Bayesian uncertainty quantification for nuclear density functional theory

Fig. 3. Prior and posterior probability distributions of parameters (diagonal), bivariate joint distributions (upper triangle), and scatter plots of posterior distributions (lower triangle)

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Fig. 4. Sensitivity analysis: the Pearson correlation coefficient between $Δ μ_{p n}^{*}$ and symmetry energy as functions of baryon density^[23]

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Fig. 5. Effective proton-neutron chemical potential differences $Δ μ_{p n}^{*}$ and symmetry energy $E_{s y m} (ρ_{0})$ under the Skyrme density functional (open circles) and RMF models(open triangles), along with Gaussian process predictions (mean values represented by solid and dashed lines, respectively, and 1σ errors bands depicted by shaded bands). The experimental values of $Δ μ_{p n}^{*}$ are indicated by the grey dashed lines.^[23]

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Fig. 6. Posterior probability density distribution (PDF) of $E_{s y m} (2 ρ_{0} / 3)$ derived from the Skyrme energy density functional, the RMF model, and Bayesian model averaging(color online)^[23]

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Table 1. Upper and lower bounds of the uniform prior distribution for parameters
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Table 1. Upper and lower bounds of the uniform prior distribution for parameters
参数Parameters 下限Lower limits 上限Upper limits
$ρ_{0}$ / fm^-3 0.145 0.165
$E_{0} (ρ_{0})$ / MeV -16.6 -15.4
$K_{0}$ / MeV 180 280
$E_{s y m} (ρ_{0})$ / MeV 28 40
$c_{ω}$ 0 0.012
$m_{D i r a c}^{*} / m$ 0.55 0.65
L / MeV 0 200
$m_{σ}$ / MeV 450 550

Table 2. Experimental data and theoretical uncertainties adopted for binding energy E_b and charge radius R_c used in Bayesian analyses^[8]. Theoretical uncertainty for each observable is obtained by multiplying the standard error listed in the second row (±0.25% for E_b and ±0.26% for R_c) by the weighting factor listed next to experimental data.

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Table 2. Experimental data and theoretical uncertainties adopted for binding energy E_b and charge radius R_c used in Bayesian analyses^[8]. Theoretical uncertainty for each observable is obtained by multiplying the standard error listed in the second row (±0.25% for E_b and ±0.26% for R_c) by the weighting factor listed next to experimental data.

原子核Atomic nucleus $_{Z}^{A} X$	结合能Binding energy E_b / MeV (±0.25%)		电荷半径Charge radius $R_{c}$ / fm (±0.26%)
$_{8}^{16} O$	-127.620	1.0	2.690	5.4
$_{20}^{40} C a$	-342.050	1.0	3.471	1.0
$_{20}^{48} C a$	-416.000	1.0	3.470	1.0
$_{28}^{68} N i$	-590.410	1.0
$_{50}^{100} S n$	-825.300	1.0
$_{50}^{132} S n$	-1 102.840	1.0	4.704	1.0
$_{82}^{208} P b$	-1 636.430	1.0	5.497	1.0

Table 3. The MAP values, means, medians, and 68% and 90% confidence intervals for the posterior distributions of the parameters

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Table 3. The MAP values, means, medians, and 68% and 90% confidence intervals for the posterior distributions of the parameters

参数Parameters	最大后验估计MAP	平均值Mean	中位数Median	68%置信区间68% C.L.	90%置信区间90% C.L.
$ρ_{0}$ / fm^-3	0.151 9	0.151 8	0.151 8	0.150 3~0.153 3	0.148 8~0.154 7
$E_{0} (ρ_{0})$ / MeV	-16.12	-16.14	-16.14	-16.22~-16.07	-16.30~15.99
$K_{0}$ / MeV	227.9	228.2	228.2	222.0~234.4	215.6~240.4
$E_{s y m} (ρ_{0})$ / MeV	34.50	36.40	36.75	33.75~38.96	31.03~39.84
$c_{ω}$	0.002 154	0.002 536	0.002 249	0.000 711 4~0.004 354	0.000 106 3~0.006 750
$m_{D i r a c}^{*} / m$	0.594 0	0.595 5	0.595 0	0.584 7~0.606 2	0.573 8~0.619 5
L / MeV	82.14	115.2	114.9	75.51~153.4	54.53~182.9
$m_{σ}$ / MeV	502.8	501.6	501.7	497.9~505.3	493.8~508.9

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Mengying QIU, Zhen ZHANG. Bayesian uncertainty quantification for nuclear density functional theory[J]. NUCLEAR TECHNIQUES, 2025, 48(5): 050006

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Paper Information

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

Received: Mar. 7, 2025

Accepted: --

Published Online: Jun. 26, 2025

The Author Email: Zhen ZHANG (张振)

DOI:10.11889/j.0253-3219.2025.hjs.48.250098

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laser devices and laser physics

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Instrumentation, Measurement and Metrology

Table 1. Upper and lower bounds of the uniform prior distribution for parameters

Table 1. Upper and lower bounds of the uniform prior distribution for parameters

Table 3. The MAP values, means, medians, and 68% and 90% confidence intervals for the posterior distributions of the parameters

Table 3. The MAP values, means, medians, and 68% and 90% confidence intervals for the posterior distributions of the parameters