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arxiv: 2606.28287 · v1 · pith:ZITDKSZ3new · submitted 2026-06-26 · ⚛️ nucl-th · cs.LG

Bridging Ab Initio Symmetries and Global Nuclear Masses with Interpretable Neural Networks

Phong Dang , Evander Espinoza , Xiaoliang Wan , Michela Negro , Jerry P. Draayer , Feng Pan , Tomas Dytrych , Daniel Langr
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David Kekejian
This is my paper

Pith reviewed 2026-06-29 01:47 UTC · model grok-4.3

classification ⚛️ nucl-th cs.LG
keywords nuclear massesSU(4) symmetryCasimir operatorsneural networksWigner symmetrynuclear binding energiesinterpretable modelsglobal mass models
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The pith

Wigner's SU(4) symmetry supplies predictive power for nuclear masses across the entire chart when encoded as Casimir operators in compact neural models.

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

The paper builds three neural-network mass models that embed the Casimir operators of SU(3) and SU(4) symmetries as explicit features. On AME2016 training and AME2020 validation sets the SU(4) operators alone reduce root-mean-square error by nearly half relative to a liquid-drop baseline, and by roughly a fifth on extrapolation. The most compact of the three models, WINN, reaches a validation RMSE of 0.430 MeV while exposing an increase in the quadratic SU(4) Casimir near the neutron dripline and a rise in the quartic operator among superheavy nuclei. These results indicate that the symmetries identified in light nuclei continue to shape binding energies globally.

Core claim

The SU(4) Casimir operators carry information beyond bulk liquid-drop properties; when used as an operator basis they cut RMSE by nearly half on train and test data and by about a fifth on extrapolation. The resulting Wigner-Informed NN (WINN) attains the lowest validation RMSE of 0.430 MeV and further shows an enhancement of the quadratic SU(4) Casimir near the neutron dripline together with an unexpected gain of the quartic operator in the superheavy region.

What carries the argument

The Wigner-Informed NN (WINN), a mass formula that takes the SU(3) and SU(4) Casimir operators directly as its operator basis.

If this is right

  • SU(4) symmetry supplies information beyond the liquid-drop model for global nuclear masses.
  • WINN reaches competitive accuracy while remaining compact enough for direct physical interpretation.
  • The quadratic SU(4) Casimir increases near the neutron dripline, indicating local restoration of Wigner's symmetry.
  • The quartic SU(4) operator gains weight in the superheavy region.

Where Pith is reading between the lines

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

  • The same operator basis could be tested on other global observables such as charge radii or quadrupole moments.
  • The approach offers a route to embed ab initio symmetry constraints into phenomenological mass tables without increasing model complexity.
  • If the quartic-term rise persists in future data, it may point to a new effective degree of freedom in superheavy nuclei.

Load-bearing premise

That the observed error reductions can be attributed specifically to the SU(3) and SU(4) Casimir operators rather than to the general flexibility of neural-network fitting on the chosen data splits.

What would settle it

A control experiment in which a neural network of comparable size and architecture, trained on identical splits but without the Casimir operators, yields RMSE reductions equal to or larger than those obtained when the operators are included.

Figures

Figures reproduced from arXiv: 2606.28287 by Daniel Langr, David Kekejian, Evander Espinoza, Feng Pan, Jerry P. Draayer, Michela Negro, Phong Dang, Tomas Dytrych, Xiaoliang Wan.

Figure 1
Figure 1. Figure 1: Color maps that show evolution of six operators, including total harmonic oscillator quanta ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Correlation matrix of nine features, namely proton number ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Neural network architectures for (a) FINN, (b) GINN, and (c) WINN. Blue nodes are input features; green nodes are hidden neurons arranged in three [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Color maps that show uncertainty of binding energy prediction ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SHAP summary plot for GINN trained on the LDM [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Contribution of each term in the WINN mass formula, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

Ab initio modeling has established Wigner's SU(4) and Elliott's SU(3) as dominant symmetries of the nuclear force in light and intermediate-mass nuclei. We ask whether they also govern nuclear binding across the entire chart. Our aim is not high-precision prediction but physical insight, through interpretable, symmetry-based models. From the SU(3) and SU(4) Casimir operators we construct three neural-network (NN) mass models: Feature-Informed NN (FINN) for point predictions, Gaussian-Informed NN (GINN) adding uncertainty quantification, and Wigner-Informed NN (WINN) -- a mass formula using the Casimirs as an operator basis. All are trained on AME2016 and validated on nuclei new to AME2020. The SU(4) operators alone cut the root-mean-square error (RMSE) by nearly half on train and test data, and by about a fifth on extrapolation, relative to the liquid-drop baseline -- showing that Wigner's symmetry carries predictive information beyond bulk properties. Despite its compact form, WINN reaches the lowest validation RMSE, 0.430 MeV -- competitive with state-of-the-art mass models -- which we read less as a benchmark than as evidence that its symmetry basis captures important physics. WINN further reveals i) an enhancement of the quadratic SU(4) Casimir near the neutron dripline, signaling restoration of Wigner's symmetry, and ii) an unexpected gain of the quartic operator in the superheavy region. We thereby elevate emergent symmetries from the hidden order within individual nuclei to a governing principle of the whole nuclear chart.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper constructs three neural-network mass models (FINN, GINN, WINN) whose inputs or basis functions are the Casimir operators of SU(3) and SU(4). Trained on AME2016 and validated on nuclei new to AME2020, the models are reported to reduce RMSE by nearly half relative to a liquid-drop baseline on train/test sets and by ~20% on extrapolation; WINN, a compact linear combination of the Casimirs, achieves the lowest validation RMSE of 0.430 MeV and is interpreted as evidence that Wigner’s symmetry governs global masses, with additional claims of quadratic-term enhancement near the dripline and quartic-term gain in the superheavy region.

Significance. If the reported RMSE reductions can be shown to arise specifically from the symmetry operators rather than from the greater expressive power of neural networks, the work would supply an interpretable, symmetry-motivated global mass formula that links ab initio findings to the entire chart. The use of a held-out AME2020 validation set and the compact operator basis of WINN are concrete strengths that would remain valuable even if the attribution to SU(4) requires further controls.

major comments (3)
  1. [Results section (RMSE tables)] Results section (RMSE tables): all quantitative claims of improvement (nearly 50% on train/test, ~20% on extrapolation) are made relative to a parametric liquid-drop model rather than to a neural network of identical architecture supplied with non-symmetry features of comparable dimensionality (e.g., polynomials in Z and N up to degree 4). Without this control, the central attribution of the RMSE reduction to SU(4) content cannot be distinguished from the general flexibility of the NN on the chosen data splits.
  2. [§4 (WINN definition)] §4 (WINN definition) and validation paragraph: WINN is a fitted linear combination whose coefficients are determined from the same data used for performance reporting; the interpretation that its basis “captures important physics” therefore rests on post-fit coefficient analysis rather than on a parameter-free derivation or an ablation that isolates the Casimir operators from other possible bases of the same dimension.
  3. [Methods and results] Methods and results: no error bars, hyperparameter sweeps, or ablation studies on the neural-network components are reported, so the quoted validation RMSE of 0.430 MeV cannot be assessed for statistical significance or sensitivity to architecture choices.
minor comments (2)
  1. Notation for the Casimir operators should be defined once with explicit expressions (including normalization conventions) rather than assumed from ab initio literature.
  2. Figure captions for the dripline and superheavy-region plots should state the exact definition of “near the dripline” and the mass-number range used for the quartic-term analysis.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important controls that would strengthen the attribution of performance gains to the symmetry operators. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: Results section (RMSE tables): all quantitative claims of improvement (nearly 50% on train/test, ~20% on extrapolation) are made relative to a parametric liquid-drop model rather than to a neural network of identical architecture supplied with non-symmetry features of comparable dimensionality (e.g., polynomials in Z and N up to degree 4). Without this control, the central attribution of the RMSE reduction to SU(4) content cannot be distinguished from the general flexibility of the NN on the chosen data splits.

    Authors: We agree that a direct comparison against a neural network supplied with non-symmetry features of matched dimensionality is the most rigorous way to isolate the contribution of the Casimir operators. The liquid-drop model was chosen as the conventional baseline for global mass formulas, but it does not control for network expressivity. We will add this control experiment (polynomial features in Z and N up to degree 4, same architecture and training protocol) to the revised Results section and update the RMSE tables accordingly. revision: yes

  2. Referee: §4 (WINN definition) and validation paragraph: WINN is a fitted linear combination whose coefficients are determined from the same data used for performance reporting; the interpretation that its basis “captures important physics” therefore rests on post-fit coefficient analysis rather than on a parameter-free derivation or an ablation that isolates the Casimir operators from other possible bases of the same dimension.

    Authors: The operator basis itself is selected on theoretical grounds from the established SU(3) and SU(4) Casimirs of ab initio calculations, which supplies the parameter-free motivation for the functional form. The subsequent linear fit then determines the relative weights and reveals systematic trends (quadratic enhancement near the dripline, quartic gain in the superheavy region). We will revise §4 to separate the theoretical basis choice from the data-driven coefficient analysis more explicitly and will add a short ablation comparing WINN performance against a linear model using an alternative four-dimensional basis of comparable size (e.g., simple powers of Z and N) to quantify the advantage of the symmetry operators. revision: partial

  3. Referee: Methods and results: no error bars, hyperparameter sweeps, or ablation studies on the neural-network components are reported, so the quoted validation RMSE of 0.430 MeV cannot be assessed for statistical significance or sensitivity to architecture choices.

    Authors: We acknowledge that the absence of uncertainty estimates and sensitivity checks limits the ability to judge robustness. In the revised manuscript we will report bootstrap-derived standard errors on all RMSE values and include a concise hyperparameter-sensitivity table (varying learning rate, hidden-layer width, and regularization within the ranges explored during training). Full ablation studies on the neural-network components of FINN and GINN will be added to the Methods section. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs neural-network mass models by training on AME2016 data with SU(3)/SU(4) Casimir operators as explicit input features and reports empirical RMSE reductions versus a liquid-drop baseline on both in-sample and held-out AME2020 nuclei. These performance numbers are direct outputs of the supervised fitting procedure rather than quantities derived by algebraic identity or self-referential definition from the same inputs. No load-bearing self-citation, uniqueness theorem, or ansatz is invoked to justify the central attribution; the comparison to the parametric baseline, while methodologically debatable, does not render the reported test-set or extrapolation errors tautological. The derivation chain therefore remains self-contained as an empirical feature-ablation study.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that SU(4)/SU(3) Casimirs remain relevant globally and on the standard assumption that neural-network training on AME2016 produces generalizable coefficients. No new physical entities are postulated.

free parameters (2)
  • Neural network weights and biases
    Hundreds to thousands of parameters fitted to AME2016 masses in FINN, GINN, and WINN.
  • WINN operator coefficients
    Explicit coefficients multiplying the Casimir operators are determined by fitting to data.
axioms (1)
  • domain assumption SU(4) and SU(3) Casimir operators encode the dominant symmetry information relevant to nuclear binding energies across the chart.
    Invoked to justify using these operators as the feature basis; drawn from ab initio results in light nuclei.

pith-pipeline@v0.9.1-grok · 5864 in / 1588 out tokens · 34065 ms · 2026-06-29T01:47:10.664615+00:00 · methodology

discussion (0)

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Reference graph

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