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REVIEW 3 major objections 5 minor 46 references

Solving two and three-body systems with deep neural networks

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A deep neural network with the Hamiltonian expectation value as its loss function solves two- and three-body bound states, reproducing the deuteron and triton properties without assuming a trial wave function.

desk verdict Solid DNN variational proof-of-principle for the deuteron and triton, with a genuinely new multipole trick for three-body systems, but the 'any potential' claim outruns the evidence and the three-body truncation is uncharacterized. read the letter →

arxiv 2507.17559 v1 pith:5JNW664X submitted 2025-07-23 hep-ph nucl-th

Ruitian Li , Xuan Luo , Hao Sun , Pablo G. Ortega This is my paper
classification hep-phnucl-th PACS 21.45.-v02.70.-c07.05.Mh24.10.Cn
keywords deepneuralnetworksSchrödingerequationtwo-bodyproblemthree-bodychiraleffectivefieldtheorymultipoleexpansiondeuterontriton
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 claims that a deep neural network, trained with the variational energy expectation as its loss function, can solve two- and three-body bound state problems for arbitrary potentials and channel counts, without making any prior assumption about the wave function's shape. The authors show that a network fed with coordinate meshes reproduces the deuteron binding energy and wave function to experimental accuracy, and reproduces the triton binding energy to within 0.6% of the Gaussian expansion method for the same three-channel truncation. If the method scales, it would provide a way to study nuclear and hadronic many-body systems without building a tailored trial basis, potentially extending to four-body systems. The key extension is a multipole expansion that rewrites the potentials and wave functions of two Jacobi channels into the coordinates of a single reference channel, so the neural network only needs to see one two-dimensional grid.

What carries the argument

The variational-principle deep neural network: a feed-forward network with softplus or tanh activations whose output is the wave function on a coordinate mesh and whose loss function is the energy expectation value $\langle \Psi | H | \Psi \rangle / \langle \Psi | \Psi \rangle$, minimized by gradient descent. The load-bearing extension for three bodies is the multipole expansion of Eqs. (A18) and (B4), which rewrites the potentials and wave functions of Jacobi channels $c=2$ and $c=3$ in terms of the coordinates of the reference channel $c=1$, reducing the three-body problem to a single $\rho$-$\lambda$ grid.

What would settle it

Compute the triton with the full 34-channel N3LO χEFT potential used in Ref. [32] using the DNN method, varying the multipole order in Eqs. (A18) and (B4) and the mesh density; if the binding energy does not approach the 8.09 MeV reference value as these parameters increase, the convergence premise is violated.

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Extended reading notes

Core claim

The authors construct a deep neural network whose input is the discretized spatial coordinates and whose output is the (coupled-channel) reduced wave function, with the variational energy expectation value $\langle H \rangle$ as the loss function. For two-body systems this solves the radial Schrödinger equation directly; for three-body systems, they pick one Jacobi channel (say $c=1$) and use a multipole expansion to express the pair potentials and wave functions of the other two channels ($c=2,3$) in terms of the $\rho$ and $\lambda$ coordinates of that channel, so the entire problem lives on a single two-dimensional grid. They validate the network on the harmonic oscillator (analytic solution), the deuteron with a chiral effective field theory N3LO NN potential (binding energy 2.2241 MeV vs 2.2246 MeV reference), and the triton in a three-channel truncation (binding energy 9.83 MeV vs 9.77 MeV from GEM, D-wave probability 6.67% vs 6.70%).

Load-bearing premise

The three-body results hinge on the multipole expansion of the potentials and wave functions from the two non-selected Jacobi channels into the reference channel's coordinates converging quickly enough with the included terms, which the paper does not test by varying the multipole order.

Editorial extensions

If this is right

  • Any potential that can be evaluated on the coordinate mesh—central, spin-orbit, tensor, or coupled channels—can be fed into the same network without building a tailored trial basis.
  • The deuteron results converge with mesh density and hidden-layer size (2.5340 → 2.2241 MeV as $M$ goes 100 → 1000), showing the method is systematically improvable rather than fixed by an ansatz.
  • The triton DNN result matches GEM to 0.6% for the same channels, so the three-body extension is validated by comparison with an established method, not only by an analytic test.
  • Because the network output is unconstrained by a trial function, states with nodes or relative phases can be represented by switching to a tanh activation, which a poor basis choice might miss.

Reading between the lines

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

  • A natural next test—not performed here—would be to increase the number of triton channels and include the $L^2$ and $(L\cdot S)^2$ terms omitted in this work; the expected shift toward the experimental 8.48 MeV would tell whether the multipole expansion stays convergent as the tensor force becomes more important.
  • The same coordinate-transformation strategy could in principle be recursed to four-body systems, where the bookkeeping of angular-momentum couplings would grow substantially but the network architecture would not need to change.
  • Since the method is unsupervised, it could be adapted to scattering or resonance problems by altering the loss function, but the paper does not explore that direction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports an unsupervised deep-neural-network variational method for bound states. In the two-body case the network output is the reduced radial wave function on a radial mesh and the loss is the normalized energy expectation; the method is tested on a harmonic oscillator and on the deuteron with a chiral-EFT NN potential in the coupled 3S1-3D1 channel. In the three-body case a single Jacobi channel is retained and the potentials and wave functions of the other rearrangement channels are re-expanded through a multipole expansion (Appendices A and B); the method is tested on a separable harmonic oscillator and on a three-channel triton model. The reported deuteron binding energy is 2.2241 MeV and the triton binding energy is 9.83 MeV versus 9.77 MeV from a GEM calculation with the same channels, with wave functions and correlation functions in visual agreement.

Significance. The variational formulation is a strength: energies are minimized, not regressed to target values, and the harmonic-oscillator benchmarks are analytic. If the triton result survives a multipole-convergence test, the method would be a useful addition to few-body techniques, since it avoids explicit trial-function ansätze and is not tied to a Gaussian or spline basis. The claimed generality to any potential and any channel number is, however, currently not evidenced; the paper validates only one realistic three-body case at one truncation. The lack of training ensembles and of a specified multipole cutoff leaves the numerical accuracy claims incompletely supported.

major comments (3)
  1. [Sec. III B2 and Appendices A-B] The central three-body result rests on the multipole expansions in Eqs. (A18)-(A20) and (B3)-(B6), but the paper never states the maximum multipole orders used for the triton or shows a convergence scan. The sums over l1, l2, L, and ell (and li in Eq. (B3)) are infinite, and the only numerical evidence is a single DNN-vs-GEM comparison at one fixed truncation. Because the NN potential has tensor and spin-orbit components, slow convergence of these expansions is a real risk; the reported 0.6% agreement could be coincidental. Please report the truncation orders and demonstrate convergence of Bt, PD, and the correlation functions with respect to both the multipole cutoffs and the (rho, lambda) mesh.
  2. [Sec. IV and Sec. III B2] The summary statement that the method is 'applicable to any type of potential and number of channels' is not supported by the calculations shown. The triton calculation is restricted to three partial waves and, as the authors state in Sec. III B2, deliberately omits the L2 and (L*S)2 terms that are part of the same N3LO potential used for the deuteron benchmark. The spin-orbit derivation in Appendix A is written for potentials of the form V(r) S*L; it is not demonstrated for the momentum-dependent and nonlocal operators present in the full potential. The claim should either be demonstrated on a case including those terms or appropriately narrowed.
  3. [Sec. III A2, Table II, Table III, and Sec. III B2] All reported energies and wave functions correspond to a single training run, while the text itself notes that the DNN results fluctuate from one training to another. Without repeated initializations and a reported spread, the stated keV-level deuteron accuracy and the triton 0.6% agreement cannot be distinguished from optimizer noise. A small ensemble of runs (for example, 5-10 seeds) with mean and standard deviation for Eb, <T>, <V>, PD, and the triton observables should be provided.
minor comments (5)
  1. [Sec. III A1] The statement that the DNN wave function 'deviates less than 2%' from the exact solution should specify the norm used; the figure suggests a pointwise comparison, but the text is ambiguous.
  2. [Sec. III B2] The caveat that the triton calculation is 'not a full accurate calculation' is important and should appear in the abstract or conclusion as well as in Sec. III B2.
  3. [Table I] The DNN matter radius rd = 1.867 fm is compared with experiment 1.971(6) fm but no chiral-EFT reference value is given and the roughly 5% discrepancy is not discussed.
  4. [Throughout] There are several typographical and wording issues, including 'discretized' in Sec. II, 'all this methods' in Sec. I, and 'pannel' in the Fig. 6 caption.
  5. [Sec. II] No code or data availability statement is provided; a statement would improve reproducibility, especially for the multipole-expansion coefficients and network hyperparameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the energies are variational outputs of a DNN loss function, and the benchmarks are external data or independent GEM calculations on the same truncated Hamiltonian.

full rationale

The paper's derivation chain is variational rather than regressive. The DNN parameters are optimized by minimizing the energy expectation ⟨H⟩ (Eq. 1), so the reported binding energies are variational outputs, not quantities fitted to benchmark values. The deuteron benchmark uses an externally published χEFT N3LO potential (Ref. [32], Saha et al.) plus experimental constants (Refs. [35,36]); none of these are the authors' own fitted results, and no self-citation is load-bearing. The triton benchmark is a GEM calculation run on the same truncated Hamiltonian with the same partial waves (Sec. III B 2), so the DNN-vs-GEM agreement tests the numerical solver rather than being constructed by matching. The multipole expansions in Appendices A and B are derived re-expressions of V(ρ_c) and ψ(c) in channel-1 coordinates; the paper does not fit the multipole coefficients to the GEM answer. The honest limitations the paper itself states—only 3 of 34 channels, omission of L2 and (L·S)2 terms, and absence of a multipole-order convergence scan—weaken the "any type of potential and number of channels" generality claim (Sec. IV), but that is a convergence/validity risk, not circularity. No equation reduces to its own input by construction, no fitted parameter is renamed a prediction, and no author-imported uniqueness theorem is invoked. Score 0.

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

The physics content is imported from Refs [30-32,38,39]; the paper contributes a numerical procedure. The ledger makes explicit that the 3-body result is only as good as the multipole truncation and the channel truncation.

free parameters (4)
  • Mesh size and range = deuteron: M=1000 over [0.01,15] fm; triton: 80x80 over [0,15] fm; HO: M=200 over [0.01,30] fm
    Convergence tested for the deuteron only; the triton uses fewer points with no convergence check.
  • Hidden-layer widths = two-body: 16 nodes x 2 layers; three-body: 8 nodes x 2 layers
    Varied in Table III; results stable for nh>=8, but no error bars from repeated runs.
  • Activation and optimizer = softplus+Adam for two-body and HO; tanh+Adamax for triton
    Chosen by hand to match expected wave function behavior, weakening the no prior assumptions claim; learning rates not reported.
  • Multipole expansion truncation = not specified
    The set of retained multipoles in Eqs. (A18) and (B4) is not stated; no convergence scan is provided.
assumptions (4)
  • domain assumption The chiral EFT NN potential of Ref [32] is a valid external representation of the nuclear force.
    The method imports the potential; the deuteron and triton numbers inherit its physics and are not derived by the network.
  • ad hoc to paper The multipole expansion of V(rho_c) and psi(c) into channel c=1 coordinates converges at the included truncation order.
    Used to build the 3-body Hamiltonian in a single Jacobi grid, but no study of truncation error is shown; the claim that the method works for any potential depends on this.
  • ad hoc to paper The three-channel truncation for the triton is an adequate method test.
    Only the 3 partial waves in Table IV are included; the paper explicitly says this is not a full triton calculation, but the abstract's wording is stronger than what is shown.
  • standard math The variational principle and second-order finite differences give the exact ground state in the limit of sufficient training.
    Standard numerical analysis; boundary conditions u(0)=0 and u(infinity)=0 are implied by the mesh and not explicitly enforced.

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Cite this review

Pith. "Pith review of Solving two and three-body systems with deep neural networks." pith.science (2026). https://pith.science/paper/5JNW664X

@misc{pith2026250717559,
  author       = {Pith},
  title        = {Pith review of: Solving two and three-body systems with deep neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5JNW664X}},
  note         = {Machine review of arXiv:2507.17559}
}
read the original abstract

We develop a new method for solving two- and three-body bound state problems using unsupervised machine learning techniques. We use a deep neural network to calculate both simple and realistic potentials, obtaining the properties of the deuteron and triton bound states for the chiral effective field theory NN potential. Our results provide significant accuracy with no prior assumptions about the behaviour of the wave function. This neural network technique, which extends from two-body to three-body, may provide insight into potential solutions to the nuclear and hadronic many-body problems.

Figures

Figures reproduced from arXiv: 2507.17559 by the authors.

Figure 1
Figure 1. FIG. 1: Diagram of the Deep Neural Network. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Jacobi coordinates in a three-body system. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: DNN-predicted wave function (red line) and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Deuteron radial wave functions Ψ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Predicted reduced radial wave function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: One-body (upper pannel) and two-body (lower [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Predicted probability density [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: One-body (upper pannel) and two-body (lower [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Works this paper leans on

46 extracted references · 33 canonical work pages

  1. [1]

    The en- ergy of the states is given by, E = ℏω n + 3 2 (29) with n = 2q + ℓ for q = 0, 1, 2,

    Study of the two-body isotropic harmonic oscillator For an initial test the two-body DNN code we will solve the 3D isotropic harmonic oscillator, whose potential is, V = 1 2 µω2r2 (27) The solution of this potential can be found in any Quantum Mechanics textbook [28], having the general solution Ψ(r) = N rℓe− 1 2 µωr2 L(ℓ+1/2) k (µωr2)Yℓm(ˆr) (28) with N ...

    work page

  2. [2]

    The deuteron consists of a proton and a neu- tron and is the simplest two-nucleon bound state system

    Study of the deuteron We now analyze the deuteron case, which involves a 3S1-3D1 coupled channels calculation and more complex potentials. The deuteron consists of a proton and a neu- tron and is the simplest two-nucleon bound state system. Until now, the deuteron has been intensively studied both experimentally and theoretically, and the relevant exper- ...

    work page

  3. [3]

    Then, the Schr¨ odinger equation of Eq

    Three-body harmonic oscillator with equal masses For the extension to three body systems, as we did for the two body, let’s start considering a simple isotropic three body harmonic oscillator with a potential is give by V = 1 2 k ρ2 1 + ρ2 2 + ρ2 3 (34) For equal masses this potential is separable, V = 3 4 kρ2 + kλ2, (35) and the solution factorizes as Ψ ...

    work page

  4. [4]

    Las nuevas tecnolog ´ ıas: com- putaci´ on cu´ antica y aprendizaje autom´ atico para estu- diar las interacciones fundamentales y sus aplicaciones a la f ´ ısica m´ edica

    Study of the Triton Finally, in this section we evaluate the validity of the DNN three-body model for coupled channels and more complex potentials involving tensor and spin-orbit terms, by exploring the three-nucleon system. We will consider a J = T = 1 2 state, that’s it, the triton case, with the same χEFT potential employed in the deuteron case of Sec....

  5. [5]

    Kamimura, Phys

    M. Kamimura, Phys. Rev. A 38, 621 (1988)

    1988

  6. [6]

    Kameyama, M

    H. Kameyama, M. Kamimura, and Y. Fukushima, Phys. Rev. C 40, 974 (1989)

    work page 1989

  7. [7]

    Hiyama, Y

    E. Hiyama, Y. Kino, and M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003)

    work page 2003

  8. [8]

    Hiyama, Few Body Syst

    E. Hiyama, Few Body Syst. 53, 189 (2012)

    work page 2012

Show all 46 references
  1. [9]

    Hiyama, PTEP 2012, 01A204 (2012)

    E. Hiyama, PTEP 2012, 01A204 (2012)

    2012

  2. [10]

    Hohenberg and W

    P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

    1964

  3. [11]

    Aspuru-Guzik, Chem

    A. Aspuru-Guzik, Chem. Rev. 119, 10856 (2019), arXiv:1812.09976 [quant-ph]

    2019 arXiv

  4. [12]

    Kirkpatrick et al

    J. Kirkpatrick et al. , Science 374, 1385 (2021)

    2021

  5. [13]

    W. L. McMillan, Phys. Rev. 138, A442 (1965)

    1965

  6. [14]

    Ceperley, G

    D. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 16, 3081 (1977)

    1977

  7. [15]

    D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980)

    1980

  8. [16]

    von der Linden, Physics Reports 220, 53 (1992)

    W. von der Linden, Physics Reports 220, 53 (1992)

    1992

  9. [17]

    S. A. Chin, Physical Review A 42, 6991 (1990)

    1990

  10. [18]

    R. J. Needs, M. D. Towler, N. D. Drummond, and P. L. R ´ ıos, Journal of Physics: Condensed Matter22, 023201 (2009)

    2009

  11. [19]

    Dunjko and H

    V. Dunjko and H. J. Briegel, Rept. Prog. Phys. 81, 074001 (2018)

    2018

  12. [20]

    Mehta, M

    P. Mehta, M. Bukov, C.-H. Wang, A. G. R. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, Phys. Rept. 810, 1 (2019), arXiv:1803.08823 [physics.comp-ph]

    2019 arXiv

  13. [21]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Rev. Mod. Phys. 91, 045002 (2019), arXiv:1903.10563 [physics.comp-ph]

    2019 arXiv

  14. [22]

    Naito, H

    T. Naito, H. Naito, and K. Hashimoto, Phys. Rev. Res. 5, 033189 (2023), arXiv:2302.08965 [physics.comp-ph]

    2023 arXiv

  15. [23]

    C. Wang, T. Naito, J. Li, and H. Liang, (2024), arXiv:2403.16819 [nucl-th]

    2024 arXiv

  16. [24]

    W.-L. Wu, L. Meng, and S.-L. Zhu, (2025), arXiv:2506.20555 [hep-ph]

    2025

  17. [25]

    J. W. T. Keeble and A. Rios, Phys. Lett. B 809, 135743 (2020), arXiv:1911.13092 [nucl-th]

    2020 arXiv

  18. [26]

    Cybenko, Math

    G. Cybenko, Math. Control Signals Syst. 2, 303 (1989)

    1989

  19. [27]

    Hornik, Neural Networks 4, 251 (1991)

    K. Hornik, Neural Networks 4, 251 (1991)

    1991

  20. [28]

    W. S. McCulloch and W. Pitts, The bulletin of mathe- matical biophysics 5, 115 (1943)

    1943

  21. [29]

    A. D. Becke, The Journal of Chemical Physics 140, 18A301 (2014), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/1.4869598/19900307/18a301 1 1.4869598.pdf

    2014 doi

  22. [30]

    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. Lev- enberg, D. Man´ e, R. Monga, S. Moore, D. Murray, C. Olah...

    2015

  23. [31]

    D. P. Kingma and J. Ba (2014) arXiv:1412.6980 [cs.LG]

    2014 arXiv

  24. [32]

    J. J. Sakurai, Modern quantum mechanics (Addison- Wesley, Reading, MA, 1994)

    1994

  25. [33]

    A. S. Householder, The bulletin of mathematical bio- physics 3, 63 (1941)

    1941

  26. [34]

    D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003), arXiv:nucl-th/0304018

    2003 arXiv

  27. [35]

    D. R. Entem and R. Machleidt, Phys. Rev. C 66, 014002 (2002), arXiv:nucl-th/0202039

    2002 arXiv

  28. [36]

    S. K. Saha, D. R. Entem, R. Machleidt, and Y. Nosyk, Phys. Rev. C 107, 034002 (2023), arXiv:2209.13170 [nucl-th]

    2023 arXiv

  29. [37]

    C. A. Bertulani, Nuclear Physics in a Nutshell (Princeton University Press, 2007)

    2007

  30. [38]

    Dumbrajs, R

    O. Dumbrajs, R. Koch, H. Pilkuhn, G. c. Oades, H. Behrens, J. j. De Swart, and P. Kroll, Nucl. Phys. B 216, 277 (1983)

    1983

  31. [39]

    Van Der Leun and C

    C. Van Der Leun and C. Alderliesten, Nuclear Physics A 380, 261 (1982)

    1982

  32. [40]

    Martorell, D

    J. Martorell, D. W. L. Sprung, and D. C. Zheng, Phys. Rev. C 51, 1127 (1995)

    1995

  33. [41]

    M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, and X. Xu, Chinese Physics C 41, 030003 (2017)

    2017

  34. [42]

    Horie and K

    H. Horie and K. Sasaki, Progress of Theoretical Physics 25, 475 (1961), https://academic.oup.com/ptp/article- pdf/25/3/475/5223037/25-3-475.pdf

    1961

  35. [43]

    Umeya and K

    A. Umeya and K. Muto, Nucl. Phys. A 955, 194 (2016)

    2016

  36. [44]

    E. J. Weniger, Journal of Mathematical Physics 26, 276 (1985), https://pubs.aip.org/aip/jmp/article- pdf/26/2/276/19107161/276 1 online.pdf

    1985

  37. [45]

    Mehrem, J

    R. Mehrem, J. T. Londergan, and M. H. Macfarlane, J. Phys. A 24, 1435 (1991)

    1991

  38. [46]

    Mehrem and A

    R. Mehrem and A. Hohenegger, J. Phys. A 43, 9 (2010), arXiv:1006.2108 [math-ph]

    2010 arXiv

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