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Bayesian Learning of (n,p) Reaction Cross Sections with Quantified Uncertainties

Arunabha Saha, Songshaptak De

Abstract

Accurate modeling of neutron-induced (n,p) reaction cross sections is essential for diverse applications in nuclear physics, including reactor design, nuclear astrophysics, and radionuclide production. However, experimental data are often sparse or incomplete, and theoretical results like TALYS Evaluated Nuclear Data Library (TENDL-2023) data may carry systematic uncertainties. In this work, we present a data-driven framework based on a Bayesian Neural Network (BNN), denoted as BNN-I6, to predict (n,p) reaction cross sections with quantified uncertainties. The model incorporates six physically motivated input features and is trained on Evaluated Nuclear Data from the ENDF/B-VIII.1 library. Leveraging stochastic variational inference, the BNN offers reliable uncertainty estimates in addition to accurate predictions. The performance of BNN-I6 is benchmarked against the TENDL-2023 library and experimental measurements across a wide range of nuclei. Additionally, SHapley Additive exPlanations (SHAP) based feature-importance analysis reveals the dominant role of theoretical cross-section inputs in driving predictions. These results highlight the potential of BNN-based approaches to enhance nuclear data evaluations and support future applications in data-scarce regimes.

Bayesian Learning of (n,p) Reaction Cross Sections with Quantified Uncertainties

Abstract

Accurate modeling of neutron-induced (n,p) reaction cross sections is essential for diverse applications in nuclear physics, including reactor design, nuclear astrophysics, and radionuclide production. However, experimental data are often sparse or incomplete, and theoretical results like TALYS Evaluated Nuclear Data Library (TENDL-2023) data may carry systematic uncertainties. In this work, we present a data-driven framework based on a Bayesian Neural Network (BNN), denoted as BNN-I6, to predict (n,p) reaction cross sections with quantified uncertainties. The model incorporates six physically motivated input features and is trained on Evaluated Nuclear Data from the ENDF/B-VIII.1 library. Leveraging stochastic variational inference, the BNN offers reliable uncertainty estimates in addition to accurate predictions. The performance of BNN-I6 is benchmarked against the TENDL-2023 library and experimental measurements across a wide range of nuclei. Additionally, SHapley Additive exPlanations (SHAP) based feature-importance analysis reveals the dominant role of theoretical cross-section inputs in driving predictions. These results highlight the potential of BNN-based approaches to enhance nuclear data evaluations and support future applications in data-scarce regimes.
Paper Structure (7 sections, 3 equations, 7 figures)

This paper contains 7 sections, 3 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Data Preparation
  4. BNN Model Implementation
  5. Results and Discussions
  6. Summary
  7. Acknowledgements

Figures (7)

  • Figure 1: (color online) A schematic diagram of our BNN-I6 model. The model consists of an input layer with six features, two hidden layers with 150 and 100 neurons respectively, and two output nodes predicting the mean and standard deviation of the target distribution.
  • Figure 2: (color online) $\sigma_{\mathrm{rms}}$ of (n,p) reaction cross sections predicted by the BNN-I6 model and those from the TENDL-2023 evaluation with respect to available experimental data, shown as a function of the proton-number range. The BNN-I6 predictions exhibit generally lower r.m.s. deviations across most of the nuclear landscape region. Minor fluctuations at intermediate and heavy mass ranges (20 $<$ Z $\le$ 30) and (50 $<$ Z $\le$ 60) reflect localized model uncertainty in those regions.
  • Figure 3: (color online) Comparison of the experimental data of (n,p) reaction cross sections (solid symbols) with predictions from BNN-I6 model (solid green lines) and the TENDL-2023 data (purple dashed lines) for medium-mass target nuclei: (a) $^{126}$Te(n,p)$^{126}$Sb, (b) $^{124}$Xe(n,p)$^{124}$I, (c) $^{160}$Gd(n,p)$^{160}$Eu, and (d) $^{164}$Er(n,p)$^{164}$Ho. The shaded region denotes the uncertainty band of the BNN-predicted distributions. The corresponding r.m.s. error metric, $\sigma_{\mathrm{rms}}$, are indicated for both TENDL-2023 and BNN-I6 in each panel. The BNN-I6 predictions follow the experimental trends closely across the full energy range, exhibiting comparable r.m.s. deviations with TENDL-2023.
  • Figure 4: (color online) Same as Fig. 2, but for heavy target nuclei: (a) $^{192}$Pt(n,p)$^{192}$Ir, (b) $^{197}$Au(n,p)$^{197}$Pt, (c) $^{204}$Tl(n,p)$^{204}$Hg, (d) $^{207}$Pb(n,p)$^{207}$Tl, (e) $^{209}$Bi(n,p)$^{209}$Pb, and (f) $^{210}$Po(n,p)$^{210}$Bi. The BNN-I6 model provides an accurate reproduction of the experimental excitation functions with narrower uncertainty bands. For nuclei in the Pb-–Bi region, where reaction thresholds are high and data are sparse, the BNN-I6 model maintains reasonable agreement with the experimental data. The $\sigma_{rms}$ values for BNN and TENDL are also shown for comparison.
  • Figure 5: (color online) Comparison of the neutron-induced (n,p) reaction cross sections obtained from the BNN-I6 model with those from the evaluated TENDL-2023 library for various target nuclei. The solid green lines represent the BNN-predicted mean values of $\ln(\sigma[\text{barn}]$), while the shaded green bands reflect the predictive uncertainty of the network. The dashed violet lines correspond to the TENDL-2023 results. The BNN-I6 model reproduces the general energy dependence of the (n,p) excitation functions and shows good agreement with TENDL-2023 predictions over most of the incident-energy range. Slight deviations and wider uncertainty intervals at higher energies primarily reflect the reduced density of training data in those regions and the increasing model uncertainty beyond the most constrained energy domain.
  • ...and 2 more figures