Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).
Google Scholar
Mardirossian, N. & M. H.-G. Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals. Mol. Phys. 115, 2315–2372 (2017).
Google Scholar
Kryachko, E. S. & Ludeña, E. V. Energy density functional theory of many-electron systems, vol. 4 (Springer Science & Business Media, 2012).
Bender, M., Heenen, P.-H. & Reinhard, P.-G. Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003).
Google Scholar
Lalazissis, G. A., Ring, P. & Vretenar, D. Extended density functionals in nuclear structure physics, vol. 641 (Springer Science & Business Media, 2004).
Meng, J. Relativistic density functional for nuclear structure, vol. 10 (World Scientific, 2016).
Epelbaum, E., Hammer, H.-W. & Meißner, U.-G. Modern theory of nuclear forces. Rev. Mod. Phys. 81, 1773–1825 (2009).
Google Scholar
Nakatsukasa, T., Matsuyanagi, K., Matsuo, M. & Yabana, K. Time-dependent density-functional description of nuclear dynamics. Rev. Mod. Phys. 88, 045004 (2016).
Google Scholar
Bulgac, A., Forbes, M. M., Jin, S., Perez, R. N. & Schunck, N. Minimal nuclear energy density functional. Phys. Rev. C. 97, 044313 (2018).
Google Scholar
Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).
Google Scholar
Brack, M., Guet, C. & Hakansson, H.-B. Selfconsistent semiclassical description of average nuclear properties-a link between microscopic and macroscopic models. Phys. Rep. 123, 275–364 (1985).
Google Scholar
Centelles, M., Schuck, P. & Viñas, X. Thomas–Fermi theory for atomic nuclei revisited. Ann. Phys. 322, 363 (2007).
Google Scholar
Colò, G. & Hagino, K. Orbital-free density functional theory: differences and similarities between electronic and nuclear systems. Prog. Theor. Exp. Phys. 2023, 103D01 (2023).
Centelles, M., Pi, M., Vinas, X., Garcias, F. & Barranco, M. Self-consistent extended Thomas-Fermi calculations in nuclei. Nucl. Phys. A 510, 397–416 (1990).
Google Scholar
Dutta, A., Arcoragi, J.-P., Pearson, J., Behrman, R. & Tondeur, F. Thomas-Fermi approach to nuclear mass formula: (i). spherical nuclei. Nucl. Phys. A 458, 77–94 (1986).
Google Scholar
Aboussir, Y., Pearson, J., Dutta, A. & Tondeur, F. Nuclear mass formula via an approximation to the hartree-fock method. At. Data Nucl. Data Tables 61, 127 – 176 (1995).
Brack, M. et al. Funny hills: the shell-correction approach to nuclear shell effects and its applications to the fission process. Rev. Mod. Phys. 44, 320–405 (1972).
Google Scholar
Brack, M. & Pauli, H. On Strutinsky’s averaging method. Nucl. Phys. A 207, 401–424 (1973).
Google Scholar
Strutinsky, V. Shell effects in nuclear masses and deformation energies. Nucl. Phys. A 95, 420–442 (1967).
Google Scholar
Strutinsky, V. “Shells” in deformed nuclei. Nucl. Phys. A 122, 1–33 (1968).
Google Scholar
Brack, M. & Quentin, P. Self-consistent average density matrices and the Strutinsky energy theorem. Phys. Lett. B 56, 421–423 (1975).
Google Scholar
Bohigas, O., Campi, X., Krivine, H. & Treiner, J. Extensions of the Thomas-Fermi approximation for finite nuclei. Phys. Lett. B 64, 381–385 (1976).
Google Scholar
Chu, Y., Jennings, B. & Brack, M. Nuclear binding energies and liquid drop parameters in the extended Thomas-Fermi approximation. Phys. Lett. B 68, 407–411 (1977).
Google Scholar
Tozer, D. J., Ingamells, V. E. & Handy, N. C. Exchange-correlation potentials. J. Chem. Phys. 105, 9200–9213 (1996).
Google Scholar
Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).
Google Scholar
Boehnlein, A. et al. Colloquium: machine learning in nuclear physics. Rev. Mod. Phys. 94, 031003 (2022).
Google Scholar
He, W. et al. Machine learning in nuclear physics at low and intermediate energies. Sci. China Phys. Mech. Astron. 66, 282001 (2023).
Google Scholar
Niu, Z. M. et al. Radial basis function approach in nuclear mass predictions. Phys. Rev. C. 88, 024325 (2013).
Google Scholar
Utama, R., Piekarewicz, J. & Prosper, H. B. Nuclear mass predictions for the crustal composition of neutron stars: a Bayesian neural network approach. Phys. Rev. C. 93, 014311 (2016).
Google Scholar
Niu, Z. M. & Liang, H. Z. Nuclear mass predictions based on Bayesian neural network approach with pairing and shell effects. Phys. Lett. B 778, 48 – 53 (2018).
Neufcourt, L., Cao, Y. C., Nazarewicz, W. & Viens, F. Bayesian approach to model-based extrapolation of nuclear observables. Phys. Rev. C. 98, 034318 (2018).
Google Scholar
Pastore, A., Neill, D., Powell, H., Medler, K. & Barton, C. Impact of statistical uncertainties on the composition of the outer crust of a neutron star. Phys. Rev. C. 101, 035804 (2020).
Google Scholar
Wu, X. H. & Zhao, P. W. Predicting nuclear masses with the kernel ridge regression. Phys. Rev. C. 101, 051301 (R) (2020).
Google Scholar
Wu, X. H., Guo, L. H. & Zhao, P. W. Nuclear masses in extended kernel ridge regression with odd-even effects. Phys. Lett. B 819, 136387 (2021).
Wu, X. H., Lu, Y. Y. & Zhao, P. W. Multi-task learning on nuclear masses and separation energies with the kernel ridge regression. Phys. Lett. B 834, 137394 (2022).
Niu, Z. M. & Liang, H. Z. Nuclear mass predictions with machine learning reaching the accuracy required by r-process studies. Phys. Rev. C. 106, L021303 (2022).
Google Scholar
Akkoyun, S., Bayram, T., Kara, S. O. & Sinan, A. An artificial neural network application on nuclear charge radii. J. Phys. G Nucl. Part. Phys. 40, 055106 (2013).
Google Scholar
Utama, R., Chen, W.-C. & Piekarewicz, J. Nuclear charge radii: density functional theory meets Bayesian neural networks. J. Phys. G Nucl. Part. Phys. 43, 114002 (2016).
Ma, Y. et al. Predictions of nuclear charge radii and physical interpretations based on the naive Bayesian probability classifier. Phys. Rev. C. 101, 014304 (2020).
Google Scholar
Wu, D., Bai, C. L., Sagawa, H. & Zhang, H. Q. Calculation of nuclear charge radii with a trained feed-forward neural network. Phys. Rev. C. 102, 054323 (2020).
Google Scholar
Ma, J.-Q. & Zhang, Z.-H. Improved phenomenological nuclear charge radius formulae with kernel ridge regression. Chin. Phys. C. 46, 074105 (2022).
Google Scholar
Dong, X.-X., An, R., Lu, J.-X. & Geng, L.-S. Nuclear charge radii in Bayesian neural networks revisited. Phys. Lett. B 838, 137726 (2023).
Niu, Z. M., Liang, H. Z., Sun, B. H., Long, W. H. & Niu, Y. F. Predictions of nuclear β-decay half-lives with machine learning and their impact on r-process nucleosynthesis. Phys. Rev. C. 99, 064307 (2019).
Google Scholar
Lovell, A. E., Nunes, F. M., Catacora-Rios, M. & King, G. B. Recent advances in the quantification of uncertainties in reaction theory. J. Phys. G Nucl. Part. Phys. 48, 014001 (2020).
Google Scholar
Wu, D., Bai, C. L., Sagawa, H., Nishimura, S. & Zhang, H. Q. β-delayed one-neutron emission probabilities within a neural network model. Phys. Rev. C. 104, 054303 (2021).
Google Scholar
Saxena, G., Sharma, P. K. & Saxena, P. Modified empirical formulas and machine learning for alpha-decay systematics. J. Phys. G Nucl. Part. Phys. 48, 055103 (2021).
Google Scholar
Neudecker, D. et al. Informing nuclear physics via machine learning methods with differential and integral experiments. Phys. Rev. C. 104, 034611 (2021).
Google Scholar
Wang, X., Zhu, L. & Su, J. Modeling complex networks of nuclear reaction data for probing their discovery processes. Chin. Phys. C. 45, 124103 (2021).
Google Scholar
Huang, T. X., Wu, X. H. & Zhao, P. W. Application of kernel ridge regression in predicting neutron-capture reaction cross-sections. Commun. Theor. Phys. 74, 095302 (2022).
Google Scholar
Jiang, W. G., Hagen, G. & Papenbrock, T. Extrapolation of nuclear structure observables with artificial neural networks. Phys. Rev. C. 100, 054326 (2019).
Google Scholar
Lasseri, R.-D., Regnier, D., Ebran, J.-P. & Penon, A. Taming nuclear complexity with a committee of multilayer neural networks. Phys. Rev. Lett. 124, 162502 (2020).
Google Scholar
Yoshida, S. Nonparametric Bayesian approach to extrapolation problems in configuration interaction methods. Phys. Rev. C. 102, 024305 (2020).
Google Scholar
Wang, X., Zhu, L. & Su, J. Providing physics guidance in Bayesian neural networks from the input layer: the case of giant dipole resonance predictions. Phys. Rev. C. 104, 034317 (2021).
Google Scholar
Bai, J., Niu, Z., Sun, B. & Niu, Y. The description of giant dipole resonance key parameters with multitask neural networks. Phys. Lett. B 815, 136147 (2021).
Neufcourt, L., Cao, Y., Nazarewicz, W., Olsen, E. & Viens, F. Neutron drip line in the CA region from Bayesian model averaging. Phys. Rev. Lett. 122, 062502 (2019).
Google Scholar
Neufcourt, L. et al. Quantified limits of the nuclear landscape. Phys. Rev. C. 101, 044307 (2020).
Google Scholar
Wang, Z.-A., Pei, J., Liu, Y. & Qiang, Y. Bayesian evaluation of incomplete fission yields. Phys. Rev. Lett. 123, 122501 (2019).
Google Scholar
Lovell, A. E., Mohan, A. T. & Talou, P. Quantifying uncertainties on fission fragment mass yields with mixture density networks. J. Phys. G-Nucl. Part. Phys. 47, 114001 (2020).
Google Scholar
Qiao, C. Y. et al. Bayesian evaluation of charge yields of fission fragments of 239U. Phys. Rev. C. 103, 034621 (2021).
Google Scholar
Keeble, J. & Rios, A. Machine learning the deuteron. Phys. Lett. B 809, 135743 (2020).
Adams, C., Carleo, G., Lovato, A. & Rocco, N. Variational Monte Carlo calculations of a ≤4 nuclei with an artificial neural-network correlator ansatz. Phys. Rev. Lett. 127, 022502 (2021).
Google Scholar
Lovato, A., Adams, C., Carleo, G. & Rocco, N. Hidden-nucleons neural-network quantum states for the nuclear many-body problem. Phys. Rev. Res. 4, 043178 (2022).
Yang, Y. & Zhao, P. A consistent description of the relativistic effects and three-body interactions in atomic nuclei. Phys. Lett. B 835, 137587 (2022).
Yang, Y. L. & Zhao, P. W. Deep-neural-network approach to solving the ab initio nuclear structure problem. Phys. Rev. C. 107, 034320 (2023).
Google Scholar
Rigo, M., Hall, B., Hjorth-Jensen, M., Lovato, A. & Pederiva, F. Solving the nuclear pairing model with neural network quantum states. Phys. Rev. E 107, 025310 (2023).
Google Scholar
Pederson, R., Kalita, B. & Burke, K. Machine learning and density functional theory. Nat. Rev. Phys. 4, 357–358 (2022).
Huang, B., von Rudorff, G. F. & von Lilienfeld, O. A. The central role of density functional theory in the ai age. Science 381, 170–175 (2023).
Google Scholar
Snyder, J. C., Rupp, M., Hansen, K., Müller, K.-R. & Burke, K. Finding density functionals with machine learning. Phys. Rev. Lett. 108, 253002 (2012).
Google Scholar
Brockherde, F. et al. By-passing the Kohn-Sham equations with machine learning. Nat. Commun. 8, 872 (2017).
Google Scholar
Nagai, R., Akashi, R. & Sugino, O. Completing density functional theory by machine learning hidden messages from molecules. npj Computational Mater. 6, 43 (2020).
Google Scholar
Bogojeski, M., Vogt-Maranto, L., Tuckerman, M. E., Müller, K.-R. & Burke, K. Quantum chemical accuracy from density functional approximations via machine learning. Nat. Commun. 11, 5223 (2020).
Google Scholar
Moreno, J. R., Carleo, G. & Georges, A. Deep learning the Hohenberg-Kohn maps of density functional theory. Phys. Rev. Lett. 125, 076402 (2020).
Google Scholar
Li, L. et al. Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics. Phys. Rev. Lett. 126, 036401 (2021).
Google Scholar
Kirkpatrick, J. et al. Pushing the frontiers of density functionals by solving the fractional electron problem. Science 374, 1385–1389 (2021).
Google Scholar
Margraf, J. T. & Reuter, K. Pure non-local machine-learned density functional theory for electron correlation. Nat. Commun. 12, 1–7 (2021).
Google Scholar
Ma, H., Narayanaswamy, A., Riley, P. & Li, L. Evolving symbolic density functionals. Sci. Adv. 8, eabq0279 (2022).
Google Scholar
Bai, Y., Vogt-Maranto, L., Tuckerman, M. E. & Glover, W. J. Machine learning the Hohenberg-Kohn map for molecular excited states. Nat. Commun. 13, 7044 (2022).
Google Scholar
Wu, X. H., Ren, Z. X. & Zhao, P. W. Nuclear energy density functionals from machine learning. Phys. Rev. C. 105, L031303 (2022).
Google Scholar
Hizawa, N., Hagino, K. & Yoshida, K. Analysis of a Skyrme energy density functional with deep learning. Phys. Rev. C. 108, 034311 (2023).
Google Scholar
Chen, Y. Y. & Wu, X. H. Machine learning nuclear orbital-free density functional based on Thomas-Fermi approach. Int. J. Mod. Phys. E 33, 2450012 (2024).
Google Scholar
Vautherin, D. Hartree-Fock calculations with Skyrme’s interaction. ii. Axially deformed nuclei. Phys. Rev. C. 7, 296–316 (1973).
Google Scholar
Lee, S.-J. et al. Relativistic Hartree calculations for axially deformed nuclei. Phys. Rev. Lett. 57, 2916–2919 (1986).
Google Scholar
Pannert, W., Ring, P. & Boguta, J. Relativistic mean-field theory and nuclear deformation. Phys. Rev. Lett. 59, 2420–2422 (1987).
Google Scholar
Dobaczewski, J., Flocard, H. & Treiner, J. Hartree-Fock-Bogolyubov description of nuclei near the neutron-drip line. Nucl. Phys. A 422, 103–139 (1984).
Google Scholar
Bender, M., Rutz, K., Reinhard, P.-G. & Maruhn, J. A. Consequences of the center–of–mass correction in nuclear mean–field models. Eur. Phys. J. A 7, 467–478 (2000).
Google Scholar
Beiner, M., Flocard, H., Van Giai, N. & Quentin, P. Nuclear ground-state properties and self-consistent calculations with the Skyrme interaction: (i). spherical description. Nucl. Phys. A 238, 29–69 (1975).
Google Scholar
Slater, J. C. A simplification of the Hartree-Fock method. Phys. Rev. 81, 385–390 (1951).
Google Scholar
Staszczak, A., Stoitsov, M., Baran, A. & Nazarewicz, W. Augmented Lagrangian method for constrained nuclear density functional theory. Eur. Phys. J. A 46, 85–90 (2010).
Google Scholar
National Nuclear Data Center (NNDC), https://www.nndc.bnl.gov/ (2025).
