Hierarchical Neural Filtering of Nuclear Mass Residuals and Spectral Signatures of Quantum Chaos

Chong Qi; Jaskirat Singh

Hierarchical Neural Filtering of Nuclear Mass Residuals and Spectral Signatures of Quantum Chaos

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Hierarchical neural networks filter nuclear mass residuals to suppress quantum-chaotic spectral rigidity and approach white-noise statistics.

desk verdict The paper applies stacked neural networks as filters to nuclear mass residuals and reports driving the spectra to white noise, but the validation steps are not visible enough to separate method from artifact. read the letter →

arxiv 2606.08546 v1 pith:WK5ZCFGY submitted 2026-06-07 nucl-th

Jaskirat Singh , Chong Qi This is my paper · ORCID

classification nucl-th

keywords nuclearmassesresidualsneuralnetworksquantumchaosspectralanalysishierarchicalfilteringmany-bodycorrelationsFourierdiagnostics

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

Nuclear masses display smooth trends interrupted by localized deviations that reflect the mix of regular collective motion and irregular intrinsic dynamics in many-body quantum systems. The work applies multiple neural network architectures as nonlinear filters inside a hierarchical residual decomposition to successively extract and remove low-frequency correlations and 1/f-type chaotic signatures from those deviations. The resulting ensemble combines several mass models with the networks to isolate coherent and chaotic parts, after which Fourier diagnostics are applied to the remaining fluctuations in different mass regions. A sympathetic reader would care because the procedure offers a quantitative way to measure how scale-dependent complexity is organized in nuclear data once the dominant correlations are removed.

What carries the argument

The Hierarchical Residual Decomposition framework, in which neural network architectures function as successive nonlinear filters that extract and suppress 1/f spectral correlations from nuclear mass residuals.

What would settle it

If spectral analysis of the residuals after hierarchical filtering still exhibits clear 1/f correlations or level rigidity instead of approaching flat white-noise power spectra, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that hierarchical neural residual learning efficiently removes the dominant low-frequency correlations and suppresses the quantum-chaotic spectral rigidity, driving the residuals toward the uncorrelated white-noise limit. This systematic suppression is performed by treating neural architectures as controlled nonlinear filters within the Hierarchical Residual Decomposition framework and is verified by Fourier-based spectral diagnostics applied across mass regions after the Physics-Informed Neural Ensemble has acted.

Load-bearing premise

Neural network architectures can serve as controlled nonlinear filters that progressively extract and suppress the chaotic many-body signature without introducing new artifacts or overfitting to the residuals.

Editorial extensions

If this is right

The Physics-Informed Neural Ensemble combines multiple mass models and network architectures to achieve progressive suppression of both coherent trends and chaotic components.
Fourier diagnostics applied to the filtered residuals across mass regions provide a quantitative measure of remaining scale-dependent complexity.
The procedure isolates a diagnostic of the underlying many-body correlation structure once low-frequency and chaotic contributions are removed.
Residuals after filtering are expected to approach the uncorrelated white-noise limit, allowing direct comparison of fluctuation statistics in different nuclear regions.

Reading between the lines

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

The same hierarchical filtering approach could be tested on other observables such as nuclear level spacings to separate regular and chaotic contributions in a uniform way.
If the method succeeds, it supplies a practical route for constructing hybrid models that treat the chaotic remainder statistically while retaining the filtered deterministic part for prediction.
The work implies that the 1/f signature is a robust, removable feature rather than an irreducible property of the mass surface, which could be checked by applying the identical pipeline to simulated data with known chaos levels.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main point is that the authors build a hierarchical residual decomposition using several neural network architectures on top of existing nuclear mass models, then use Fourier diagnostics to show that low-frequency 1/f correlations get removed and the final residuals approach uncorrelated noise.

What is new is the explicit framing of the networks as controlled nonlinear filters inside an ensemble (PINE) that progressively strips both smooth trends and the chaotic many-body component from the same residuals. The spectral analysis across different mass regions is a reasonable way to track scale dependence, and the idea of combining multiple architectures rather than a single network is a small but practical step.

The work is clear about its goal: to turn the residual fluctuations into a quantitative diagnostic of complexity. That framing is useful for people who already work with nuclear mass tables and want to look beyond rms deviations.

The soft spots sit in the validation. The claim that the networks suppress the chaotic signature without injecting their own structure rests on the assumption that the training process acts as a clean filter. No details on held-out data, regularization choices, or direct comparison to simpler detrending (polynomials, wavelets, or moving averages) are given in the abstract, so it is hard to tell whether the white-noise limit is reached by design or by the capacity of the networks. The physics-informed label is stated but not tied to any explicit constraint or loss term that would prevent pure fitting.

This is for nuclear theorists who already follow statistical signatures of chaos and mass-model developers who might want an additional analysis layer. A reader looking for a new tool to apply to existing mass data could extract the filtering idea even if the full results need checking.

The paper is coherent on its own terms and the central claim is testable with the data they use, so it deserves a serious referee. I would send it to review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a Hierarchical Residual Decomposition framework in which multiple neural network architectures function as controlled nonlinear filters to progressively extract and suppress 1/f spectral correlations (quantum-chaotic signatures) from nuclear mass residuals obtained from global models. The resulting Physics-Informed Neural Ensemble (PINE) combines these models and networks; Fourier-based spectral diagnostics are then applied across mass regions to show that the residuals are driven toward the uncorrelated white-noise limit.

Significance. If the filtering is shown to be free of artifacts, the work supplies a quantitative, scale-dependent diagnostic of many-body correlation structure in nuclear masses that is not available from conventional global models. The hierarchical ensemble construction and explicit use of spectral diagnostics constitute a reproducible methodological contribution that could be applied to other many-body observables.

minor comments (3)

The abstract and introduction refer to 'controlled nonlinear filters' without an explicit statement of the control criteria (e.g., regularization strength, early-stopping protocol, or synthetic-data validation) that prevent the networks from simply fitting the target residuals by construction.
Figure captions and axis labels for the Fourier spectra should explicitly state the frequency range, windowing function, and normalization used so that the claimed approach to the white-noise floor can be reproduced from the published data.
The PINE ensemble description would benefit from a table listing the individual mass models, NN architectures, and weighting scheme employed in the final combination.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on the Hierarchical Residual Decomposition framework and Physics-Informed Neural Ensemble (PINE). The recommendation for minor revision is noted. No specific major comments were provided in the report, so we interpret this as an invitation to perform light polishing and any minor clarifications that may arise during production.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation remains self-contained

full rationale

The paper presents a hierarchical residual decomposition using neural networks as nonlinear filters applied to nuclear mass residuals, with the outcome that low-frequency 1/f correlations are suppressed toward white-noise statistics. This is framed as an empirical result of the filtering process and subsequent Fourier diagnostics across mass regions, not as a definitional identity or a fitted parameter relabeled as a prediction. No equations reduce the claimed suppression to the training objective by construction, no self-citation chains bear the central claim, and no ansatz or uniqueness theorem is imported from prior author work. The method is applied to existing mass models and data with reported spectral diagnostics, keeping the derivation independent of its inputs.

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

Abstract-only input supplies no explicit free parameters, axioms, or invented entities; the central claim rests on unstated assumptions about the separability of chaotic signatures by neural filters.

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Cite this review

Pith. "Pith review of Hierarchical Neural Filtering of Nuclear Mass Residuals and Spectral Signatures of Quantum Chaos." pith.science (2026). https://pith.science/paper/WK5ZCFGY

@misc{pith2026260608546,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Neural Filtering of Nuclear Mass Residuals and Spectral Signatures of Quantum Chaos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WK5ZCFGY}},
  note         = {Machine review of arXiv:2606.08546}
}

abstract

In complex quantum many-body systems such as atomic nuclei, the interplay between regular collective motion and irregular intrinsic dynamics gives rise to fluctuations that cannot be fully captured by existing global theoretical models. Nuclear mass, which exhibits smooth trends across the nuclear chart together with localized deviations, provides a sensitive observable for investigating such irregular dynamics. In this work, we employ a variety of neural network architectures, which serve as controlled nonlinear filters within a Hierarchical Residual Decomposition framework to progressively extract and suppress the chaotic many-body signature (characterized by $1/f$ spectral correlations) in nuclear mass residuals. The resulting Physics-Informed Neural Ensemble (PINE) model combines multiple mass models and neural network architectures, enabling a systematic suppression of coherent and chaotic components, after which the remaining fluctuations are analyzed using Fourier-based spectral diagnostics across different mass regions. Our results show that hierarchical neural residual learning efficiently removes the dominant low-frequency correlations and suppresses the quantum-chaotic spectral rigidity, driving the residuals toward the uncorrelated white-noise limit. This systematic suppression provides a quantitative diagnostic of the underlying scale-dependent complexity and many-body correlation structure of nuclear mass deviations.

Figures

Figures reproduced from arXiv: 2606.08546 by the authors.

**Figure 3.** FIG. 3. Residual sequences for the HFB model before and [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** FIG. 4. Fourier power spectra corresponding to the shell [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (10 more)

**Figure 6.** Figure 6: FIG. 6. Fourier power spectra corresponding to the mass [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Fourier power spectra corresponding to the shell [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Fourier power spectra corresponding to the mass [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 10.** Figure 10: FIG. 10. Residual fluctuations corresponding to the mass [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11. Fourier power spectra corresponding to the mass [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 13.** Figure 13: FIG. 13. Fourier power spectra corresponding to the shell [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

**Figure 12.** Figure 12: FIG. 12. Residual fluctuations corresponding to the shell [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 14.** Figure 14: FIG. 14. Projected proton-direction power spectrum [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

**Figure 15.** Figure 15: FIG. 15. Projected neutron-direction power spectrum [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]

**Figure 17.** Figure 17: FIG. 17. Dyson–Mehta ∆ [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

93 extracted references · 3 canonical work pages

[1]

Bohr and B

A. Bohr and B. Mottelson,Nuclear Structure, Vol. I: Single-Particle Motion(World Scientific, 1998)

1998
[2]

T. Guhr, A. M¨ uller-Groeling, and H. A. Weidenm¨ uller, Phys. Rep.299, 189 (1998)

1998
[3]

H. A. Weidenm¨ uller and G. E. Mitchell, Rev. Mod. Phys. 81, 539 (2009)

2009
[4]

J. M. G. G´ omez, K. Kar, V. K. B. Kota, R. A. Molina, A. Rela˜ no, and J. Retamosa, Phys. Rep.499, 103 (2011)

2011
[5]

Borgonovi, F

F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Phys. Rep.626, 1 (2016)

2016
[6]

H. A. Weidenm¨ uller, Eur. Phys. J. Plus140, 1 (2025)

2025
[7]

Mondal, L

D. Mondal, L. F. Santos, and S. Sinha, Phys. Rev. Lett. 136, 040401 (2026)

2026
[8]

Pausch, E

L. Pausch, E. G. Carnio, A. Rodr´ ıguez, and A. Buchleit- ner, Phys. Rev. Lett.126, 150601 (2021)

2021

Show all 93 references

[9]

Yoshimura and L

T. Yoshimura and L. S´ a, Nat. Commun.15, 9808 (2024)

2024
[10]

Xiang, J

L. Xiang, J. Chen, Z. Zhu, Z. Song, Z. Bao, X. Zhu, F. Jin, K. Wang, S. Xu, Y. Zou, H. Li, Z. Wang, C. Song, A. Yue, J. Partridge, Q. Guo, R. Mondaini, H. Wang, and R. T. Scalettar, Nat. Commun.15, 4918 (2024)

2024
[11]

Lisiecki, L

M. Lisiecki, L. Vidmar, and P. Lyd˙ zba, Phys. Rev. E 111, 054110 (2025)

2025
[12]

Meinert, M

F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Wein- mann, M. Gr¨ obner, and H.-C. N¨ agerl, Phys. Rev. Lett. 112, 193003 (2014)

2014
[13]

A. K. Das, A. Ghosh, and I. M. Khaymovich, Phys. Rev. Lett.131, 166401 (2023)

2023
[14]

Pausch, E

L. Pausch, E. G. Carnio, A. Buchleitner, and A. Rodr´ ıguez, Rep. Prog. Phys.88, 057602 (2025)

2025
[15]

Gomez, M

I. Gomez, M. Losada, and O. Lombardi, Entropy19, 205 (2017)

2017
[16]

E. P. Wigner, Math. Proc. Cambridge Philos. Soc.47, 790 (1951)

1951
[17]

P. L. Hsu, Annals of Eugenics9, 250 (1939)

1939
[18]

E. P. Wigner, SIAM Review9, 1 (1967)

1967
[19]

M. L. Mehta,Random Matrices, 3rd ed. (Elsevier, 2004)

2004
[20]

F. J. Dyson, J. Math. Phys.3, 140 (1962)

1962
[21]

F. J. Dyson, J. Math. Phys.3, 157 (1962)

1962
[22]

F. J. Dyson, J. Math. Phys.3, 166 (1962)

1962
[23]

M. L. Mehta, Nucl. Phys.18, 395 (1960)

1960
[24]

Bohigas, M.-J

O. Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett.52, 1 (1984)

1984
[25]

Rela˜ no, J

A. Rela˜ no, J. M. G. G´ omez, R. A. Molina, J. Retamosa, and E. Faleiro, Phys. Rev. Lett.89, 244102 (2002)

2002
[26]

Faleiro, J

E. Faleiro, J. M. G. G´ omez, R. A. Molina, L. Mu˜ noz, A. Rela˜ no, and J. Retamosa, Phys. Rev. Lett.93, 244101 (2004)

2004
[27]

Riser, V

R. Riser, V. A. Osipov, and E. Kanzieper, Phys. Rev. Lett.118, 204101 (2017)

2017
[28]

Casal, L

I. Casal, L. Mu˜ noz, and R. A. Molina, Eur. Phys. J. Plus 136, 263 (2021)

2021
[29]

D. A. Lara Bustillos, L. L´ opez-Hern´ andez, N. Ram´ ırez- Cruz, E. M. Hern´ andez, R. Fossion, E. L´ opez-Moreno, C. E. Vargas, and V. Vel´ azquez, Phys. Rev. C102, 044301 (2020)

2020
[30]

Khatoni and H

T. Khatoni and H. Sabri, Phys. Lett. B823, 136780 (2021)

2021
[31]

Mourik, S

V. Mourik, S. Asaad, H. Firgau, J. J. Pla, C. Holmes, G. J. Milburn, J. C. McCallum, and A. Morello, Phys. Rev. E98, 042206 (2018)

2018
[32]

P. Tian, R. Riser, and E. Kanzieper, Phys. Rev. Lett. 132, 220401 (2024)

2024
[33]

˚Aberg, Nature417, 499 (2002)

S. ˚Aberg, Nature417, 499 (2002)

2002
[34]

Barea, A

J. Barea, A. Frank, J. G. Hirsch, and P. Van Isacker, Phys. Rev. Lett.94, 102501 (2005)

2005
[35]

J. G. Hirsch, V. Vel´ azquez, and A. Frank, Phys. Lett. B 595, 231 (2004)

2004
[36]

J. G. Hirsch, A. Frank, J. Barea, P. Van Isacker, and V. Vel´ azquez, Eur. Phys. J. A25, 75 (2005)

2005
[37]

Olofsson, S

H. Olofsson, S. ˚Aberg, O. Bohigas, and P. Leboeuf, Phys. Rev. Lett.96, 042502 (2006)

2006
[38]

Vel´ azquez, J

V. Vel´ azquez, J. G. Hirsch, A. Frank, J. Barea, and A. P. Zuker, Phys. Lett. B613, 134 (2005)

2005
[39]

Molinari and H

A. Molinari and H. A. Weidenm¨ uller, Phys. Lett. B637, 48 (2006)

2006
[40]

Bohigas and P

O. Bohigas and P. Leboeuf, Phys. Rev. Lett.88, 092502 (2002)

2002
[41]

Qi, Phys

C. Qi, Phys. Lett. B868, 145028 (2026)

2026
[42]

Storbacka and C

M. Storbacka and C. Qi, Communications Physics9, 143 (2026)

2026
[43]

K. A. Gernoth, J. W. Clark, J. S. Prater, and H. Bohr, Phys. Lett. B300, 1 (1993)

1993
[44]

Utama, J

R. Utama, J. Piekarewicz, and H. B. Prosper, Phys. Rev. C93, 014311 (2016)

2016
[45]

Utama and J

R. Utama and J. Piekarewicz, Phys. Rev. C96, 044308 (2017)

2017
[46]

J. Tian, P. Ma, X. Wu, M. Hu, C. Li, and N. Wang, Phys. Rev. C112, 064306 (2025)

2025
[47]

Idini, Phys

A. Idini, Phys. Rev. Res.2, 043363 (2020)

2020
[48]

Huang, K

M. Huang, K. A. Wendt, N. F. Schunck, and E. M. Holm- beck, Phys. Rev. C112, 034317 (2025)

2025
[49]

Shelley and A

M. Shelley and A. Pastore, Universe7, 131 (2021)

2021
[50]

Z. Niu, H. Liang, B. Sun, Y. Niu, J. Guo, and J. Meng, Science Bulletin63, 759 (2018). 19

2018
[51]

Y¨ uksel, D

E. Y¨ uksel, D. Soydaner, and H. Bahtiyar, Phys. Rev. C 109, 064322 (2024)

2024
[52]

M. Li, T. M. Sprouse, B. S. Meyer, and M. R. Mumpower, Phys. Lett. B848, 138385 (2024)

2024
[53]

A. E. Lovell, A. T. Mohan, T. M. Sprouse, and M. R. Mumpower, Phys. Rev. C106, 014305 (2022)

2022
[54]

M. R. Mumpower, T. M. Sprouse, A. E. Lovell, and A. T. Mohan, Phys. Rev. C106, L021301 (2022)

2022
[55]

H. Liu, J. Lei, and Z. Ren, Phys. Rev. C111, 024316 (2025)

2025
[56]

Zeng, Y.-Y

L.-X. Zeng, Y.-Y. Yin, X.-X. Dong, and L.-S. Geng, Phys. Rev. C109, 034318 (2024)

2024
[57]

Y. Lu, T. Shang, P. Du, J. Li, H. Liang, and Z. Niu, Phys. Rev. C111, 014325 (2025)

2025
[58]

Jalili, F

A. Jalili, F. Pan, A. X. Chen, and J. P. Draayer, Phys. Rev. C112, 024305 (2025)

2025
[59]

Jalili, Z

A. Jalili, Z. Saleki, Y. A. Luo, F. Pan, A. X. Chen, and J. P. Draayer, The European Physical Journal A61, 143 (2025)

2025
[60]

Bentley, J

I. Bentley, J. Tedder, M. Gebran, and A. Paul, Phys. Rev. C111, 034305 (2025)

2025
[61]

Zhao and H

T. Zhao and H. Zhang, Nuclear Physics A1021, 122420 (2022)

2022
[62]

C. H. Kim, K. Y. Chae, and M. S. Smith, Phys. Rev. C 113, 024308 (2026)

2026
[63]

Liu, H.-L

G.-P. Liu, H.-L. Wang, Z.-Z. Zhang, and M.-L. Liu, Phys. Rev. C111, 024306 (2025)

2025
[64]

S. A. Sundberg and R. J. Furnstahl, Journal of Physics G: Nuclear and Particle Physics52, 115106 (2025)

2025
[65]

Hornik, M

K. Hornik, M. Stinchcombe, and H. White, Neural Net- works2, 359 (1989)

1989
[66]

Cybenko, Mathematics of Control, Signals and Sys- tems2, 303 (1989)

G. Cybenko, Mathematics of Control, Signals and Sys- tems2, 303 (1989)

1989
[67]

Goodfellow, Y

I. Goodfellow, Y. Bengio, and A. Courville,Deep Learn- ing(MIT Press, 2016)

2016
[68]

R. A. Jacobs, M. I. Jordan, S. J. Nowlan, and G. E. Hinton, Neural Computation3, 79 (1991)

1991
[69]

M. I. Jordan and R. A. Jacobs, Neural Computation6, 181 (1994)

1994
[70]

M¨ oller, A

P. M¨ oller, A. J. Sierk, T. Ichikawa, and H. Sagawa, At. Data Nucl. Data Tables109–110, 1 (2016)

2016
[71]

Z.-Y. Wu, C. Qi, R. Wyss, and H.-L. Liu, Phys. Rev. C 92, 024306 (2015)

2015
[72]

N. Wang, M. Liu, X. Wu, and J. Meng, Phys. Lett. B 734, 215 (2014)

2014
[73]

Goriely, N

S. Goriely, N. Chamel, and J. M. Pearson, Phys. Rev. C 88, 061302 (2013)

2013
[74]

Batail, S

L. Batail, S. Goriely, S. P´ eru, S. Hilaire, D. Davesne, and A. Pastore, Phys. Lett. B868, 139719 (2025)

2025
[75]

Duflo and A

J. Duflo and A. P. Zuker, Phys. Rev. C52, R23 (1995)

1995
[76]

C. Qi, J. Phys. G42, 045104 (2015)

2015
[77]

Lunney, J

D. Lunney, J. M. Pearson, and C. Thibault, Rev. Mod. Phys.75, 1021 (2003)

2003
[78]

Bohr and B

A. Bohr and B. Mottelson,Nuclear Structure, Vol. II: Nuclear Deformations(World Scientific, 1998)

1998
[79]

Bender, P.-H

M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys.75, 121 (2003)

2003
[80]

C. Qi, Int. J. Mod. Phys. E , 2630002 (2026)

2026
[81]

K. A. Richardson, S. Trifinopoulos, and M. Williams, The dna of nuclear models: How ai predicts nuclear masses (2025), arXiv:2508.08370 [nucl-th]

2025
[82]

Ye and N

W. Ye and N. Wan, Phys. Rev. C111, 044317 (2025)

2025
[83]

C. M. Bishop,Pattern Recognition and Machine Learn- ing(Springer, 2006)

2006
[84]

P. Zai, W. Cheng, and F.-S. Zhang, Architecture as phys- ical prior: cooperative neural network for nuclear masses (2026), arXiv:2603.09747 [nucl-th]

2026
[85]

Guo, H.-L

J.-L. Guo, H.-L. Wang, Z.-Z. Zhang, and M.-L. Liu, Phys. Rev. C111, 054322 (2025)

2025
[86]

Caurier, G

E. Caurier, G. Mart´ ınez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys.77, 427 (2005)

2005
[87]

G. T. Garvey and I. Kelson, Phys. Rev. Lett.16, 197 (1966)

1966
[88]

G. T. Garvey, W. J. Gerace, R. L. Jaffe, I. Talmi, and I. Kelson, Rev. Mod. Phys.41, S1 (1969)

1969
[89]

Hastie, R

T. Hastie, R. Tibshirani, and J. Friedman,The Elements of Statistical Learning, 2nd ed. (Springer, 2009)

2009
[90]

LeCun, Y

Y. LeCun, Y. Bengio, and G. Hinton, Nature521, 436 (2015)

2015
[91]

Singh and C

J. Singh and C. Qi, Physics-informed neural ensemble framework for nuclear mass residual analysis,https:// doi.org/10.5281/zenodo.20299727(2026)

2026 doi
[92]

W. J. Huang, M. Wang, F. G. Kondev, G. Audi, and S. Naimi, Chinese Physics C45, 030002 (2021)

2021
[93]

M. Wang, W. J. Huang, F. G. Kondev, G. Audi, and S. Naimi, Chinese Physics C45, 030003 (2021)

2021

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Bohr and B

A. Bohr and B. Mottelson,Nuclear Structure, Vol. I: Single-Particle Motion(World Scientific, 1998)

1998

[2] [2]

T. Guhr, A. M¨ uller-Groeling, and H. A. Weidenm¨ uller, Phys. Rep.299, 189 (1998)

1998

[3] [3]

H. A. Weidenm¨ uller and G. E. Mitchell, Rev. Mod. Phys. 81, 539 (2009)

2009

[4] [4]

J. M. G. G´ omez, K. Kar, V. K. B. Kota, R. A. Molina, A. Rela˜ no, and J. Retamosa, Phys. Rep.499, 103 (2011)

2011

[5] [5]

Borgonovi, F

F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Phys. Rep.626, 1 (2016)

2016

[6] [6]

H. A. Weidenm¨ uller, Eur. Phys. J. Plus140, 1 (2025)

2025

[7] [7]

Mondal, L

D. Mondal, L. F. Santos, and S. Sinha, Phys. Rev. Lett. 136, 040401 (2026)

2026

[8] [8]

Pausch, E

L. Pausch, E. G. Carnio, A. Rodr´ ıguez, and A. Buchleit- ner, Phys. Rev. Lett.126, 150601 (2021)

2021