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

Bayesian neural network (BNN) approach is employed to improve the nuclear mass predictions of various models. It is found that the noise error in the likelihood function plays an important role in the predictive performance of the BNN approach. By including a distribution for the noise error, an appropriate value can be found automatically in the sampling process, which optimizes the nuclear mass predictions. Furthermore, two quantities related to nuclear pairing and shell effects are added to the input layer in addition to the proton and mass numbers. As a result, the theoretical accuracies are significantly improved not only for nuclear masses but also for single-nucleon separation energies. Due to the inclusion of the shell effect, in the unknown region, the BNN approach predicts a similar shell-correction structure to that in the known region, e.g., the predictions of underestimation of nuclear mass around the magic numbers in the relativistic mean-field model. This manifests that better predictive performance can be achieved if more physical features are included in the BNN approach.


Bayes' theorem
where x n and t n (n = 1, 2, ..., N) are input and output data, N is the number of data; p(D|ω) is the likelihood function, which contains the information about parameters ω derived from the observations; p(ω|D) is the probability distribution of parameters ω after the data D are considered, which is called the posterior distribution; p(D) is a normalization constant, which ensures the posterior distribution is a valid probability density and integrates to one.
For the likelihood function p(D|ω), a Gaussian distribution, p(D|ω) = exp(−χ 2 /2), is usually employed, where the objective function χ 2 reads Here, the standard deviation parameter ∆t n is the associated noise error related to the nth observable. For the BNN approach, the function S(x; ω) is described with a neural network, which is where x = {x i } and ω = {a, b j , c j , d ji }, and H and I are the numbers of neurons in the hidden layer and the number of input variables, respectively. In total, the number of parameters in this neural network is 1 + (2 + I) * H.
For the prior distributions p(ω) of model parameters, they are usually set as Gaussian distributions with zero means. However, the precisions (inverse of variances) of these Gaussian distributions are not set as fixed values by hand. We set them as gamma distributions so that the precisions can vary over a large range and hence the BNN approach can search the optimal values of precisions in the sampling process automatically.
After specifying the likelihood function p(D|ω) and the prior distribution p(ω), the posterior distribution p(ω|D) of model parameters is known in principle. One can then make predictions based on this posterior distribution, Since the model parameters are described with a probability distribution, an estimate of uncertainty in theoretical predictions is obtained naturally as Note that Eq. (4) involves a high-dimensional integral in the whole parameter space. For that, we will employ the Monte Carlo integral algorithm, where the posterior distribution p(ω|D) is sampled using the flexible Bayesian model developed by Neal [43], in which the Markov chain Monte Carlo algorithm is employed.
In this work, we will employ the BNN approach to reconstruct mass residuals between experimental data M exp and mass predictions M th of various models, i.e., As in Refs. [44,45], the inputs are usually taken as x = (Z, A). However, we will consider more physical information into the BNN approach, so two extra inputs δ and P related to nuclear pairing and shell effects are also included, which are Here, ν p and ν n are the differences between the actual nucleon numbers Z and N and the nearest magic numbers (8,20,28,50,82 , 126 for protons and 8, 20, 28, 50, 82, 126, 184 for neutrons) [30]. For simplicity, we will use BNN-I2 and BNN-I4 to denote the BNN For the theoretical mass models, we take two microscopic (RMF [27] and HFB-31 [14]), two macroscopic-microscopic (WS4 [12] and FRDM12 [11]), and two macroscopic (BW [30] and BW2 [30]) mass models as examples.
The noise errors in Eq. (2) were usually taken as a fixed value estimated from mass differences between experimental data and model predictions [44,45]. A more elegant way is to set it as a distribution, and the sampling process can search an appropriate value automatically, which can optimize the nuclear mass predictions. In this work, we will use a gamma distribution for the noise precision (inverse of squared noise error 1/∆t 2 ), because the gamma distribution is the conjugate prior distribution of the precision of Gaussian distribution, which can make calculations easier in mathematics [37]. Table I gives the root-mean-square (rms) deviations of nuclear mass with respect to the experimental data in the learning sets for various mass models and their counterparts improved by the BNN-I2 approaches. Clearly, the BNN approach can significantly improve the mass predictions even with a fixed noise precision. By using a gamma distribution, the improvements are further enhanced and the reduction in the rms deviations even approaches 40% for the RMF and BW2 models. In the following, all calculations will be performed with a gamma distribution for the noise precision.
It is well known that nuclear pairing and shell effects play very important roles in mass predictions [1]. For further improving the mass deviations related to such effects, two extra inputs δ and P are included in addition to Z and A. The corresponding rms deviations for various mass models improved by the BNN-I4 approach are given in Table II. The results of original models and those improved by BNN-I2 are also shown for comparison.
As the best example, the liquid-drop BW mass model only includes the volume, surface, symmetry, and Coulomb terms, while both pairing and shell effects are fully neglected [30].
Improved by BNN-I2, its posterior rms deviation is still much larger than those of other mass models. However, with the BNN-I4 approach, its posterior rms deviation is significantly reduced from 850 to 266 keV.
In general, improved by the BNN-I4 approach, the rms deviations of all mass models are significantly reduced, e.g., exceeding 90% for the BW model. It can be seen clearly in Fig. 1(a). In addition, from the rms deviations for the validation set shown in Table II, one can evaluate the predictive performance of the BNN approach. Although the rms deviations for the validation set are slightly larger than those for the learning set, the improvements on the original models are still significant.
The single-nucleon separation energies are related to the derivatives of nuclear mass surface. They are also very important to nucleon-capture reactions in astrophysics.
Therefore, it is interesting to investigate the improvements of single-nucleon separation energies with the BNN approaches. Previous studies found that the RBF approach is one of the powerful techniques to improve the mass predictions of nuclear models [31][32][33], but its improvement in overall mass predictions even deteriorates the description of single-nucleon separation energy (S n or S p ) unless the RBF is done twice separately [34]. Table III shows the rms deviations of S n and S p with respect to the data in the learning and validation sets for various mass models and their counterparts improved by the BNN approaches. For completeness, the two-neutron (S 2n ) and two-proton (S 2p ) separation energies are given together. The results for the learning set are shown in Fig. 1(b). It is clear that the BNN approach can improve the predictions of nuclear masses and the single-nucleon separation energies simultaneously, remarkably for the BNN-I4 approach. This indicates the BNN-I4 approach is more effective to simultaneously improve the descriptions of nuclear mass surface and its derivatives than the BNN-I2 approach.
The rms deviation provides only a gross assessment of the accuracy of a nuclear mass model. To show some details, we present the mass differences between the experimental III: Rms deviations (in MeV) of single-neutron (S n ), single-proton (S p ), two-neutron (S 2n ), and two-proton (S 2p ) separation energies with respect to the experimental data in the learning and validation sets for various mass models and their counterparts improved by the BNN-I2 and BNN-I4 approaches.

Model
Model+BNN Learning set also found in the HFB mass models with Skyrme force [46] or Gogny force [47], which are generally explained as being due to the physics missing from the energy density functionalsthe so-called "beyond mean-field" physics. The idea of the BNN approach is to employ a neural network for simulating such kinds of missing physics in nuclear mass models, so it is expected that the mass predictions of nuclear models can be improved. Panel (b) Quantitatively, the resulting rms deviation is reduced from 2.263 to 0.451 MeV. However, the remaining differences still show some odd-even staggering structures, i.e., smaller and larger differences appear alternately. In addition, from the structure outside the contour lines in panel (b), the BNN approach predicts a systematic overestimation (underestimation) of nuclear mass in the neutron-rich (neutron-deficient) region except for heavy neutron-rich nuclei. It is different from the structure in the known region inside the contour lines, which holds richer features and predicts an overestimation of nuclear masses for nuclei around the magic numbers.
It is well known the odd-even staggering and local structures around magic numbers are related to nuclear pairing correlation and shell effect, respectively. Therefore, the inclusion well improved by the BNN approaches, especially by BNN-I4. This further manifests the BNN-I4 approach achieves better predictive performance than the BNN-I2 approach.
Apart from improving the mass predictions of nuclear models, the BNN approach also provides the uncertainties in mass predictions, which are shown in Figs In summary, we have employed the Bayesian neural network approach to improve the nuclear mass predictions of various models. By using a distribution for the noise error in likelihood function, the BNN approach can find the optimal value of the noise error automatically, which improve nuclear mass predictions remarkably. To better describe nuclear pairing and shell effects on mass predictions, we further include two relevant quantities in addition to the proton and mass numbers, keeping the number of parameters unchanged. It is found that the present BNN approach not only eliminates the smooth mass deviations significantly but also remarkably reduces the odd-even staggering in mass deviations. As a result, the accuracies of all mass models considered here are significantly improved not only for the nuclear masses but also for the separation energies. Furthermore, the mass corrections with the present BNN approach show more structure features, e.g., it predicts an overestimation of nuclear masses for nuclei towards (Z, N) = (28, 82) and (50, 126) in the RMF mass model. This manifests better predictive performance can be achieved not only in the known region but also in the unknown region far from the βstability line, if more physical features are included in the BNN approach.
It is known that there exists an exact universal energy density functional for nuclear ground-state properties, though it is very difficult even impossible to construct it. If one is able to find an accurate energy density functional with the BNN approach by taking various densities as the direct inputs, one can make reliable predictions for various nuclear groundstate properties. Works along this line are now in progress. In addition, one can also apply the BNN approach to improve other nuclear properties with many experimental data, such as nuclear charge radii, β-decay half-lives, and so on.