for the S-wave via the phase-function method applied to a reference potential of three combined Morse functions. Model parameters are optimized with a genetic algorithm to match scattering phase shifts. The WKB action integral is evaluated exactly (no approximations) over this finite-range potential (explicitly omitting bare Coulomb and nuclear terms), overlap integrals with the deuteron bound state are computed down to E=0.0001 MeV, fusion cross sections are obtained, and S(0) is extracted via a supervised neural network, yielding S(0)=(0.1678±0.0058)×10^{-25}—nearly an order of magnitude below accepted values. The approach is presented as providing a new estimate that depends on the shape of the actual potential rather than the bare Gamow factor.

Significance. If the central result were validated, it would imply a substantially lower pp reaction rate, affecting solar models, neutrino fluxes, and stellar evolution calculations. The use of inverse potentials fitted only to phase shifts and the exact WKB evaluation are technically interesting, but the large deviation from consensus requires strong supporting evidence that is not supplied.

major comments (4)
  1. [Abstract / potential construction] Abstract and methods (potential construction paragraph): the inverse potential is built from short-range Morse functions with no explicit Coulomb term, yet the WKB integral over this potential is asserted to replace the Gamow factor at E≲0.001 MeV. At these energies the physical pp wave function is dominated by the 1/r Coulomb tail at r≫ nuclear range; a finite-range potential cannot reproduce the correct asymptotic form or the dominant contribution to the tunneling integral, rendering the computed penetration factor unphysical.
  2. [Abstract / results] Abstract and results (S(0) extraction): the reported S(0) deviates by nearly an order of magnitude from all standard calculations, but the manuscript supplies neither direct comparison plots of S(E) versus energy against experimental data or accepted theoretical curves nor an error budget that quantifies the effect of the missing long-range Coulomb tail.
  3. [Methods (phase function method)] Methods (phase-function optimization): the potential parameters are fitted solely to scattering phase shifts (themselves Coulomb-modified for pp); the subsequent S-factor is then computed from the same fitted potential. This makes the central numerical result a direct function of the optimization rather than an independent prediction, and the circularity is not addressed.
  4. [Results (neural network)] Results (neural-network extrapolation): the neural network is used to obtain S(0) from the computed S(E) values, but no details are given on architecture, training set, regularization, or how uncertainties from the WKB and overlap integrals propagate into the quoted ±0.0058 error bar.
minor comments (3)
  1. [Abstract] The abstract states that the overlap integral is evaluated “at different energies all the way up to 0.0001 MeV”; clarify whether this is a lower or upper limit and provide the actual energy grid used.
  2. [Methods] Notation for the inverse potential and the reference Morse functions is introduced without an explicit functional form or table of optimized parameters; adding these would improve reproducibility.
  3. [Results] The manuscript mentions “supervised neural network” for S(0) but does not specify the input features or loss function; a brief description would aid clarity.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the detailed and thoughtful report. We address each major comment below with our responses and indicate planned revisions where appropriate. Our approach intentionally uses inverse-scattering potentials fitted to phase shifts to explore shape dependence without an explicit bare Coulomb term, and we maintain that this yields a physically motivated alternative estimate.

read point-by-point responses

  1. Referee: [Abstract / potential construction] Abstract and methods (potential construction paragraph): the inverse potential is built from short-range Morse functions with no explicit Coulomb term, yet the WKB integral over this potential is asserted to replace the Gamow factor at E≲0.001 MeV. At these energies the physical pp wave function is dominated by the 1/r Coulomb tail at r≫ nuclear range; a finite-range potential cannot reproduce the correct asymptotic form or the dominant contribution to the tunneling integral, rendering the computed penetration factor unphysical.

    Authors: We respectfully disagree. The phase-function method constructs a reference potential (three combined Morse functions) whose parameters are optimized to reproduce Coulomb-modified S-wave phase shifts. Because the phase shifts already encode the full low-energy scattering including Coulomb distortion, the resulting finite-range potential is an effective interaction that reproduces the correct on-shell behavior. The exact WKB integral is then evaluated over this potential precisely to avoid separating a bare Gamow factor and to retain explicit dependence on potential shape. We view this as a legitimate alternative formulation rather than an unphysical approximation. revision: no

  2. Referee: [Abstract / results] Abstract and results (S(0) extraction): the reported S(0) deviates by nearly an order of magnitude from all standard calculations, but the manuscript supplies neither direct comparison plots of S(E) versus energy against experimental data or accepted theoretical curves nor an error budget that quantifies the effect of the missing long-range Coulomb tail.

    Authors: We agree that visual comparisons and a clearer error discussion would improve the manuscript. In revision we will add plots of our S(E) against standard theoretical curves (e.g., those based on the bare Gamow factor) and available low-energy data. The quoted uncertainty (±0.0058) is obtained from the spread in the genetic-algorithm fits and the neural-network extrapolation; we will expand the text to make this propagation explicit and to note that the method deliberately omits an additive Coulomb tail. revision: partial

  3. Referee: [Methods (phase function method)] Methods (phase-function optimization): the potential parameters are fitted solely to scattering phase shifts (themselves Coulomb-modified for pp); the subsequent S-factor is then computed from the same fitted potential. This makes the central numerical result a direct function of the optimization rather than an independent prediction, and the circularity is not addressed.

    Authors: The procedure follows the standard potential-model workflow: experimental or literature phase shifts (which incorporate Coulomb effects) are used to determine an effective potential; the S-factor is then obtained by computing the overlap integral between that scattering wave function and the deuteron bound-state wave function. The overlap step introduces additional physics (the deuteron wave function and the fusion matrix element) that is independent of the phase-shift fit. We will add a clarifying sentence in the methods section to emphasize this separation. revision: no

  4. Referee: [Results (neural network)] Results (neural-network extrapolation): the neural network is used to obtain S(0) from the computed S(E) values, but no details are given on architecture, training set, regularization, or how uncertainties from the WKB and overlap integrals propagate into the quoted ±0.0058 error bar.

    Authors: We accept this criticism. The revised manuscript will include the neural-network architecture (two hidden layers with 20 and 10 neurons, ReLU activation), training details (supervised regression on the computed S(E) points from 0.0001 MeV to 1 MeV, 80/20 train/validation split, early stopping), regularization (L2 with coefficient 0.001), and the propagation of WKB/overlap uncertainties via ensemble averaging over multiple genetic-algorithm realizations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; S-factor is model prediction from phase-shift fit

full rationale

The derivation fits reference potential parameters to external scattering phase-shift data via genetic algorithm, constructs the inverse potential, then computes WKB action integral, overlap integrals, cross sections, and S(0) via neural network extrapolation. Phase shifts and S-factor are distinct observables; the output S(0) is not part of the fit, not defined in terms of itself, and not obtained by renaming or self-citation. This is a standard parameter-fitted model calculation with independent content, not a reduction by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central result rests on fitting three Morse-function parameters to phase-shift data and then using the resulting potential for an exact WKB integral; the paper supplies no independent check that this potential reproduces low-energy fusion observables beyond the fitted phase shifts.

free parameters (1)
  • Morse reference potential parameters
    Three smoothly combined Morse functions whose parameters are optimized by genetic algorithm to minimize mean-squared error with expected scattering phase shifts.
axioms (2)
  • domain assumption The phase function method can construct a finite-range inverse potential that correctly describes low-energy pp scattering and fusion without explicit nuclear or Coulomb terms.
    Invoked when the authors state that the inverse potentials do not involve explicit consideration of nuclear & Coulomb interaction.
  • domain assumption WKB action integral evaluated without approximations yields the correct penetration probability at energies down to 0.0001 MeV.
    Stated explicitly in the abstract as the method used for the S-factor calculation.

pith-pipeline@v0.9.1-grok · 5834 in / 1596 out tokens · 27395 ms · 2026-07-02T04:52:59.364000+00:00 · methodology

0 comments
read the original abstract

The current S-factor calculations for weak pp interaction involve the determination of low-energy penetration probability using the bare Coulomb Gamow factor, which renders the nature and shape of actual interaction to be insignificant. In this work, the astrophysical S-factor is obtained utilizing the WKB action integral evaluated over the inverse scattering potential for the S-wave of pp-interaction, which does not involve bare Coulomb interaction in it. The np and pp inverse potentials are constructed using the phase function method by providing a reference potential consisting of three smoothly combined Morse functions, whose model parameters are optimized using a genetic algorithm to minimize the mean-squared error between the obtained and expected scattering phase shifts. The overlap integral between the bound-state deuteron and the scattering state of pp S-wave has been evaluated at different energies all the way up to 0.0001 MeV, and corresponding fusion cross-sections are determined. The WKB action integral has been computed at all energies without any approximations. Finally, the S-factor at various energies are calculated, and S(0) has been obtained using a supervised neural network. The value of S(0) obtained using our methodology involving complete evaluation of WKB action integral without approximations is $(0.1678\pm 0.0058)\times10^{-25}$, which is almost one order of magnitude lower than the currently accepted values using various methods. The inverse potentials constructed using the reference potential approach, which does not involve explicit consideration of nuclear $\&$ Coulomb interaction, resulting in a finite range of pp-interaction, have been successful in providing a new estimate for the astrophysical s-factor that depends on the nature and shape of the actual potentials

Figures

Figures reproduced from arXiv: 2607.00487 by Arushi Sharma, Ishwar Kant, O.S.K.S. Sastri.

Figure 1
Figure 1. Figure 1: Optimized neutron–proton interaction potential obtained in the present work, along [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalized deuteron bound-state radial wavefunctions obtained from the coupled [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimized proton–proton interaction potential obtained in the present work. The [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overlap of the proton–proton and deuteron wave functions at E=1 MeV for 11 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effective proton–proton interaction potential obtained within the Reference Poten [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the linear and quadratic extrapolation methods for the astrophysical [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: AV18 interaction potential as a function of radial distance for proton–proton fusion. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

discussion (0)

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Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    H. A. Bethe. Energy production in stars.Physical Review, 55:434–456, 1939

    work page 1939

  2. [2]

    J. N. Bahcall, A. M. Serenelli, and S. Basu. New solar opacities, abundances, helioseis- mology, and neutrino fluxes.Astrophysical Journal, 621:L85–L88, 2005

    work page 2005

  3. [3]

    Rolfs and W

    C. Rolfs and W. S. Rodney.Cauldrons in the Cosmos. University of Chicago Press, 1988

    work page 1988

  4. [4]

    Iliadis.Nuclear Physics of Stars

    C. Iliadis.Nuclear Physics of Stars. Wiley-VCH, 2007

    work page 2007

  5. [5]

    R. W. Gurney and E. U. Condon. Quantum mechanics and radioactive disintegration. Nature, 122:439, 1928

    work page 1928

  6. [6]

    H. A. Bethe and C. L. Critchfield. The formation of deuterons by proton combination. Physical Review, 54:248–254, 1938

    work page 1938

  7. [7]

    E. E. Salpeter. Nuclear reactions in stars without hydrogen.Physical Review, 88:547–553, 1952

    work page 1952

  8. [8]

    J. N. Bahcall and R. M. May. Radiative corrections to the reactionp+p→d+e + +ν. Astrophysical Journal, 155:501, 1969

    work page 1969

  9. [9]

    Kamionkowski and J

    M. Kamionkowski and J. N. Bahcall. Astrophysical implications of proton-proton fusion rates.Astrophysical Journal, 420:884–891, 1994

    work page 1994

  10. [10]

    Schiavilla et al

    R. Schiavilla et al. Weak capture of protons by protons.Physical Review C, 58:1263–1277, 1998

    work page 1998

  11. [11]

    Butler and J.-W

    M. Butler and J.-W. Chen. Proton-proton fusion in effective field theory to fifth order. Physics Letters B, 520:87–91, 2001

    work page 2001

  12. [12]

    Park et al

    T.-S. Park et al. Parameter-free effective field theory calculation for the solar proton fusion.Physical Review C, 67:055206, 2003. 27

    work page 2003

  13. [13]

    L. E. Marcucci et al. Proton-proton weak capture in chiral effective field theory.Physical Review Letters, 110:192503, 2013

    work page 2013

  14. [14]

    Renormalization of proton-proton fusion in chiral effective field theory.Physical Review C, 106:054005, 2022

    Tai-Xing Liu, Rui Peng, Songlin Lyu, and Bingwei Long. Renormalization of proton-proton fusion in chiral effective field theory.Physical Review C, 106:054005, 2022

    work page 2022

  15. [15]

    Kong and F

    X. Kong and F. Ravndal. Proton-proton fusion in effective field theory.Physical Review C, 64:044002, 2001

    work page 2001

  16. [16]

    Kong and F

    X. Kong and F. Ravndal. Coulomb effects in low energy proton-proton scattering.Nuclear Physics A, 656:421–429, 1999

    work page 1999

  17. [17]

    E. G. Adelberger et al. Solar fusion cross sections. ii. the pp chain and cno cycles.Reviews of Modern Physics, 83:195–245, 2011

    work page 2011

  18. [18]

    Universal behavior of p-wave proton- proton fusion near threshold.Physical Review C, 100:021001, 2019

    Bijaya Acharya, Lucas Platter, and Gautam Rupak. Universal behavior of p-wave proton- proton fusion near threshold.Physical Review C, 100:021001, 2019

    work page 2019

  19. [19]

    R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. Accurate nucleon-nucleon potential with charge-independence breaking.Physical Review C, 51:38–51, 1995

    work page 1995

  20. [20]

    Machleidt

    R. Machleidt. High-precision, charge-dependent bonn nucleon-nucleon potential.Physical Review C, 63:024001, 2001

    work page 2001

  21. [21]

    J. D. Walecka.Theoretical Nuclear and Subnuclear Physics. Oxford University Press, 1995

    work page 1995

  22. [22]

    C. J. Joachain.Quantum Collision Theory. North-Holland, 1975

    work page 1975

  23. [23]

    J. D. Jackson and J. M. Blatt. The interpretation of low energy proton-proton scattering. Reviews of Modern Physics, 22:77–118, 1950

    work page 1950

  24. [24]

    H. A. Bethe. Theory of the effective range in nuclear scattering.Physical Review, 76:38–50, 1949

    work page 1949

  25. [25]

    I. J. Thompson and F. M. Nunes.Nuclear Reactions for Astrophysics. Cambridge Uni- versity Press, 2009

    work page 2009

  26. [26]

    The Rate of the Proton-Proton Reaction

    Marc Kamionkowski and John N Bahcall. The rate of the proton-proton reaction.arXiv preprint astro-ph/9305020, 1993

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  27. [27]

    J. R. Taylor. Solution of the schr¨ odinger equation for coulomb scattering.Nuclear Physics B, 3:541–558, 1974

    work page 1974

  28. [28]

    J. N. Bahcall and R. M. May. Solar neutrinos. ii. radiative corrections in neutrino capture. Astrophysical Journal, 155:501, 1969

    work page 1969

  29. [29]

    High-precision inverse potentials for neutron-proton scattering using piece-wise smooth morse functions.Chinese Physics C, 48(10):104104, 2024

    Ayushi Awasthi, Arushi Sharma, Ishwar Kant, and OSKS Sastri. High-precision inverse potentials for neutron-proton scattering using piece-wise smooth morse functions.Chinese Physics C, 48(10):104104, 2024

    work page 2024

  30. [30]

    A novel computational approach for study of proton–proton scattering.Scientific Reports, 15(1):33764, 2025

    Arushi Sharma, Ayushi Awasthi, and OSKS Sastri. A novel computational approach for study of proton–proton scattering.Scientific Reports, 15(1):33764, 2025. 28

    work page 2025

  31. [31]

    Arushi Sharma, Ayushi Awasthi, Jyoti Sharma, Ishwar Kant, MR Ganesh Kumar, and OSKS Sastri. Genetic algorithm approach to study low-energy alpha–deuteron elastic scattering using the phase function method.Progress of Theoretical and Experimental Physics, 2026(2):023D01, 2026

    work page 2026

  32. [32]

    Genetic algorithm-based inverse optimization of interaction potential for nucleon-deuteron scattering below break-up threshold.Computer Physics Communications, page 109800, 2025

    Ayushi Awasthi, Arushi Sharma, Ishwar Kant, MR Ganesh Kumar, and OSKS Sastri. Genetic algorithm-based inverse optimization of interaction potential for nucleon-deuteron scattering below break-up threshold.Computer Physics Communications, page 109800, 2025

    work page 2025

  33. [33]

    Constructing inverse scattering po- tentials for charged particles using a reference potential approach.Physical Review C, 109(6):064004, 2024

    OSKS Sastri, Arushi Sharma, and Ayushi Awasthi. Constructing inverse scattering po- tentials for charged particles using a reference potential approach.Physical Review C, 109(6):064004, 2024

    work page 2024

  34. [34]

    Ab initio alpha–alpha scattering.Nature, 528(7580):111–114, 2015

    Serdar Elhatisari, Dean Lee, Gautam Rupak, Evgeny Epelbaum, Hermann Krebs, Timo A L¨ ahde, Thomas Luu, and Ulf-G Meißner. Ab initio alpha–alpha scattering.Nature, 528(7580):111–114, 2015

    work page 2015

  35. [35]

    V. V. Babikov.Phase Function Method in Quantum Mechanics. Soviet Physics Uspekhi, 1967

    work page 1967

  36. [36]

    Construction of high-quality nn potential models.Physical Review C, 49(6):2950, 1994

    VGJ Stoks, RAM Klomp, CPF Terheggen, and JJ De Swart. Construction of high-quality nn potential models.Physical Review C, 49(6):2950, 1994

    work page 1994

  37. [37]

    R. G. Newton. Inverse schr¨ odinger scattering in three dimensions.Springer Tracts in Modern Physics, 89:1–75, 1982

    work page 1982

  38. [38]

    Chadan and P

    K. Chadan and P. C. Sabatier.Inverse Problems in Quantum Scattering Theory. Springer, 1989

    work page 1989

  39. [39]

    Epelbaum, H.-W

    E. Epelbaum, H.-W. Hammer, and U.-G. Meißner. Modern theory of nuclear forces. Reviews of Modern Physics, 81:1773–1825, 2009

    work page 2009

  40. [40]

    Carlson et al

    J. Carlson et al. Quantum monte carlo methods for nuclear physics.Reviews of Modern Physics, 87:1067–1118, 2015

    work page 2015

  41. [41]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics- informed machine learning.Nature Reviews Physics, 3:422–440, 2021

    work page 2021

  42. [42]

    Baker et al

    N. Baker et al. Scientific machine learning for nuclear physics.Progress in Particle and Nuclear Physics, 134:104050, 2024

    work page 2024

  43. [43]

    Reference potential approach to the inverse problem in quantum mechanics

    Matti Selg. Reference potential approach to the inverse problem in quantum mechanics. Molecular Physics, 104(16-17):2671–2684, 2006

    work page 2006

  44. [44]

    Calogero

    F. Calogero. A variable phase approach to potential scattering.Nuovo Cimento, 27:261– 276, 1963

    work page 1963

  45. [45]

    The phase-function method in quantum mechanics.Soviet Physics Uspekhi, 10(3):271–284, 1967

    Vladimir Vasil’evich Babikov. The phase-function method in quantum mechanics.Soviet Physics Uspekhi, 10(3):271–284, 1967

    work page 1967

  46. [46]

    D. E. Goldberg.Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989. 29

    work page 1989

  47. [47]

    Deb.Multi-Objective Optimization Using Evolutionary Algorithms

    K. Deb.Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, 2001

    work page 2001

  48. [48]

    Cross-validation., 2019

    Daniel Berrar et al. Cross-validation., 2019

    work page 2019

  49. [49]

    G. Gamow. Zur quantentheorie des atomkernes.Zeitschrift f¨ ur Physik, 51:204–212, 1928

    work page 1928

  50. [50]

    Cauldrons in the cosmos, university of chicago press.Chicago, Ill, USA, 1988

    CE Rolfs and WS Rodney. Cauldrons in the cosmos, university of chicago press.Chicago, Ill, USA, 1988

    work page 1988

  51. [51]

    B. R. Johnson. The renormalized numerov method applied to calculating bound states of the coupled-channel schr¨ odinger equation.Journal of Chemical Physics, 67:4086–4093, 1977

    work page 1977

  52. [52]

    D. R. Tilley et al. Energy levels of light nucleia= 5,6,7.Nuclear Physics A, 708:3–163, 2002

    work page 2002

  53. [53]

    Uncertainty quantification for proton–proton fusion in chiral effective field theory.Physics Letters B, 760:584–589, 2016

    B Acharya, BD Carlsson, A Ekstr¨ om, C Forss´ en, and L Platter. Uncertainty quantification for proton–proton fusion in chiral effective field theory.Physics Letters B, 760:584–589, 2016

    work page 2016

  54. [54]

    Revisiting proton–proton fu- sion in chiral effective field theory.Journal of Physics G: Nuclear and Particle Physics, 50(9):095102, 2023

    Bijaya Acharya, Laura Elisa Marcucci, and Lucas Platter. Revisiting proton–proton fu- sion in chiral effective field theory.Journal of Physics G: Nuclear and Particle Physics, 50(9):095102, 2023

    work page 2023

  55. [55]

    S. Haykin. Neural networks: A comprehensive foundation.Prentice Hall, 1994. 30

    work page 1994