Predictions of charge density distributions for nuclei with $Z \geq 8$

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.

The Hohenberg-Kohn theorem establishes that, in principle, the ground state (GS) charge density contains all GS information of a many-electron system, such that all GS observables can be expressed as functionals of the GS charge density. Conventional Kohn-Sham density functional theory requires iterative solution of the self-consistent-field equations at substantial computational cost, motivating the development of deep learning surrogates for electronic structure calculations and, in turn, accelerating computer-aided materials design. Here, we propose AIDEN, an Atomic-Interaction Density Equivariant Network for solving real-space charge density. AIDEN separates the element-dependent one-center density from environment-induced density redistribution and represents the latter through complementary atom- and edge-centered tensor correlations. A continuous low-rank Gaussian decoder then reconstructs the density at arbitrary spatial coordinates while reusing atomic encodings independently of the evaluation grid. AIDEN achieves state-of-the-art accuracy on periodic crystal benchmarks while remaining competitive for molecular systems, and further demonstrates zero-shot transferability across several structurally distinct out-of-distribution case studies. Furthermore, AIDEN provides substantially faster inference than both baseline models and full SCF calculations, enabling efficient charge density reconstruction for large-scale electronic structure calculations.

The prediction of masses of atomic nuclei using machine learning can complement theoretical models and advance the exploration of poorly known domains of the nuclear chart. We propose a machine learning technique based on gated recurrent units (GRU), which have demonstrated competitive performance in nuclear-mass prediction by exploiting long-term dependencies. By integrating multiplicative interactions and product-unit transformations within recurrent units, we report significant improvements in nuclear-mass prediction. Computations are performed in the complex domain to jointly capture amplitude and phase dynamics. For interpolation and temporal-extrapolation tasks based on the atomic mass evaluation (AME2016 and AME2020), the complex additive-multiplicative product-unit gated recurrent unit (AM-PU-GRU) model consistently achieves the lowest prediction errors, with an interpolation RMSE of 0.227 $\pm$ 0.004 MeV and an extrapolation RMSE of 0.179 $\pm$ 0.015 MeV. These results surpass other state-of-the-art machine learning models and also outperform the real-valued GRU baseline and product-unit ablation variants, while remaining robust to different theoretical priors, including WS4 and SEMF. Our findings establish complex-valued product-unit recurrent networks as a new benchmark for sequence-based nuclear-mass prediction.