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Learning Nuclear Structure with AI: Radii and Collectivity

Giuliano Giacalone, Sokratis Trifinopoulos, Mike Williams

arXiv:2609.17838v1nucl-thcs.AIcs.LGnucl-ex

TL;DR

NuCLR uses held-out multi-task ensembles to learn charge radii and B(E2) from the shared nuclear-data record. Multi-task learning reduces RMS errors relative to single-task learning, while comparisons identify regions where new measurements could distinguish data-driven from theory-driven extrapolations.

  • Problem

    Learning how nuclear information is organized across observables and nuclei can provide an empirical baseline for extrapolations and experimental design, while charge radii and B(E2) test distinct structural patterns.

    Source: pp. 1–2

  • Method

    Held-out NuCLR ensembles use a shared, task-conditioned representation to predict charge radii and B(E2), with prediction intervals calibrated and evaluated on held-out measured nuclei.

    Source: p. 2

  • Results

    MTL reduces charge-radius RMS from 0.0760 to 0.0147 fm and B(E2) RMS from 0.295 to 0.192 e^2b^2 relative to STL.

    Source: p. 3

  • Takeaways & Limitations

    Comparisons with nuclear models show where measured trends are reproduced and where new measurements could distinguish empirical from theory-driven extrapolations.

    Source: p. 5

    The work has not established how correlations responsible for the charge-radius and B(E2) predictions emerge within the learned representation.

    Source: p. 5

Abstract

from arXiv · show

Low-energy nuclear structure is encoded in a broad body of experimental information across the chart of nuclides. Learning how this information is organized across observables and nuclei can provide a data-driven empirical baseline for theoretical extrapolations and experimental design. Here, we develop held-out ensembles based on NuCLR (Nuclear Co-Learned Representations), a multi-task model of nuclear data, to study charge radii and electric-quadrupole transition strengths. Out-of-fold (OOF) validation shows that shared representation improves performance over single-task learning, yielding a charge-radius $\mathrm{RMS}$ deviation of $0.0147~{\rm fm}$ and a $\mathrm{B(E2)}$ $\mathrm{RMS}$ deviation of $0.192~e^2{\rm b}^2$ across hundreds of nuclides, competitive with state-of-the-art nuclear models. Our error bars estimate the expected prediction accuracy across the nuclear chart, highlighting regions where new data would encode information beyond the learned patterns. NuCLR thus serves as a data-driven surveyor of nuclear structure and a step toward a shared, multi-observable foundation model of the nuclear chart.

S1. ADDITIONAL ISOTOPIC CHAINS

This section introduces additional isotope-chain results for charge radii and quadrupole transitions.

S1. ADDITIONAL ISOTOPIC CHAINS

  • Additional isotope-chain results cover charge radii and quadrupole transitions.The charge-radius chains are Ca, Sn, and Hg; the B(E2) chains are Zr, Sn, Xe, and Th.Source: S1. ADDITIONAL ISOTOPIC CHAINS · p. 9

A. Charge radii

The additional charge-radius chains show both learned trends and regions where NuCLR does not recover measured structure. Pm predictions also agree with independent isotope-shift data.

A. Charge radii

  • NuCLR intervals are too broad to establish whether the Ca arch at 20 < N < 28 is learned reliably; for smaller N its predictions agree better with data than nuclear models.For N ≳20, NuCLR is consistent with data and the BSkG3 and 5DCH predictions.Source: A. Charge radii · p. 9, Figure S1 · p. 9
  • In Sn, NuCLR predicts kinks around N = 82 that are absent from measurements, and its broader intervals mark this region as less supported by learned data patterns.Nuclear models reproduce the smooth measured trend.Source: A. Charge radii · p. 9, Figure S1 · p. 9
  • NuCLR's narrow Hg intervals remain smooth and miss the measured odd-even staggering and non-monotonic behavior associated with shape coexistence.The passage identifies this as a case where the shared representation does not recover structure present in the measured observable.Source: A. Charge radii · p. 9, Figure S1 · p. 9
  • NuCLR's Pm trend agrees with BSkG3 and independently measured isotope shifts.The isotope shifts are aligned to a common 147Pm NuCLR anchor in Fig. S2.Source: A. Charge radii · p. 10, Figure S2 · p. 10

B. Quadrupole transitions

The additional B(E2) chains show where NuCLR captures distinct trends and where its data-driven extrapolations diverge from measurements or nuclear models.

B. Quadrupole transitions

  • NuCLR fails to reproduce the sharp rise in Zr collectivity between N = 58 and N = 60, a transition it cannot reliably infer from the observables in its shared representation.The BSkG3 and 5DCH models provide only a qualitative description of the observed trends.Source: B. Quadrupole transitions · p. 10, Figure S3 · p. 11
  • NuCLR distinguishes flat B(E2) evolution in Sn from the shell-driven pattern in Xe, although both chains span the same major neutron shell.All considered models reproduce Xe's broad trends, while the nuclear models also reproduce Sn's flat measured evolution.Source: B. Quadrupole transitions · p. 10, Figure S3 · p. 11
  • Beyond the measured Th region, NuCLR predicts decreasing collectivity while EDF models remain strongly collective, making new measurements a sharp discriminator between the continuations.Both NuCLR and EDF models predict suppressed strengths around N = 126 followed by a rapid return to large collectivity below the measured sequence.Source: B. Quadrupole transitions · pp. 10–11, Figure S3 · p. 11

A. Network Architecture

NuCLR represents each supervised example with nucleus coordinates and an observable label, then uses a shared residual network and task-specific readout.

A. Network Architecture

  • Each supervised row is represented by discrete learned embeddings for proton number Z, neutron number N, and observable label q.The embeddings are concatenated before entering the network.Source: A. Network Architecture · p. 12
  • A SiLU-activated affine layer feeds two residual fully connected blocks, each with two 1024-dimensional ReLU linear layers and an identity skip connection.The campaigns use hidden width 1024 and no dropout or normalization.Source: A. Network Architecture · p. 12
  • The readout produces task components, and the requested observable's component supplies predictions and loss after affine mapping to its target range.Training uses full-batch MuAdamW at learning rate 10^-4 and weight decay 10^-2 on non-bias parameters; the final checkpoint is retained without a learning-rate schedule or early stopping.Source: A. Network Architecture · p. 12

B. Pretraining and Target Masking

NuCLR uses separate charge-radius and B(E2) campaigns with task-specific pretraining and target masking. Both produce held-out ensembles for measured nuclei and full-ensemble predictions for unmeasured nuclei.

B. Pretraining and Target Masking

  • The radius campaign pretrains without Rch, then finetunes with radius and nine support tasks active across 868 nuclei in ten folds.Four independently seeded models are trained per fold; binding-energy and BW2-residual supervision remains active during adaptation.Source: B. Pretraining and Target Masking · p. 12
  • The B(E2) campaign hides its target throughout pretraining and masks same-fold quadrupole proxies during both pretraining and finetuning.Its ten folds are stratified by B(E2) magnitude and broad Z region; four models per fold are finetuned for 30,000 epochs.Source: B. Pretraining and Target Masking · pp. 12–13
  • Positive B(E2) targets are modeled in log10 space, with α = 0.75 weighting the log-scale term before predictions are transformed back to e^2b^2.Non-positive entries are excluded.Source: B. Pretraining and Target Masking · p. 13
  • Each campaign yields 40 finetuned models, using only the four held-out models for measured targets and the full ensemble for unmeasured targets.The held-out predictions prevent a model that saw a target value from contributing to that target's measured prediction.Source: B. Pretraining and Target Masking · p. 13

C. Auxiliary Observables

The campaigns use auxiliary mass, separation-energy, decay-energy, and other nuclear observables as support tasks, with different selections for the B(E2) leakage-safe protocol.

C. Auxiliary Observables

  • The leakage-safe B(E2) campaign masks quadrupole proxies, while auxiliary observables include Z, N, separation and decay energies, BW2 residual, and Rch.BW2 is the residual associated with a Bethe-Weizsaecker-type liquid-drop binding-energy baseline; Sn and Sp are separation energies, and Qα, Qβ−, and QEC are decay energies.Source: C. Auxiliary Observables · p. 13
  • Auxiliary mass, separation-energy, and decay-energy labels come from evaluated tables that retain entries flagged as systematic estimates, so not every auxiliary label is a direct measurement.The stated tables are from AME2020.Source: C. Auxiliary Observables · p. 13

D. Uncertainty Interval Model

The interval model learns local prediction difficulty from out-of-fold residuals, then calibrates widths using separate measured-nucleus groups. Its coverage is evaluated empirically, especially under regional withholding.

D. Uncertainty Interval Model

  • The interval model learns local error scales from genuinely OOF residuals without changing NuCLR's central predictions.Three disjoint measured-nucleus groups train the scale model, calibrate its width, and evaluate coverage, with roles rotated over repeated partitions.Source: D. Uncertainty Interval Model · p. 13
  • A gradient-boosted regressor predicts relative residual scale from nuclear-structure and chart-location features, excluding prediction magnitude, replica spread, and distance to measured targets.The selected model uses 200 depth-2 trees; adding replica-spread or nearest-target support features did not improve local calibration.Source: D. Uncertainty Interval Model · pp. 13–14
  • Across each random five-fold partition, three folds train hϕ, one disjoint calibration fold sets the interval multiplier, and one held-out test fold evaluates coverage; all 20 ordered calibration/test assignments yield 100 residual-scale fits.Source: D. Uncertainty Interval Model · p. 14
  • The reported interval half-width is the calibrated multiplier times the local scale, and the lower endpoint is clipped at zero.The calibration factor converts relative scale to an interval with empirically tested marginal coverage; it is not added to ensemble spread.Source: D. Uncertainty Interval Model · p. 14
  • Local calibration is measured as RMS deviation of cellwise 68% coverage from 0.68 across translated 8 × 8 chart regions with at least eight test nuclei.Lower scores indicate more uniform local calibration; regional diagnostics hold out only the residual-scale model.Source: D. Uncertainty Interval Model · p. 15
  • Randomly held-out nuclei show near-nominal overall coverage, but withholding contiguous regions degrades coverage, so unmeasured-nucleus widths are not interpreted as calibrated intervals.Open-point bars show learned OOF residual scale for unmeasured nuclei; no monotonic increase with distance from measured data is imposed.Source: D. Uncertainty Interval Model · p. 15
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