Nuclear mass predictions based on Bayesian neural network approach with pairing and shell effects

Background: The mass, or binding energy, is the basis property of the atomic nucleus. It determines its stability and reaction and decay rates. Quantifying the nuclear binding is important for understanding the origin of elements in the universe. The astrophysical processes responsible for the nucleosynthesis in stars often take place far from the valley of stability, where experimental masses are not known. In such cases, missing nuclear information must be provided by theoretical predictions using extreme extrapolations. To take full advantage of the information contained in mass model residuals, i.e., deviations between experimental and calculated masses, one can utilize Bayesian machine-learning techniques to improve predictions. Purpose: To improve the quality of model-based predictions of nuclear properties of rare isotopes far from stability, we consider the information contained in the residuals in the regions where the experimental information exist. As a case in point, we discuss two-neutron separation energies S2n of even-even nuclei. Through this observable, we assess the predictive power of global mass models towards more unstable neutron-rich nuclei and provide uncertainty quantification of predictions. Methods: We consider 10 global models based on nuclear density functional theory with realistic energy density functionals as well as two more phenomenological mass models. The emulators of S2n residuals and credibility intervals (Bayesian confidence intervals) defining theoretical error bars are constructed using Bayesian Gaussian processes and Bayesian neural networks. We consider a large training dataset pertaining to nuclei whose masses were measured before 2003. For the testing datasets, we considered those exotic nuclei whose masses have been determined after 2003. By establishing statistical methodology and parameters, we carried out extrapolations toward the 2n dripline. Results: While both Gaussian processes and Bayesian neural networks reduce the root-mean-square (rms) deviation from experiment significantly, GP offers a better and much more stable performance. The increase in the predictive power of microscopic models aided by the statistical treatment is quite astonishing: The resulting rms deviations from experiment on the testing dataset are similar to those of more phenomenological models. We found that Bayesian neural networks results are prone to instabilities caused by the large number of parameters in this method. Moreover, since the classical sigmoid activation function used in this approach has linear tails that do not vanish, it is poorly suited for a bounded extrapolation. The empirical coverage probability curves we obtain match very well the reference values, in a slightly conservative way in most cases, which is highly desirable to ensure honesty of uncertainty quantification. The estimated credibility intervals on predictions make it possible to evaluate predictive power of individual models and also make quantified predictions using groups of models. Conclusions: The proposed robust statistical approach to extrapolation of nuclear model results can be useful for assessing the impact of current and future experiments in the context of model developments. The new Bayesian capability to evaluate residuals is also expected to impact research in the domains where experiments are currently impossible, for instance, in simulations of the astrophysical r process.

Bayesian approach to model-based extrapolation of nuclear observables

The region of heavy calcium isotopes forms the frontier of experimental and theoretical nuclear structure research where the basic concepts of nuclear physics are put to stringent test. The recent discovery of the extremely neutron-rich nuclei around $^{60}\mathrm{Ca}$ O. B. Tarasov et al. [Phys. Rev. Lett. 121, 022501 (2018)] and the experimental determination of masses for $^{55--57}\mathrm{Ca}$ S. Michimasa et al. [Phys. Rev. Lett. 121, 022506 (2018)] provide unique information about the binding energy surface in this region. To assess the impact of these experimental discoveries on the nuclear landscape's extent, we use global mass models and statistical machine learning to make predictions, with quantified levels of certainty, for bound nuclides between Si and Ti. Using a Bayesian model averaging analysis based on Gaussian-process-based extrapolations we introduce the posterior probability ${p}_{\mathrm{ex}}$ for each nucleus to be bound to neutron emission. We find that extrapolations for drip-line locations, at which the nuclear binding ends, are consistent across the global mass models used, in spite of significant variations between their raw predictions. In particular, considering the current experimental information and current global mass models, we predict that $^{68}\mathrm{Ca}$ has an average posterior probability ${p}_{\mathrm{ex}}\ensuremath{\approx}76%$ to be bound to two-neutron emission while the nucleus $^{61}\mathrm{Ca}$ is likely to decay by emitting a neutron (${p}_{\mathrm{ex}}\ensuremath{\approx}46%$).

Neutron Drip Line in the Ca Region from Bayesian Model Averaging.

Nuclear $\ensuremath{\beta}$ decay is a key process to understand the origin of heavy elements in the universe, while the accuracy is far from satisfactory for the predictions of $\ensuremath{\beta}$-decay half-lives by nuclear models to date. In this work, we pave a novel way to accurately predict $\ensuremath{\beta}$-decay half-lives with the machine learning based on the Bayesian neural network, in which the known physics has been explicitly embedded, including the ones described by the Fermi theory of $\ensuremath{\beta}$ decay, and the dependence of half-lives on pairing correlations and decay energies. The other potential physics, which is not clear or even missing in nuclear models nowadays, will be learned by the Bayesian neural network. The results well reproduce the experimental data with a very high accuracy and further provide reasonable uncertainty evaluations in half-life predictions. These accurate predictions for half-lives with uncertainties are essential for the $r$-process simulations.

Predictions of nuclear β -decay half-lives with machine learning and their impact on r -process nucleosynthesis

Fission product yields are key infrastructure data for nuclear applications in many aspects. It is a challenge both experimentally and theoretically to obtain accurate and complete energy-dependent fission yields. We apply the Bayesian neural network (BNN) approach to learn existing fission yields and predict unknowns with uncertainty quantification. We demonstrated that the BNN is particularly useful for evaluations of fission yields when incomplete experimental data are available. The BNN evaluation results are quite satisfactory on distribution positions and energy dependencies of fission yields.

Bayesian Evaluation of Incomplete Fission Yields.

Resorting to a neural network approach we refined several representative and sophisticated global nuclear mass models within the latest atomic mass evaluation (AME2012). In the training process, a quite robust algorithm named the Levenberg–Marquardt (LM) method is employed to determine the weights and biases of the neural network. As a result, this LM neural network approach demonstrates a very useful tool for further improving the accuracy of mass models. For a simple liquid drop formula the root mean square (rms) deviation between the predictions and the 2353 experimental known masses are sharply reduced from 2.455 MeV to 0.235 MeV, and for the other revisited mass models, the rms is remarkably improved by about 30%.

https://iopscience.iop.org/article/10.1088/1361-6471/aa5d78/pdf

Performance of the Levenberg-Marquardt neural network approach in nuclear mass prediction

By incorporating with the radial basis function correction, the Weizsacker-Skyrme-type nuclear mass formula was further improved. The root-mean-square (rms) deviation of the binding energies between the theoretical calculations and 2267 experimental masses has been significantly reduced from 493 keV to 323 keV. The alpha-decay energies Q(alpha) obtained from the binding energy by this hybrid formula also become more precise, i.e. the rms deviations for the Z = 74-118 even-even isotopes and for the 46 superheavy nuclei fall from 420 keV to 161 keV and from 501 keV to 230 keV, respectively. With the above calculated Q(alpha) values as inputs, the calculated alpha-decay half-lives for the even-even (Z = 74-118) nuclei agree with the experimental ones very well, especially for the superheavy nuclei. Thus the further improved Weizsacker-Skyrme-type nuclear mass formula is useful for predicting nuclear properties associated with mass evaluation systematically.

https://iopscience.iop.org/article/10.1088/0954-3899/42/9/095107/pdf

Peng Hui Chen

Papers

Performance of the Levenberg-Marquardt neural network approach in nuclear mass prediction

Weizsäcker–Skyrme-type mass formula by considering radial basis function correction