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

This paper introduces BNN-I6, a Bayesian Neural Network framework trained on ENDF/B-VIII.1 data that accurately predicts (n,p) reaction cross sections with quantified uncertainties, outperforming TENDL-2023 benchmarks while identifying theoretical cross-sections as the dominant predictive feature.

The Gist

Imagine you are trying to predict the weather. You have a supercomputer model that tries to guess if it will rain, but sometimes it gets it wrong because the data is missing or the model is a bit rigid. Now, imagine you have a second "weather forecaster" who not only predicts the rain but also says, "I'm 90% sure it will rain, but there's a small chance I'm wrong."

That is essentially what this paper is about, but instead of weather, they are predicting nuclear reactions.

Here is a simple breakdown of the research by Arunabha Saha and Songshaptak De:

1. The Problem: The "Missing Puzzle Pieces"

In the world of nuclear physics, scientists need to know exactly how neutrons (tiny particles) hit atoms and turn them into different elements (a process called an (n,p) reaction). This is crucial for:

• Building safe nuclear reactors.
• Creating medicines for cancer treatment.
• Understanding how stars make energy.

The Issue: We don't have data for every single atom in the universe. It's like trying to solve a giant jigsaw puzzle, but you are missing half the pieces. The existing computer models (like TENDL-2023) try to fill in the gaps, but they often make mistakes and, more importantly, they don't tell you how unsure they are about those guesses.

2. The Solution: The "Uncertainty-Aware" AI

The authors created a new tool called BNN-I6. Think of this not as a standard calculator, but as a Bayesian Neural Network.

• Standard AI: "I predict the cross-section (the likelihood of the reaction) is 5.0." (It acts like it knows everything).
• Bayesian AI (BNN-I6): "I predict the cross-section is 5.0, but I'm only 80% sure, so the real number could be between 4.5 and 5.5."

This is a game-changer. In nuclear physics, knowing how much you don't know is just as important as the prediction itself. It's like a doctor saying, "I think you have a cold, but I'm not 100% sure, so let's run one more test," rather than just prescribing medicine blindly.

3. How They Trained the AI

To teach this AI, they didn't just throw random numbers at it. They gave it six specific "clues" (inputs) to help it learn:

1. Proton Count (Z): How many protons are in the atom?
2. Neutron Count (N): How many neutrons are there?
3. Pairing Effect (δ): Are the particles paired up nicely (like shoes in a pair) or are they lonely?
4. Energy Gap (∆E): How much extra energy does the incoming neutron have?
5. The "Old Guess" (TENDL): They fed the AI the prediction from the old, standard model (TENDL-2023) as a starting point.
6. Isospin Ratio: A fancy way of measuring the balance between protons and neutrons.

They trained the AI on a massive library of known data (8,110 data points) and then tested it on atoms it had never seen before.

4. The Results: Better than the Old Way

When they compared their new AI to the old standard model (TENDL-2023):

• Accuracy: The AI was generally more accurate, especially for heavy atoms where data is scarce.
• Confidence: The AI's "uncertainty bands" (the range of possible answers) were tight and realistic. It knew when it was guessing and when it was sure.
• The "Secret Sauce": They used a tool called SHAP (like a detective's magnifying glass) to see which clues mattered most. They found that the AI relied heavily on the "Old Guess" (the TENDL data) as its primary guide, but it used the other clues (like energy and particle counts) to fine-tune the answer and fix the mistakes of the old model.

5. Why This Matters

Imagine you are designing a nuclear reactor wall. If you use a model that is wrong and doesn't tell you it's wrong, the wall might crack under radiation. If you use this new BNN-I6 model, you get a prediction plus a safety warning: "Hey, this prediction is a bit shaky because we don't have much data for this specific atom."

In a nutshell:
This paper introduces a smarter, more honest AI for nuclear physics. It doesn't just guess the numbers; it tells us how confident it is in those guesses. This helps scientists make safer designs for reactors, better medicines, and understand the universe, even when they are working with incomplete information. It's like upgrading from a weatherman who just says "It might rain" to one who says, "It will likely rain, but here is the exact probability and the range of possibilities."

Technical Summary

Here is a detailed technical summary of the paper "Bayesian Learning of (n,p) Reaction Cross Sections with Quantified Uncertainties" by Arunabha Saha and Songshaptak De.

1. Problem Statement

Accurate modeling of neutron-induced $(n,p)$ reaction cross sections is critical for nuclear reactor design, nuclear astrophysics, and the production of medical radionuclides. However, current approaches face significant challenges:

• Data Scarcity: Experimental data are often sparse, incomplete, or non-existent for isotopes far from stability or at high incident energies.
• Systematic Uncertainties: Theoretical models (e.g., Hauser-Feshbach formalism in TALYS) and evaluated libraries (e.g., TENDL-2023, ENDF/B-VIII.1) rely on numerous nuclear parameters (optical potentials, level densities) that introduce systematic uncertainties.
• Lack of Uncertainty Quantification: Traditional machine learning (ML) models often provide point estimates without reliable error bounds, which is insufficient for safety-critical nuclear applications.

2. Methodology

The authors propose a data-driven framework using a Bayesian Neural Network (BNN), specifically named BNN-I6, to predict $(n,p)$ cross sections with quantified uncertainties.

Data Preparation

• Source: Evaluated Nuclear Data File (ENDF/B-VIII.1).
• Scope: Covers target nuclei with proton numbers $Z \approx 8–77$ and incident neutron energies up to $\sim 20$ MeV.
• Dataset Size: 8,110 data points split into training (70%), validation (15%), and testing (15%) sets.
• Input Features ($x$): The model utilizes six physically motivated features:
  1. Proton number ($Z$)
  2. Neutron number ($N$)
  3. Pairing effect ($\delta$): Coded as $1$(even-even),$0$(odd-$A$),$-1$ (odd-odd).
  4. Energy offset ($\Delta E = E - E_{thresh}$): Incident energy relative to the reaction threshold.
  5. Theoretical cross-section ($\ln(\sigma_{TENDL})$): Logarithmic value from the TENDL-2023 library.
  6. Isospin asymmetry: $(N-Z)/A$.
• Target: The natural logarithm of the experimental cross-section, $\ln(\sigma_{ENDF})$.

Model Architecture (BNN-I6)

• Structure:
  • Input Layer: 6 neurons.
  • Hidden Layers: Two fully connected layers with 150 and 100 neurons, respectively, using Exponential Linear Unit (ELU) activation functions.
  • Output Layer: Two nodes predicting the mean ($\mu$) and standard deviation ($\sigma$) of the target distribution.
• Theoretical Foundation:
  • Unlike standard ANNs with fixed weights, BNNs treat weights as random variables with probability distributions.
  • Inference: Uses Stochastic Variational Inference (SVI) to approximate the posterior distribution of weights $P(w|D)$ by minimizing the Evidence Lower Bound (ELBO).
  • Loss Function: The ELBO combines the expected log-likelihood (data fit) and the Kullback-Leibler (KL) divergence (regularization against the prior), preventing overfitting.
• Training Strategy: Two-stage training with an initial phase (8,000 epochs, learning rate $\sim 10^{-3}$) followed by fine-tuning (1,000 epochs, learning rate $\sim 2 \times 10^{-4}$).

Interpretability Analysis

• SHAP (SHapley Additive exPlanations): Used to rank the importance of input features and understand the model's decision-making process.

3. Key Contributions

1. Uncertainty Quantification: The primary contribution is the ability of the BNN-I6 to provide not just a central prediction but also credible intervals (uncertainty bounds) for cross-section predictions, addressing a major gap in traditional nuclear data evaluation.
2. Hybrid Data-Driven Framework: The model successfully integrates physical inputs (nuclear structure, energy) with theoretical priors (TENDL-2023) to correct systematic biases in evaluated libraries.
3. Robustness in Data-Scarce Regions: The framework demonstrates the ability to generalize to heavy nuclei and regions with limited experimental data, where traditional models often struggle.
4. Feature Attribution: The study explicitly identifies that the theoretical cross-section input ($\sigma_{TENDL}$) is the dominant driver of predictions, while nuclear parameters play a secondary, corrective role.

4. Results

• Performance Metrics: The model's accuracy was evaluated using the root mean square (r.m.s.) error ($\sigma_{rms}$) against experimental data (ENDF/B-VIII.1).
  • Overall: BNN-I6 consistently outperformed TENDL-2023, showing lower $\sigma_{rms}$ values across most proton number ranges ($Z$).
  • Specific Nuclei:
    • Medium-mass (e.g., $^{126}$Te, $^{160}$Gd): BNN-I6 closely matched experimental excitation functions, whereas TENDL-2023 showed deviations near thresholds and at saturation.
    • Heavy-mass (e.g., $^{192}$Pt, $^{209}$Bi): Despite data scarcity, BNN-I6 maintained reasonable agreement with experimental data and provided narrow, physically meaningful uncertainty bands.
• Comparison with TENDL-2023:
  • BNN-I6 reproduced the global trends of TENDL-2023 but corrected systematic over/under-predictions.
  • In the Pb-Bi region, where reaction thresholds are high, BNN-I6 avoided the over-prediction seen in TENDL-2023.
• Feature Importance (SHAP):
  • $\ln(\sigma_{TENDL})$ was identified as the most influential feature, indicating the model relies heavily on the theoretical baseline.
  • Other features ($\Delta E$, $\delta$, $Z$, $N$, isospin) had smaller but significant impacts, serving to refine the theoretical baseline based on specific nuclear properties.

5. Significance and Conclusion

This work establishes a robust, uncertainty-quantified alternative to traditional nuclear reaction modeling. The BNN-I6 framework offers several critical advantages:

• Reliability: By providing uncertainty estimates, it allows researchers to gauge the reliability of predictions in regions where experimental data is missing.
• Complementarity: It does not replace physical theories but enhances them by learning complex, nonlinear dependencies from data, effectively "correcting" evaluated libraries like TENDL.
• Future Applications: The approach is highly valuable for designing radiation-resistant materials for fusion reactors, optimizing medical isotope production, and improving nuclear astrophysics simulations, particularly for isotopes where experimental measurements are currently impossible or too costly.

The study concludes that Bayesian Neural Networks are a promising tool for the next generation of evaluated nuclear data libraries, bridging the gap between sparse experimental data and theoretical predictions.