Continuous Bayesian probability estimator in predictions of nuclear masses

Fig. 1. (a, c, e) Raw residuals $δ^{p r e}$ of the theoretical nuclear masses for the HFB model with SkP parameter set, the RMF model with NLSH parameter set and the DZ10 model, which includes 868 even-even nuclei in AME2020. (b, d, f) The corresponding corrected residuals by the CBP method on the entire set (color online)

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Fig. 2. Extrapolation of the two-neutron separation energy $S_{2 n}$ for even-N calcium isotopes using the BMA method. The solid line represents the average prediction by the BMA method, while the shaded bands indicate the associated uncertainty $σ^{B M A} (Z, N)$ . The inset shows the probability of existence of the calcium isotopes. The limit of zero separation energy and the $P_{e x} = 0.5$ limit are indicated by the dotted lines.

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Table 1. The standard deviation from the theoretical models and after the CBP refinements in the entire set
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Table 1. The standard deviation from the theoretical models and after the CBP refinements in the entire set
模型
Models
HFB RMF Semiempirical
SkP SLy4 NL3* NLSH WS* DZ10
$σ_{p r e}$ / MeV 4.20 6.06 2.9 3.67 0.60 1.26
$σ_{p o s t}$ / MeV 0.63 0.72 0.84 0.85 0.30 0.35
$Δ σ / σ_{p r e}$ / % 85 88 71 77 50 72

Table 2. The standard deviations σ_pre and σ_post for the learning and validation sets, both before and after applying the CBP estimator (MeV)

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Table 2. The standard deviations σ_pre and σ_post for the learning and validation sets, both before and after applying the CBP estimator (MeV)

模型 Model		HFB		RMF		Semiempirical
模型 Model		SkP	SLy4	NL3*	NLSH	WS*	DZ10
训练集 Learning set	$σ_{p r e}$ / MeV	4.13	5.86	2.70	3.66	0.52	1.07
	$σ_{p o s t}$ / MeV	0.64	0.68	0.85	0.87	0.31	0.35
	$Δ σ / σ_{p r e}$ / %	85	88	69	76	41	67
验证集 Validation set	$σ_{p r e}$ / MeV	4.67	7.44	3.47	3.78	0.98	2.21
	$σ_{p o s t}$ / MeV	1.33	1.51	1.61	1.72	0.58	0.73
	$Δ σ / σ_{p r e}$ / %	72	80	53	55	41	67

Table 3. The corrected residuals $δ^{c o r r}$ of six benchmark nuclei: ¹⁴O, ²⁸Ne, ⁸²Ru, ¹¹²Zr, ¹⁷⁸Pt, and ²⁸⁰Ds, and the weights of six models by BMA are based on $δ^{c o r r}$ of these six benchmark nuclei (MeV)

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Table 3. The corrected residuals $δ^{c o r r}$ of six benchmark nuclei: ¹⁴O, ²⁸Ne, ⁸²Ru, ¹¹²Zr, ¹⁷⁸Pt, and ²⁸⁰Ds, and the weights of six models by BMA are based on $δ^{c o r r}$ of these six benchmark nuclei (MeV)

模型 Models	¹⁴O	²⁸Ne	⁸²Ru	¹¹²Zr	¹⁷⁸Pt	²⁸⁰Ds	权重 Weight
Sly4	-1.20	-1.45	-2.12	-0.45	-0.15	0.40	0.03
SkP	0.94	-1.53	-2.41	-1.75	0.08	0.08	0.01
NL3*	1.51	-0.58	-1.62	-1.26	-0.19	0.36	0.06
NLSH	1.19	-0.26	-1.13	-0.05	-0.31	0.02	0.21
DZ10	-0.19	-0.60	-0.30	-0.55	-0.02	-0.60	0.37
WS*	-1.20	-0.15	-0.39	-0.06	-0.13	0.22	0.32

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Kaizhong TAN, Wanqing GAO, Jian LIU. Continuous Bayesian probability estimator in predictions of nuclear masses[J]. NUCLEAR TECHNIQUES, 2025, 48(5): 050010

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Paper Information

Category: Special Topics on Applications of Machine Learning in Nuclear Physics and Nuclear Data

Received: Mar. 11, 2025

Accepted: --

Published Online: Jun. 26, 2025

The Author Email: Jian LIU (刘健)

DOI:10.11889/j.0253-3219.2025.hjs.48.250104

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laser devices and laser physics

Lasers and Laser Optics

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laser manufacturing

Instrumentation, Measurement and Metrology

Table 1. The standard deviation from the theoretical models and after the CBP refinements in the entire set

Table 1. The standard deviation from the theoretical models and after the CBP refinements in the entire set

Table 2. The standard deviations σpre and σpost for the learning and validation sets, both before and after applying the CBP estimator (MeV)

Table 2. The standard deviations σpre and σpost for the learning and validation sets, both before and after applying the CBP estimator (MeV)

Table 3. The corrected residuals δcorr of six benchmark nuclei: 14O, 28Ne, 82Ru, 112Zr, 178Pt, and 280Ds, and the weights of six models by BMA are based on δcorr of these six benchmark nuclei (MeV)

Table 3. The corrected residuals δcorr of six benchmark nuclei: 14O, 28Ne, 82Ru, 112Zr, 178Pt, and 280Ds, and the weights of six models by BMA are based on δcorr of these six benchmark nuclei (MeV)

Table 2. The standard deviations σ_pre and σ_post for the learning and validation sets, both before and after applying the CBP estimator (MeV)

Table 2. The standard deviations σ_pre and σ_post for the learning and validation sets, both before and after applying the CBP estimator (MeV)

Table 3. The corrected residuals $δ^{c o r r}$ of six benchmark nuclei: ¹⁴O, ²⁸Ne, ⁸²Ru, ¹¹²Zr, ¹⁷⁸Pt, and ²⁸⁰Ds, and the weights of six models by BMA are based on $δ^{c o r r}$ of these six benchmark nuclei (MeV)

Table 3. The corrected residuals $δ^{c o r r}$ of six benchmark nuclei: ¹⁴O, ²⁸Ne, ⁸²Ru, ¹¹²Zr, ¹⁷⁸Pt, and ²⁸⁰Ds, and the weights of six models by BMA are based on $δ^{c o r r}$ of these six benchmark nuclei (MeV)