Bridging Ab Initio Symmetries and Global Nuclear Masses with Interpretable Neural Networks Phong Dang
a,∗
, Evander Espinozaa , Xiaoliang Wanb , Michela Negro a , Jerry P. Draayer Dytrych f,a,c , Daniel Langr g,c , David Kekejian c
a,c
, Feng Pan
d,e,a
, Tomáš
a Department of Physics and Astronomy, Louisiana State University, Baton Rouge, 70803, LA, USA b Department of Mathematics, Louisiana State University, Baton Rouge, 70803, LA, USA c Quantum CodeX, Baton Rouge, 70808, LA, USA d Department of Physics, School of Science, Huzhou Normal University, Huzhou, 313000, China e Department of Physics, Liaoning Normal University, Dalian, 116029, China f Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Prague, 25068, Řež, Czech Republic
arXiv:2606.28287v1 [nucl-th] 26 Jun 2026
g Department of Computer Systems, Faculty of Information Technology, Czech Technical University, Prague, 16000, Czech Republic
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. Keywords: Nuclear mass, Machine learning, Interpretable neural network, Wigner’s SU(4) symmetry, Elliott’s SU(3) symmetry
1. Introduction Advances in computational hardware and data availability have catalyzed the adoption of machine learning (ML) in nuclear physics [1]. In particular, neural networks (NNs) have shown strong performance in modeling nuclear mass—a central quantity for r-process nucleosynthesis [2] and the composition of neutron star crusts [3]. Architectures explored to this end include light gradient boosting machine [4], mixture density network [5], fully connected NNs [6], decision trees [7], convolutional NN [8], Bayesian models [9–12], Gaussian processes and support vector regression [13, 14], as well as a hybrid between NN and density functional theory (DFT) [15]. The advantage of ML lies in its ability to capture recurring patterns that underpin physical systems. Atomic nuclei are particularly intriguing in this regard. Despite the complexity of the nuclear force, low-lying spectra are consistently dominated by two spatial symmetries: Elliott’s SU(3) [16–18] and symplectic
∗ Corresponding author
Email address: [email protected] (Phong Dang )
Sp(3, R) [19] symmetries. Over the last two decades, ab initio nuclear theory has unveiled the dominance of these symmetries in light and intermediate-mass nuclei [20–23], leveraged them to model clustering phenomena and reactions [24, 25], and probed new physics beyond the Standard Model [26, 27]. Prior to the introduction of the SU(3) symmetry, Wigner proposed the SU(4) [or U(4)] symmetry in 1937 [28, 29], for which he was awarded the Nobel Prize in 1963, alongside the pioneers of the nuclear shell model—Mayer [30, 31] and Jensen, Haxel and Suess [32]. Also known as the supermultiplet symmetry, it unifies the spin and isospin symmetries into a single framework. While the spin-orbit force breaking the symmetry in heavy nuclei casted doubt on the theory in the 1940s-50s, the 1960s saw renewed interest following the first observation of isobaric analog states in medium and heavy nuclei [33] and the formulation of a nuclear mass formula involving the second-order Casimir invariant of the SU(4) group as a test of this symmetry [34, 35]. This model was later extended with higher-order Casimirs and shell corrections [36–38], exceeding the Bethe-Weizsäcker formula [39], which treats nuclei as liquid drops (LDM). Approaching its nine-decade milestone, Wigner’s symmetry has remained at the forefront of nuclear physics. First-
Figure 1: Color maps that show evolution of six operators, including total harmonic oscillator quanta (Nℏω ), SU(3) Casimir operators (C2 , C3 ), SU(4) Casimir operators (C2 , C3 , C4 ), across the entire nuclear chart. The dotted lines correspond to closed harmonic oscillator shells for protons and neutrons.
relevance of these features through two NNs, whose results inspire the interpretable mass formula presented in Sec. 4. Key findings and future directions are summarized in Sec. 5.
principles studies recently establish SU(4) as a dominant symmetry of the nuclear force, which arises naturally from the strong interaction in effective field theories [40, 41], is strongly manifested in light nuclei [42, 43], and governs the coupling of nuclear structure to the weak force [42]. These insights have been exploited to tame the sign problem of nuclear lattice simulations [44, 45] and to compress no-core shell-model spaces via the ab initio Symmetry Adapted Model (SAM) [43]. As with Elliott’s SU(3), however, this evidence has been gathered mostly in light and intermediate-mass nuclei, where ab initio calculations are computationally tractable.
2. Data collection 2.1. Atomic mass datasets We collect nuclear masses from two widely used datasets, namely AME2016 [47, 48] and AME2020 [49, 50], excluding those extrapolated from theory. Following Refs. [4, 6, 13], we randomly split AME2016 into a train set (1908 nuclei) and a test set (478 nuclei) in an 80:20 ratio, and take the nuclei first reported in AME2020 as the validation set (71 nuclei). Their roles are distinct: the train set fixes the network parameters; the test set, drawn from the same distribution but unseen during training, monitors overfitting (see Sec. S3 in Supplemental Material (SM) and Ref. [51]); and the validation set probes our NNs’ ability to predict completely unseen masses. We also exclude light nuclei (Z < 8 and N < 8) from our calculations.
In this paper, we take a deliberate leap by asking whether the SU(3) and SU(4) symmetries also govern binding across the nuclear landscape, well into regions of exotic isotopes, which are being synthesized beyond the dripline at frontier facilities like FRIB, TRIUMF, GSI, GANIL and potentially provide input for NASA’s Compton Spectrometer and Imager (COSI) mission [46], which is planned for launch in 2027. Traditionally, these emergent symmetries have guided the construction of the many-body basis of individual nuclei. Here we broaden their reach by turning the same symmetries, grounded in first principles, into a lens for probing and inferring nuclear systematics across the entire chart. We build a family of NN mass models—FINN, GINN, and WINN—from a compact set of features derived from the SU(3) and SU(4) Casimir invariants. The culmination is WINN, an interpretable bilinear formula whose eight symmetry-motivated terms reproduce known masses with a validation root-mean-square error (RMSE) of 0.430 MeV. This perspective opens new frontiers—the effectiveness of such a minimal symmetry basis invites broader extensions in both symmetry content and the range of observables for future work.
2.2. Symmetry-inspired features In this work, we employ nine physics-informed features which are grouped into three categories. LDM features—To describe bulk properties of nuclear binding, we consider features appearing in the LDM. The most conspicuous choices are the proton number Z, which encodes Coulomb repulsion, and A2/3 for the surface energy. While one could include A for the volume term, we use the neutron number N instead, since A is perfectly colinear with N and Z; choosing N and Z also lets the models somewhat resolve the neutron and proton shell closures. Finally, we include T z = (N − Z)/2 to capture the asymmetry of the nuclear chart. SU(4) features—Inspired by former works [36–38], we consider the SU(4) Casimir operators (see Sec. S1 in SM and Refs.
The remainder of this paper is organized as follows. In Sec. 2, we describe how mass data are collected and split into train, test and validation sets, and introduce the symmetryinspired features used in our models. Section 3 establishes the 2
mation coefficient (MIC) [60], which captures both linear and nonlinear dependence, rather than the Pearson or Spearman coefficients [61]. First, Nℏω is strongly correlated with N and Z (MIC ≈ 0.9), indicating that it largely tracks the mass number A, consistent with its evolution in Fig. 1. Second, the SU(3) invariants are tightly coupled to each other (MIC = 0.93), as both encode quadrupole collectivity, while sharing only moderate information with the bulk features (MIC ≈ 0.6-0.7). Third, the even SU(4) Casimirs (C2 and C4 ) are perfectly correlated with T z (MIC = 1.00). That means, for the dominant irrep, their eigenvalues reduce to smooth even functions of T z (see Sec. S1 in SM), implying that C2 and C4 add no statistical information beyond the isospin asymmetry. However, their role is to supply the specific power functions of T z that embody the supermultiplet structure as a physically motivated inductive bias. Fourth, C3 [SU(4)] is weakly correlated with every other feature (MIC ≤ 0.32 with the bulk and SU(3) sectors1 ), reflecting the sign fluctuation noted above. Section S6 in SM demonstrates that C3 [SU(4)] is an irrelevant feature and is excluded henceforth.
3. Symmetry-informed neural networks
Figure 2: Correlation matrix of nine features, namely proton number (Z), neutron number (N), total harmonic oscillator quanta (Nℏω ), SU(3) Casimir operators (C2 , C3 ), SU(4) Casimir operators (C2 , C3 , C4 ), and isospin projection (T z = (N − Z)/2). (A2/3 is not included.) Along the diagonal are distributions of the features; the upper half of the matrix contains the maximal information coefficients, whereas the lower half displays scatter plots for all feature pairs.
3.1. Architecture We first employ two NN architectures, which learn a direct functional map y = F (x) from features to binding energy, to determine whether the symmetry features are relevant for nuclear binding. This is carried out by testing four combinations of features (Table 1): the LDM baseline, and the LDM augmented with SU(3) and/or SU(4) features. If the symmetry features are indeed important, one must see improvement on RMSE across the train, test and validation datasets. The two NNs are i) a single-output NN (called FINN for Feature-Informed NN) that treats binding energy (BE) as an absolute output [Fig. 3(a)], ii) a two-output NN (called GINN for Gaussian-Informed NN) that models BE as a statistical variable with expectation value (µBE ) and uncertainty (σBE ) [Fig. 3(b)]. The loss function subsumes two terms, L = L0 + λL1 with the weight λ chosen to be 2.0 for the FINN and 4.0 for the GINN. Here, L0 is an architecture-based loss, i.e., the mean-squared error (MSE) for the FINN and the negative log-likelihood (NLL) loss [⟨(BE − µBE )2 /2σ2BE + log σBE ⟩] for the GINN; L1 is a physics-informed loss adapted from Refs. [5, 13, 62] and based on the Garvey-Kelson (GK) relations [63]. Both the NNs are trained on mini batches of size 64 for 2000 epochs; at every mini-batch step, L1 is evaluated on the whole train set. We also use the SOAP optimizer [64], a second-order method with learning rate of 10−3 that we find outperforms the Adam optimizer [65] on the validation benchmark. To facilitate learning, all features and target BE are standardized to zero mean and unit variance as described in Sec. S4 of SM.
[52–56]) to be features. These include the two-, three- and fourbody operators, C2 , C3 and C4 , that describe spin-isospin polarization forces. Since these operators depend on SU(4) irreducible representations (irreps) and several irreps are allowed for a given nucleus, we hypothesize that the preference toward the lowest-C2 irrep revealed by the SAM’s first-principles calculations for light nuclei [43] also extends to the entire nuclear chart. Figure 1 shows evolution of these operators. While the quadratic and quartic operators gradually increase for heavier nuclei, the cubic Casimir exhibits a sign fluctuation between odd-mass nuclei, which becomes increasingly pronounced for very neutron-rich isotopes. SU(3) features—We select three features stemming from the three-dimensional harmonic oscillator (3DHO) shells. One is the number operator Nℏω that counts the total number of oscillator quanta of a particular distribution of nucleons across the shells. The others are the second- and third-order invariants of the SU(3) group (see Sec. S2 in SM and Ref. [57]). The former, C2 , signifies the magnitude of the mass quadrupole moment, whereas the latter, C3 , indicates whether the nucleus is spherical (zero), prolate (positive), or oblate (negative) [58, 59]. For each nucleus, an SU(3) irrep is selected such that it combines antisymmetrically with the SU(4) irrep and maximizes deformation [21]. Across the chart, these operators peak at mid-shells where nuclei tend to deform and get suppressed at shell closures where nuclei are mostly spherical (Fig. 1). Feature correlations—Before training our models, we examine the pairwise relationships among the features. As the scatter plots in Fig. 2 show, these relationships are predominantly nonmonotonic; hence, we quantify them with the maximal infor-
1 The Pearson and Spearman coefficients are zero for these relationships, showing that they are unable to detect these complex correlations, depsite the visible trends in the scatter plots.
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 layers of 64, 128, and 64 units; red nodes are outputs. (a) FINN produces a single binding-energy prediction BE. (b) GINN shares the same backbone as FINN but branches into two independent linear heads after the last hidden layer, yielding a predicted mean µBE and an uncertainty σBE . (c) WINN admits only (Z, N) as inputs and outputs eight (N, Z)-dependent coupling strengths {a, . . . , h}; the binding energy is computed as the sum of the coupling strengths multiplied by the corresponding physics feature. We apply rectified linear unit (ReLU) activation function at each hidden layer of WINN and adopt its parametric version (PReLU) for FINN and GINN, see Sec. S5 in SM.
Feature set LDM LDM+SU(4) LDM+SU(3) LDM+SU(4)+SU(3)
Train 0.944 0.521 0.915 0.411
FINN Test Validation 1.111 1.270 0.588 1.024 1.306 1.603 0.552 0.624
Train 0.869 0.604 1.036 0.400
GINN Test Validation 1.149 1.304 0.654 1.046 1.404 1.280 0.488 0.793
Train
WINN Test Validation
0.400
0.447
0.430
Table 1: Root-mean-square error (RMSE, in MeV) of FINN, GINN and WINN across four feature sets. The train and test sets are drawn from AME2016 (80:20 random split); the validation set consists of nuclei first measured in AME2020 and serves as the extrapolation benchmark. WINN is reported only for the full LDM+SU(4)+SU(3) feature set, as its bilinear structure is defined over the complete set of Casimir operators to encode as much physics as possible.
tural: in the NLL loss the gradient on µBE is weighted by 1/σ2BE , thus the network can lower the loss by inflating σBE , trading point accuracy for wider intervals. To counter this, we enlarge λ to 4.0 for a stronger constraint and adopt the β-NLL weighting [67] that rescales each loss by a stop-gradient factor σ2β BE with β = 0.5 to tighten point accuracy. The FINN, minimizing the MSE directly, is immune to this trade-off and tends to retain an RMSE edge. Nonetheless, the GINN’s value lies not in point accuracy but in the calibrated per-nucleus uncertainty σBE (Fig. 4), which the FINN cannot supply.
3.2. Results and interpretation Table 1 reports the NN’s performance across four feature sets. For the FINN, the LDM baseline with only the macroscopic inputs (Z, N, A2/3 , T z ) achieves a validation RMSE of 1.270 MeV, consistent with the inherent limitation of a purely bulk description. Adding the SU(4) Casimirs reduces the error sharply—by nearly half on the train and test sets and by about a fifth on validation (to 1.024 MeV)—demonstrating that Wigner’s symmetry carries substantial predictive information beyond the liquid-drop terms. Expanding the feature set further with the SU(3) invariants yields the best result, 0.624 MeV on the validation set. By contrast, the SU(3) features alone on top of the LDM do not improve the validation error (1.603 MeV), confirming that SU(4) is the dominant contributor [66] and that SU(3) becomes beneficial only in combination with it. The GINN follows a similar trend with respect to feature richness. The full feature set continues to achieve the lowest validation RMSE (0.793 MeV). Moreover, by examining the uncertainty σBE across the nuclear chart (Fig. 4), one can infer the confidence of the model. With the LDM features, the model is more confident for heavy nuclei that behave mostly like a fluid and displays a lack of confidence for intermediatemass nuclei where microscopic effects are important, which are addressed by the SU(3) features. The SU(4) features increase confidence broadly across the chart with the most notable improvement for exotic isotopes near the driplines. On all datasets, the FINN attains a lower RMSE than the GINN for most cases (Table 1). This systematic offset is struc-
Figure 4: Color maps that show uncertainty of binding energy prediction (σBE ) from GINN trained with four combinations of features listed in Table 1.
4
two architectures. This weight controls how strongly the model is penalized for violating local nuclear-chart consistency. For the FINN and GINN, which have no built-in knowledge of how BE scales with N and Z, a large weight is essential to enforce this locality. For the WINN, the operator basis already encodes this regularity, so the GK term acts as a lightweight refinement rather than a structural constraint, and a small weight suffices. 4.2. Results and interpretation The WINN’s performance is summarized in Table 1. This guided approach surpasses the other NNs: it not only achieves the best validation RMSE (0.430 MeV) but also maintains an excellent balance across the train, test, and validation sets. Notably, with only eight operators, the WINN is comparable with other modern mass models, see Table 3 in Ref. [4]. We also find that the pairing effect is captured without an explicit term: the BE residuals exhibit no systematic sign inversion between even-even and odd-odd nuclei and no staggering for odd-A isotopes (see Sec. S7 in SM). To assess WINN’s predictive capability beyond the known isotopes, we compare its BE predictions against two wellestablished mass models: the macroscopic-microscopic WS3 model [70] and the fully microscopic Hartree-Fock-Bogoliubov model HFB-26 [71], evaluated on almost 6000 nuclei that each model predicts to be bound beyond those in AME2020. This comparison is asymmetric by construction: both reference models are global fits with dozens of parameters, whereas WINN relies on a bilinear formula with only eight symmetry operators. Stratifying the comparison by distance from the AME2020 boundary within each isotopic chain, we find good agreement near the known region: for nuclei within 5 neutrons of the last measured isotopes, WINN matches WS3 (HFB-26) within 5 MeV for 78% (71%) of cases, with a mean absolute deviation (MAD) of 3.6 (4.6) MeV. Agreement degrades systematically deeper into unexplored territory, the MAD rising to 6.0 (6.9) MeV at 5-10 neutrons, 10.5 (11.1) MeV at 10-20 neutrons, and 26 (24) MeV beyond 20 neutrons from the last known nuclei. The near-identical trends against two independent models confirm that this degradation reflects the genuine limits of extrapolating a compact eight-term model. We regard the near-boundary performance as encouraging evidence that the SU(3) ⊗ SU(4) operator basis captures the dominant physics up to roughly 10 neutrons from the known chart. Reliable WINN predictions deeper into terra incognita will require extending Eq. (1) with additional operators or incorporating microscopic inputs beyond the current work. As noted above, the purpose of the WINN is to expose the role that each feature plays across the chart of nuclides. By analyzing the fractional contribution (Fig. 6) of each term in Eq. (1), we observe that a is the primary driver of nuclear binding, accounting for more than 45% of the mass for all nuclei, followed by Nℏω at about 25-50% level. The term Z 2 mimics Coulomb energy, lowering BE on the neutron-deficient side, whereas the term T z2 becomes vital toward the neutron dripline and in superheavy elements. The model also successfully learns the shell structure encoded in the SU(3) Casimirs. Near shell closures, where defor-
Figure 5: SHAP summary plot for GINN trained on the LDM+SU(4)+SU(3) feature set and evaluated on the test set. Each point represents one nucleus; the horizontal axis gives the SHAP value, i.e., the contribution of a feature to the predicted binding energy relative to the mean prediction. Features are ranked by decreasing mean SHAP value from top to bottom. The color encodes the feature value. A positive (negative) SHAP value indicates that the feature pushes the prediction above (below) the mean.
We also probe feature influence using the Shapley additive explanations (SHAP) values [68]. The summary plot for the full-set GINN (Fig. 5) confirms that nuclear masses are primarily driven by the bulk LDM features. Notably, Nℏω ranks unexpectedly high—comparable to the LDM ones—likely because it grows monotonically across the chart (Fig. 1) and correlates strongly with N and Z (Fig. 2), acting as a proxy for the omitted mass number A. In addition, we witness that the SU(4) features outrank the SU(3) ones, consistent with the results in Table 1 and Fig. 4. 4. An interpretable symmetry-informed neural network 4.1. Architecture While the NNs of the preceding section establish the relevance of the symmetry features, they obscure how each feature shapes binding across different regions of the nuclear landscape. Motivated by those results and earlier works [36, 38], we propose a four-body bilinear mass formula, BE = a + bZ 2 + cT z2 + dNℏω + eC2 [SU(3)] + f C3 [SU(3)] + gC2 [SU(4)] + hC4 [SU(4)],
(1)
where the coupling strengths {a, b, c, d, e, f, g, h} are functions of N and Z. The role of a is to include the “leftover” of nuclear binding beyond the symmetry descriptors. As mentioned in the correlation analysis (Sec. 2.2), for the leading irrep, the even SU(4) Casimirs reduce to smooth functions of |T z |; therefore, adding them on top of T z2 allows the model to learn the physics associated with the widely known Wigner energy [69]. We design a NN, termed WINN (Wigner-Informed NN), to learn the evolution of the coupling strengths across the nuclear chart [Fig. 3(c)]. The WINN is trained with a loss similar to the FINN, but with only 1000 epochs and weight decay of 0.01 to prevent overfitting. We also find that the GK weight must be set much smaller for the WINN (λ = 0.15) than for the other 5
5. Summary and future outlooks We have presented three symmetry-informed NN mass models built on the Casimir operators of Wigner’s SU(4) and Elliott’s SU(3) symmetries, trained on AME2016 and validated on nuclei first reported in AME2020 as a genuine test of extrapolation. The SU(4) Casimirs alone reduce RMSE by nearly half on the train and test sets and by about a fifth on validation relative to the LDM baseline; and the SHAP analysis ranks the SU(4) features above the SU(3) ones, establishing Wigner’s symmetry as the dominant microscopic contributor beyond bulk properties [66]. Combining both sectors yields validation RMSEs of 0.624 and 0.793 MeV for the FINN and GINN, while the eight-term WINN does best at 0.430 MeV—evidence that embedding physics directly in the architecture is more effective than supplying it as input features alone. We emphasize that high-precision mass prediction was never our goal: WINN’s competitive RMSE matters less as a benchmark than as proof that a compact symmetry basis encodes essential physics. The WINN coupling strengths further illuminate the physics of different mass regions. The SU(3) Casimirs capture deformation-driven shell effects, peaking at mid-shell and suppressed at shell closures. The quadratic SU(4) Casimir is enhanced near the neutron dripline (Z ≈ 8-60), which we associate with the restoration of Wigner’s symmetry under spinorbit quenching. The significant rise of the quartic Casimir C4 [SU(4)] in the neutron-rich superheavy region, where αsystematics becomes dominant, warrants a dedicated study and will be pursued in follow-up work. We highlight that, throughout this work, the SU(3) and SU(4) symmetries are assumed to hold exactly across the chart, with irrep selection inspired by ab initio results. In reality they are broken, a difficulty that has historically been addressed through pseudo and proxy schemes [79–84]. Our results carry a twofold implication: i) the traditional algebraic solutions may improve the NNs through better choices of irreps, and ii) the NN architecture may itself offer a pathway for resolving the symmetry breaking, owing to its capacity to absorb the associated nonlinearities. Both possibilities will be explored in future work. Finally, beyond the novel results, this work opens a new door in how we engage the emergent symmetries of the nuclear force. Where they once organized the structure of individual nuclei, ML now lets them speak to the systematics of the global nuclear landscape. In particular, extending the framework to other observables, e.g., β-decay, is a natural step toward a datadriven, symmetry-informed description of nuclear structure.
Figure 6: Contribution of each term in the WINN mass formula, Eq. (1), relative to the binding energy across the nuclear chart.
mation is low, BE is largely unaffected by these two operators; at mid-shell, C2 lowers the masses by a few percent while C3 contributes less than 1%. Moreover, the signs of these coupling strengths are consistent with those determined from spectra of α-like nuclei in Refs. [72, 73]. Lastly, the SU(4) Casimirs exhibit two notable behaviors. The quadratic C2 contributes at the few-percent level across most of the nuclear chart; however, a pronounced enhancement—reaching about 10-40% of BE and competing with the shell-driven Nℏω —emerges near the neutron dripline in the region Z ≈ 8-60. This behavior points to the long-held expectation that Wigner’s symmetry is restored in neutron-rich isotopes [35, 37] due to the suppression of the spin-orbit splitting [74–76]. (The degree of SU(4) restoration has been assessed for calcium isotopes [77] and more recently for heavy and superheavy nuclei through Gamow-Teller resonances [78].) A similar dripline enhancement is observed for the fourth-order Casimir, though roughly an order of magnitude smaller. Interestingly, C4 gains substantial strength in the superheavy region (N ≳ 126), becoming competitive with C2 . Since C4 is a fourbody operator sensitive to quartet spin-isospin correlations, it might be worth investigating the microscopic origin of this amplification and its connection to the dominance of α-systematics in this regime.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We thank the LSU AI journal club for facilitating collaboration. This work was supported by LSU Sponsored Research 6
Rebate Program and LSU Foundation’s Distinguished Research Professorship Program. PD acknowledges support from the Quad Fellowship of the Institute of International Education and the LSU Physics & Astronomy Department. A part of the work of FP was supported by the National Natural Science Foundation of China under Grant No. 12175097.
[7] G.-P. Liu, H.-L. Wang, Z.-Z. Zhang, M.-L. Liu, Modelrepair capabilities of tree-based machine-learning algorithms applied to theoretical nuclear mass models, Physical Review C 111 (2025) 024306. [8] Y. Lu, T. Shang, P. Du, J. Li, H. Liang, Z. Niu, Nuclear mass predictions based on a convolutional neural network, Physical Review C 111 (2025) 014325.
Data availability
[9] Z. Niu, H. Liang, Nuclear mass predictions based on bayesian neural network approach with pairing and shell effects, Physics Letters B 778 (2018) 48–53.
The code used to construct the features, train the neuralnetwork models, and reproduce results and figures of this work, together with data supporting our findings, is publicly available on this GitHub repository. We invite interested researchers and students to contribute and enrich the project.
[10] R. Utama, J. Piekarewicz, Validating neural-network refinements of nuclear mass models, Physical Review C 97 (2018) 014306. [11] S. Qu, J.-Y. Zhang, M. Bao, Nuclear mass predictions with a bayesian neural network, Chinese Physics C 49 (10) (2025) 104106.
Declaration of generative AI usage During the preparation of this manuscript the author(s) used Claude by Anthropic and Prism by OpenAI in order to polish sentences and enhance coherence of the article. After using this tool/service, the author(s) reviewed thoroughly and edited the content as needed and take(s) full responsibility for the content of the published article.
[12] V. Kejzlar, L. Neufcourt, W. Nazarewicz, Local Bayesian Dirichlet mixing of imperfect models, Scientific Reports 13 (2023) 19600. [13] E. Yüksel, D. Soydaner, H. Bahtiyar, Nuclear mass predictions using machine learning models, Physical Review C 109 (2024) 064322. [14] W. Ye, N. Wan, Understanding on prediction differences among theoretical mass models with machine learning techniques, Physical Review C 111 (2025) 044317.
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10
Supplemental Material S1. Eigenvalues of SU(4) Casimir invariants Explicit formulas for the Casimir invariants (C2 , C3 and C4 ) of the SU(4) group were derived for Wigner’s convention of SU(4) labels by Burdet, Maguin and Partensky [52], and by Danos and Gillet [53]. In this work, we adopt an alternative convention of SU(4) irreps by Draayer [54], P = n14 − n24 , P′ = n24 − n34 and P′′ = n34 − n44 , (S1) P4 where [n14 , n24 , n34 , n44 ] is a U(4) irrep [56] that sums up to the total number of nucleons, i.e., A = i=1 ni4 . The eigenvalues of the Casimir operators are derived following a general technique presented in Appendix B of Ref. [55] and applicable to any unitary group. Since the Casimir invariants only depend on the irrep labels, their eigenvalues can be computed in any subgroup chain that is most convenient, which we choose to be the canonical chain U(4) ⊃ U(3) ⊃ U(2) ⊃ U(1). The U(4) Casimir operators are defined in terms of the canonical generators Ei j [56] as follows C2 = Ei j E ji , C3 = Ei j E jk Eki and C4 = Ei j E jk Ekl Eli ,
(S2)
where summations over double indices are implied according to Einstein’s convention. Another property of the Casimirs is their independence from any state used to evaluate the eigenvalue; hence, we choose the highest weight state that enables automatic annihilation of half of the terms in Eqs. (S2). Once the eigenvalues of the U(4) invariants are attained, those of the SU(4) Casimirs are straightforward by a simple conversion from U(4) to SU(4) quantum numbers [55]. One last step is to invoke a linear mapping that preserves their algebraic properties C2 = 2C2 , C3 = 4(C3 − C2 ) and C4 = C4 − C3 ,
(S3)
so that the following symmetry relations for the Casimirs under irrep conjugation are satisfied C2 (PP′ P′′ ) = C2 (P′′ P′ P), C3 (PP′ P′′ ) = −C3 (P′′ P′ P) and C4 (PP′ P′′ ) = C4 (P′′ P′ P).
(S4)
Using the matrix elements given in Ref. [56], we arrive at the following expression for the Casimirs’ eigenvalues 3 ⟨C2 ⟩ = (P2 + P′′2 ) + 2P′2 + 2(P + P′′ )P′ + PP′′ + 6(P + P′′ ) + 8P′ , 2 3 ⟨C3 ⟩ = (P − P′′ )[P2 + P′′2 + 2(PP′ + P′ P′′ + PP′′ ) + 2(3P + 2P′ + 3P′′ + 4)], 2 1 ⟨C4 ⟩ = [21(P4 + P′′4 ) + 16P′4 + 4P3 (14P′ + 7P′′ + 42) + 4P′′3 (14P′ + 7P + 42) 64 + 32P′3 (4 + P + P′′ ) + 72P′2 (P2 + P′′2 ) + 30P2 P′′2 + 24PP′ P′′ (3P + 2P′ + 3P′′ )
(S5) (S6)
+ (288P′2 + 896P′ + 960)(P + P′′ ) + 24P2 (16P′ + 9P′′ ) + 24P′′2 (16P′ + 9P) + 384PP′ P′′ + 576(P2 + P′′2 ) + 512P′2 + 640PP′′ + 1024P′ ].
(S7)
For the dominant lowest-C2 irrep adopted throughout this work, the labels (PP′ P′′ ) according to Draayer’s convention [54] are fixed by proton and neutron numbers, thus the even Casimirs reduce to smooth even functions of the isospin projection T z = (N − Z)/2, which is consistent with their correlation coefficient (MIC) with T z shown in Fig. 2 of the main text. Note that explicit determination of the leading SU(4) irrep analytically from T z is provided in Refs. [37, 38]; however, we did not adopt these formulas since there are different conventions of the SU(4) labels and it is not clear which convention these references followed. We instead construct a Python code to obtain the U(4), then SU(4), irrep for each nucleus that can decompose to T z [56] and has the lowest C2 ; in this way, we can conveniently determine the U(3) irrep and its SU(3) reduction as well. It is very much possible that subleading irreps would not be constrained by N and Z in the same way, i.e., their Casimir eigenvalues are no longer smooth functions of T z , which we will investigate in future work. Since C2 and C4 are quadratic and quartic in their degree and are always non-negative, we fit their values against |T z | to detect their dependence exactly (Fig. S1). Our polynomial fit yields the following results ⟨C2 ⟩ ≃ 2.00 |T z |2 + 7.97 |T z | + 3.20,
(R2 = 0.99998),
(S8)
⟨C4 ⟩ ≃ 0.25 |T z |4 + 1.90 |T z |3 + 11.9 |T z |2 + 15.5 |T z | + 22.9,
(R2 = 0.99991)
(S9)
where R2 is the coefficient of determination. Crucially, the linear |T z | ∝ |N − Z| term in both C2 and C4 represents the Wigner energy [69], which is known as the cusp at N = Z. Therefore, the inclusion of C2 on top of T z2 in the WINN mass formula is equivalent to adding the Wigner energy, whereas C4 brings in higher orders of isospin asymmetry. 11
Figure S1: Eigenvalues of the even SU(4) Casimirs C2 (left) and C4 (right) versus |T z | = |N − Z|/2 for the dominant lowest-C2 irrep, with polynomial fits (red). The near-exact fits show that the Casimirs reduce to a quadratic (C2 ) and a quartic (C4 ) function of the isospin asymmetry.
S2. Eigenvalues of SU(3) Casimir invariants We follow the convention for the eigenvalues of the quadratic (C2 ) and cubic (C3 ) Casimir operators belonging to the SU(3) group by Draayer and Rosensteel [57], which can be derived similarly to those of the SU(4) group and are given as ⟨C2 ⟩ = λ2 + µ2 + λµ + 3λ + 3µ, 1 ⟨C3 ⟩ = (λ − µ)[2λ2 + 2µ2 + 5λµ + 9(λ + µ + 1)]. 9
(S10) (S11)
These operators satisfy the following property under irrep conjugation C2 (λµ) = C2 (µλ) and C3 (λµ) = −C3 (µλ).
(S12)
S3. Training dynamics and the absence of overfitting As described in Sec. 2.1 of the main text, the 20% test set is held out from training but drawn from the same distribution as the train data; it therefore serves as an overfitting monitor, distinct from the AME2020 validation set that measures extrapolation. Figure S2 shows the per-epoch train and test losses for the FINN across all feature sets. In every case the two curves descend together and plateau at essentially the same level, with no upturn of the test loss at late epochs. The small train-test gap confirms that the networks are not memorizing the training nuclei [51]. Consequently, the larger error on the AME2020 validation set (Table 1 of the main text) reflects genuine extrapolation to newly measured, often more exotic nuclei rather than overfitting. The flatness of the test curve also indicates that early stopping offers no benefit here, as the GK loss already regularizes the training effectively.
Figure S2: Per-epoch training (blue) and test (orange) loss, in normalized-target units, for the FINN on each feature set. Train and test curves remain together throughout, with no late-epoch rise in the test loss, demonstrating evidence that the models do not overfit.
S4. Normalization of the data The features used in our models span widely different scales: the bulk inputs (Z, N, A2/3 , T z ) range over tens to hundreds, the harmonic oscillator quanta Nℏω grow into the hundreds, and the SU(4) and SU(3) Casimir eigenvalues grow even faster with nucleon 12
number. Feeding such heterogeneous magnitudes directly into a neural network is detrimental: features with large numerical values dominate the gradients irrespective of their physical relevance, the optimization becomes ill-conditioned, and the inputs are pushed into the saturated regions of the activation functions. Hence, we standardize every input feature and the target binding energy to zero mean and unit variance through z-score normalization, x̃i =
xi − µi , σi + ϵ
ỹ =
y − µy , σy + ϵ
(S13)
where µi , σi are the mean and standard deviation of feature i, and µy , σy are those of the binding energy. Here, ϵ = 10−8 is a small constant that guards against division by zero for (near-)constant columns. Crucially, the statistics {µ, σ} are computed on the train set only and then applied to the test and validation sets. This prevents information from the held-out nuclei from leaking into the preprocessing, so that the test and validation errors remain honest measures of generalization and extrapolation. Since the target is also standardized, the network learns in normalized units; all predictions are mapped back to physical units (MeV) through the inverse transform y = ỹ σy + µy before RMSE is reported. After standardization the features are of comparable magnitude, which balances the gradients, accelerates and stabilizes convergence, and ensures that the relative importance the models assign to each feature reflects its predictive value. S5. Smooth versus disruptive activation functions The choice of activation function is usually regarded as a minor architectural detail, but for nuclear masses it carries physical meaning. The binding energy is not a smooth function of N and Z, specifically, at every magic number the nuclear surface stiffens and the binding is sharply enhanced, producing a near-discontinuity in the first derivative of BE. Accordingly, the two-neutron separation energy S 2n (Z, N) = BE(Z, N) − BE(Z, N − 2) drops abruptly upon crossing a neutron shell closure. A network built from disruptive activation functions [e.g., the rectified linear unit, ReLU(x) = max(0, x)] is a piecewise function of its inputs and can represent such kinks, whereas a network built from smooth, infinitely differentiable activations [e.g., the hyperbolic tangent, Tanh(x), or the sigmoid linear unit, SiLU(x) = x/(1 + e−x )] must approximate every kink with a rounded transition. To test the consequence of this distinction, we conduct an experiment by retraining the FINN four times, identical in every aspect—same architecture, feature set, optimizer, loss, and training schedule—differing only in the hidden activation. In particular, we compare two piecewise-linear activations—ReLU and its parametric variant PReLU [PReLU(x) = x for x > 0 and αx for x < 0, with a learnable slope α]—against two smooth ones, Tanh and SiLU. The random seed is also fixed so the models share an identical weight initialization and mini-batch ordering. Table S1: RMSE (MeV) for the four activation functions on the train, test and validation datasets.
Activation ReLU PReLU Tanh SiLU
Train 0.444 0.411 0.525 0.611
Test 0.555 0.552 0.579 0.677
Validation 0.894 0.624 0.811 0.826
Figure S3: Training loss as a function of epoch for the four activation functions. The piecewise-linear ReLU and PReLU networks settle to a consistently lower loss than the smooth Tanh and SiLU networks for the bulk of training.
13
Figure S3 shows the training loss versus epoch. All four converge within a few hundreds of epochs, but the piecewise-linear networks thereafter settle to a consistently lower loss. Moreover, as shown in Table S1, the train and test RMSEs are the lowest with the ReLU and PReLU activations, and the PReLU network achieves the best extrapolation on the validation set. That the ReLU is outperformed by its parametric version can be explained by the fact that one of the FINN’s features is the isospin projection T z , which is positive (negative) for neutron(proton)-rich nuclei. The ReLU, which only returns positive output, tends to gravitate toward positive inputs and ignore negative ones, whereas the PReLU takes into account both and simultaneously generates the desired discontinuity.
Figure S4: Two-neutron separation energy S 2n along three isotopic chains crossing a major neutron shell closure (dotted line): 48 Ca (N = 28), 132 Sn (N = 82), and 208 Pb (N = 126). The experimental S 2n (black) drops sharply just past each closure; the piecewise-linear activations (ReLU, PReLU) track the drop, while the smooth ones (Tanh, SiLU) round it off.
To probe deeper the advantage of each activation function at the shell closures, we compare the predicted S 2n along three isotopic chains that each cross a major neutron magic number, namely 48 Ca (N = 28), 132 Sn (N = 82), and 208 Pb (N = 126). The experimental S 2n falls sharply just past each closure, which is tracked closely by the piecewise-linear ReLU and PReLU networks as demonstrated in Fig. S4, whereas the smooth activations smear it out. Quantitatively, the shell gap ∆S 2n = S 2n (Nmagic )−S 2n (Nmagic + 2) is reproduced best by the piecewise-linear activations at the two heavy, well-developed closures (Table S2); specifically, the PReLU and ReLU respectively recover 4.89 MeV and 4.88 MeV at 208 Pb against the experimental 4.98 MeV, and 5.68 MeV and 5.82 MeV at 132 Sn against 6.53 MeV, while Tanh and SiLU capture less than two thirds of each gap. Table S2: Shell gap ∆S 2n (MeV) at three neutron closures: experiment versus the FINN trained with each activation. Values closer to experiment indicate a sharper, better-resolved closure.
Chain 48 Ca (N = 28) 132 Sn (N = 82) 208 Pb (N = 126)
Exp. 5.72 6.53 4.98
ReLU 2.96 5.82 4.88
PReLU 3.52 5.68 4.89
Tanh 3.23 4.13 2.51
SiLU 3.27 4.30 2.59
S6. Inclusion of the third-order SU(4) Casimir operator In this section, we provide justification for the exclusion of the cubic Casimir C3 [SU(4)] from the feature set. As given in Sec. S1, the expression for ⟨C3 ⟩ eigenvalue carries the overall prefactor (P − P′′ ), thus ⟨C3 [SU(4)]⟩ = 0 when P = P′′ , i.e., whenever the leading irrep of the nucleus is self-conjugate. For the dominant lowest-C2 irrep, the labels (P, P′ , P′′ ) are fixed by the nucleon numbers and fall into four categories: Category even-even odd-odd Z even, N odd Z odd, N even
(P, P′ , P′′ ) self-conjugate self-conjugate (1, |N − Z|/2 − 1/2, 0) (0, |N − Z|/2 − 1/2, 1)
⟨C3 ⟩ 0 0 >0 <0
Both even-mass families (even-even and odd-odd) are self-conjugate (P = P′′ ), so every even-A nucleus has ⟨C3 [SU(4)]⟩ ≡ 0. Only odd-A nuclei carry a nonzero value, and its sign is set entirely by which nucleon species is unpaired. The two neighbors (N, Z) and (N − 1, Z + 1) share the same odd mass number but swap these roles, which leads to the sign fluctuation noted in Sec. 2.2 of the main text and is reflected quantitatively via the low maximal information coefficients of C3 [SU(4)] with respect to the bulk and SU(3) features (see Fig. 2 of the main text). 14
This structure is borne out by the data: across the 2457 nuclei in our dataset, all 1227 even-A nuclei have ⟨C3 [SU(4)]⟩ = 0 identically, whereas the 1230 odd-A nuclei are nonzero and split by sign into the Z-even/N-odd (mean ≈ +120) and Z-odd/N-even (mean ≈ −118) classes. This sign assignment holds exactly for N ≥ Z; the only exceptions (∼ 4% of odd-A nuclei) lie deep on the proton-rich side (Z > N), where the (P − P′′ ) prefactor reverses as protons outnumber neutrons. It is unlikely for a feature that vanishes over half the nuclear chart and alternates in sign over the remainder to encode any smooth systematic trend in the binding energy. In particular, after normalization, this produces a feature that oscillates between approximately −1 and +1, which can make it difficult for the neural network to extract a smooth physical trend. Moreover, when this feature is introduced in the training, the SHAP ranking of feature importance shows a negligible contribution to the learning, as if it were effectively very noisy information. Nevertheless, the physical information encoded in the third-order SU(4) Casimir could be important for nuclear binding, as there is an overall trend of the intensity of the fluctuations, with sharper ones for heavier species and for lower Z nuclei along isotonic chains. In order to retain the information carried by this operator, we split it in two feature maps, smotting the oscillation separating the positive and negative values by defining two nonnegative derived features as follows C+3 [SU(4)] = max (C3 [SU(4)], 0) ,
(S14)
C−3 [SU(4)] = max (−C3 [SU(4)], 0) .
(S15)
This transformation preserves the magnitude and sign information of the original feature, since C3 [SU(4)] = C+3 [SU(4)]−C−3 [SU(4)], but represents the positive and negative branches as two separate smooth, nonnegative channels. This is more convenient for training because the model no longer has to learn a single rapidly sign-changing feature. Instead, it can assign independent weights to the positive and negative sectors, making it easier to identify whether each branch carries useful physical information for the BE prediction. Figure S5 shows the two new features obtained with this procedure.
Figure S5: C+3 [SU(4)] and C−3 [SU(4)] features derived from the cubic SU(4) Casimir operator. All values are non-negative.
Activation PReLU (λ = 1.0) PReLU (λ = 2.0) PReLU (λ = 4.0) ReLU (λ = 1.0) ReLU (λ = 2.0) ReLU (λ = 4.0)
Train RMSE 0.479 0.714 0.485 0.493 0.466 0.400
Test RMSE 0.662 0.841 0.700 0.796 0.721 0.666
Validation RMSE 1.145 1.311 1.039 1.218 1.218 1.535
Table S3: Performance of the FINN when the derived third-order SU(4) Casimir operators, C+3 [SU(4)] and C−3 [SU(4)], are included in the full feature set [LDM+SU(4)+SU(3)]. RMSE values are reported in MeV for two activation functions with various GK weights.
We train the FINN’s full feature set [i.e., LDM+SU(4)+SU(3)] augmented with these two derived features, first without any modification to the network architecture (PReLU activation, 10−3 learning rate, GK weight λ = 2.0, no weight decay), followed by testing with an alternative activation function and different GK weights. The results are reported in Table S3. Overall, we observe a deterioration to the model’s performance as demonstrated by higher RMSEs. In particular, most errors are above the FINN’s RMSE for the full feature set reported in Table 1 of the main text, especially on the validation set (all exceeding 1.0 MeV), which implies that the model generalizes badly on unseen data. These results lead to our conclusion that adding the proxy non-negative features derived from the cubic SU(4) Casimir does not supply more reliable predictive information to the model. Even though the information carried by this third-order operator may help capture odd-A structure distinct from that of even-A nuclei, it is probably either too localized or too discontinuous to improve a global mass model, hence, it should be ultimately excluded from the feature set of nuclear binding. 15
S7. Pairing effect To check whether the WINN misses pairing correlations, we examine the WINN mass residuals, BEpred − BEexp , separated into three classes of even-even, odd-A, and odd-odd nuclei (Fig. S6). If the pairing effect were not fully captured, one would observe a systematic underbinding for even-even nuclei, a systematic overbinding for odd-odd nuclei, and a noticeable odd-even oscillation of the residuals. We observe none of these; in fact, the class-averaged residuals are small (within ∼ 0.25 MeV) and of the same negative sign, with no even-even/odd-odd inversion and no odd-even staggering. The pairing systematics are therefore already absorbed by the model through the term a in Eq. (1) of the main text, as it is trained to learn the (N, Z)-dependent “leftover” of binding from the symmetry descriptors, which could include both the volume energy and pairing effect. Note that we find a degradation on the validation RMSE upon adding an explicit pairing term [δ(N, Z)]. One may wonder how it is possible for the NN to learn the pairing contribution that is not a smooth function of N and Z. We attribute our model’s ability to capture this jagging to the ReLU activation function, which is not a smooth one. In principle, one can use a large and deep NN to approximate any function—with sufficient data, the model should be able to pick up a functional representation of the output in terms of the input features. However, since there are only two and a half thousand nuclei discovered to date, the choice of the activation function helps in approximating the fine detail of the (N, Z) dependence, including the breaking of smoothness of BE.
Figure S6: WINN mass residuals BEpred − BEexp versus mass number A, separated into even-even, odd-A, and odd-odd nuclei. The absence of a systematic eveneven/odd-odd sign inversion or odd-even oscillation indicates that pairing correlations are already captured.
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