⚛️ nuclear theory

Evaluation of U-235 and U-238 Fission Product Yields Using Bayesian Neural Networks: Comparison of Baseline and Physics-Informed Models

This paper demonstrates that a physics-informed Bayesian neural network (BNN3), which incorporates physical features like the odd-even effect, beta-decay energy, and isospin, significantly outperforms a baseline model in predicting U-235 and U-238 fission product yields with higher accuracy, better agreement with experimental data, and narrower confidence intervals.

Original authors: Chun-Yuan Qiao, Ya-Xuan Wang, Jun-Chen Pei, Chun-Wang Ma, Yong-Jing Chen, Jin-Gen Chen, Jie Pu, Kai-Xuan Cheng, Yu-Ting Wang, Ya-Fei Guo, Xiang Chen

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chun-Yuan Qiao, Ya-Xuan Wang, Jun-Chen Pei, Chun-Wang Ma, Yong-Jing Chen, Jin-Gen Chen, Jie Pu, Kai-Xuan Cheng, Yu-Ting Wang, Ya-Fei Guo, Xiang Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict exactly how a giant Lego tower will break apart when you knock it over. In the world of nuclear physics, this "tower" is an atom like Uranium-235 or Uranium-238, and the "knock" is a neutron hitting it. When it breaks (fissions), it doesn't just fall into two random piles; it splits into specific smaller atoms (fission products) in very predictable patterns. Knowing exactly which pieces you get and how many of them is crucial for building safe nuclear reactors and managing nuclear waste.

However, predicting these patterns is like trying to guess the outcome of a complex game of dice where the dice change their behavior depending on how hard you throw them. Scientists have "evaluated libraries" (like a giant rulebook called JENDL) that contain the best-known answers, but these books only have answers for a few specific "throw strengths" (neutron energies). For everything in between, or for cases where data is missing, we need a way to fill in the gaps accurately.

The "Smart Guessing" Machine

The authors of this paper built a "Smart Guessing Machine" using a type of artificial intelligence called a Bayesian Neural Network (BNN).

Think of a standard AI as a student who memorizes a textbook and tries to guess the answers to new questions based purely on patterns it has seen. It's good, but it can be a bit rigid and doesn't always understand why the answer is what it is.

A Bayesian AI is like a student who not only memorizes the textbook but also keeps a "confidence score" for every answer. Instead of saying "The answer is X," it says, "The answer is likely X, and I'm 95% sure it's between X and Y." This is vital in nuclear physics because knowing the uncertainty is just as important as the prediction itself.

The Two Models: The Baseline vs. The Physics-Expert

The researchers built two versions of this AI to see which one was better:

The Baseline Model (BNN0): This is the "standard student." It was fed the basic facts: the number of protons, the number of neutrons, and the energy of the incoming neutron. It learned the patterns from the data but didn't know any specific rules of nuclear physics.
The Physics-Informed Model (BNN3): This is the "expert student." The researchers gave it the same basic facts, but they also handed it a cheat sheet of physics rules. Specifically, they added three key concepts:
- The Odd-Even Effect: In the nuclear world, particles prefer to be in pairs (like socks). An atom with an even number of protons or neutrons is more stable than one with an odd number. This model learned to respect this "pairing rule."
- Beta-Decay Energy: After the atom splits, the pieces are often unstable and try to fix themselves by changing a neutron into a proton (beta decay). This model learned the energy cost of that process.
- Isospin: This is a fancy way of describing the balance between protons and neutrons, helping the model understand the "personality" of the atom.

The Results: A Clear Winner

The researchers tested both models against the "rulebook" (JENDL) and real-world experimental data. Here is what they found, using simple analogies:

Smoother, Sharper Predictions: The basic model (BNN0) was okay at seeing the big picture, but it tended to "blur" the details. It missed the tiny wiggles and bumps in the data that are caused by the odd-even pairing rules. The expert model (BNN3) didn't just see the big hills and valleys; it could see the small rocks and pebbles too. It reproduced the "wiggles" in the data much more accurately.
Confidence in the Answer: The expert model didn't just guess better; it was more confident. The range of possible answers (the "confidence interval") became much narrower. Imagine the basic model saying, "The answer is somewhere between 10 and 50," while the expert model says, "The answer is between 28 and 32." Both might be right, but the expert is much more useful.
Handling the "Hard" Cases: The biggest improvement happened with the "low-yield" fragments—the rare, tiny pieces that are hard to predict. The basic model struggled here, but the expert model reduced its errors by over 90% in these difficult areas. It was like the expert student finally figuring out the trick questions that stumped the standard student.
Generalizing to New Situations: They tested the models on a neutron energy level (4.6 MeV) that the AI had never seen before. The expert model (BNN3) held its ground, predicting the patterns much closer to the real experimental data than the basic model did.

The Bottom Line

The paper concludes that by teaching the AI the "rules of the game" (physics principles) rather than just letting it memorize the scores, the model became significantly smarter. It didn't just get the numbers right; it understood the underlying structure of how atoms break apart.

This means we can now generate more accurate and reliable data for nuclear reactors and waste management, filling in the gaps where experimental data is missing or sparse, all while knowing exactly how much we can trust those predictions.

Technical Summary: Evaluation of 235,238U Fission Product Yields Using Bayesian Neural Networks

Problem Statement
Accurate evaluation of fission product yields for $^{235}\text{U}$ and $^{238}\text{U}$ is critical for advanced reactor design, nuclear waste management, and fundamental nuclear physics studies. However, existing international nuclear data libraries (e.g., JENDL, ENDF) typically provide evaluated yields only at discrete incident neutron energies (thermal, 0.5 MeV, 14 MeV). Experimental data are often sparse, exhibit incomplete isotopic coverage, and contain significant uncertainties. Furthermore, phenomenological models (e.g., Brosa, GEF) rely heavily on parameter calibration against experimental data; when such data are limited, model assumptions become weakly constrained, leading to increased uncertainties and reduced predictive reliability. There is a need for a method capable of reconstructing continuous energy-dependent yield distributions with quantified uncertainties, particularly for isotopic chains where fine nuclear structure effects (e.g., odd-even staggering) are prominent.

Methodology
The authors employ a Bayesian Neural Network (BNN) framework to evaluate cumulative fission yields. Unlike traditional neural networks that output deterministic values, the BNN treats model parameters as probability distributions, enabling the simultaneous prediction of yields and their associated uncertainties.

Model Architecture: The baseline model (BNN0) consists of a two-hidden-layer network with 20 neurons per layer, utilizing the tanh activation function. The input vector initially includes the proton number ( $Z$ ), mass number ( $A$ ) of the fission fragments, and the excitation energy of the compound nucleus ( $E_{in} = E_n + S_n$ ).
Physics-Informed Enhancement: To improve predictive capability, the authors developed an enhanced model (BNN3) by incorporating specific physics-informed features as additional inputs:
- Odd-even effect ( $\delta_{np}$ ): Quantifies nucleon pairing correlations, crucial for describing stability and yield distributions.
- Isospin ( $T_3$ ): Defined as $(Z-N)/50$ , characterizing neutron richness.
- $\beta$ -decay energy ( $Q_\beta$ ): Introduced to account for decay chain contributions to cumulative yields.
Training and Validation: The models were trained on a dataset combining experimental data from the EXFOR database and evaluated data from the JENDL library.
- For $^{235}\text{U}$ , data at thermal, 0.5 MeV, and 14 MeV were used for training.
- For $^{238}\text{U}$ , data at 0.5 MeV and 14 MeV were used for training, while the 4.6 MeV dataset (containing the most comprehensive experimental coverage) was reserved as an independent validation set to test generalization.
- The training process utilized Markov Chain Monte Carlo (MCMC) methods to infer the posterior distribution of network parameters, calculating predictive means and 95% confidence intervals (CI).

Key Contributions

Development of BNN3: The integration of physical constraints (odd-even effect, isospin, $\beta$ -decay energy) into the BNN architecture.
Ablation Study: A systematic analysis demonstrating that the odd-even effect and isospin provide complementary information, significantly reducing the Mean Squared Error (MSE) for independent yields by 35.8% compared to the baseline.
Continuous Energy Evaluation: The generation of energy-dependent yield predictions across a continuous range (0–20 MeV for $^{235}\text{U}$ , 0–15 MeV for $^{238}\text{U}$ ), bridging gaps between discrete experimental points.
Uncertainty Quantification: The provision of 95% confidence intervals that are narrower than those of the baseline model while maintaining coverage close to the nominal level.

Results
Comparative analyses between the baseline model (BNN0) and the physics-informed model (BNN3) reveal significant improvements:

Global Accuracy: For $^{235}\text{U}$ , the global MSE was reduced by 49.8% (from 0.00367 to 0.00184). For $^{238}\text{U}$ , the MSE decreased by 49.2% (from 0.00571 to 0.00290).
Local Structure Reconstruction: BNN3 successfully reproduces fine structures in isotopic chain yields, such as odd-even staggering and local fluctuations near magic numbers, which BNN0 tends to smooth over or miss. This is particularly evident in the symmetric fission region and near peak yield distributions.
Energy Dependence: BNN3 demonstrates better agreement with both experimental data and evaluated libraries regarding the energy dependence of 16 representative fission products. It captures trends such as the rapid decrease in yields at low energies and the behavior near 14–15 MeV more accurately than BNN0.
Uncertainty Reduction: The Mean Prediction Interval Width (MPIW) decreased by approximately 25% for both isotopes, indicating sharper predictions. The 95% CI coverage remained robust (97.20% for $^{235}\text{U}$ and 95.89% for $^{238}\text{U}$ ).
Low-Yield Fragments: The most dramatic improvement occurred in the low-yield region ( $Y < 0.001$ ), where prediction errors were reduced by over 92% for both isotopes, suggesting enhanced capability to learn weak-yield structures.
Generalization: On the independent 4.6 MeV validation set for $^{238}\text{U}$ , BNN3 maintained consistency with experimental trends and reproduced local isotopic structures more faithfully than BNN0, despite not being trained on this specific energy point.

Significance
The paper concludes that incorporating relevant physical information into Bayesian neural networks significantly enhances the model's sensitivity to underlying fission mechanisms. The BNN3 model offers a more accurate and robust approach for evaluating cumulative fission-yield distributions, effectively reproducing both macroscopic patterns and microscopic nuclear-structure characteristics. By providing narrower confidence intervals without sacrificing reliability, this methodology offers a foundation for the development of next-generation nuclear data libraries, addressing the challenges of sparse experimental data and the need for continuous energy-dependent evaluations in reactor physics and nuclear waste management.

The "Smart Guessing" Machine

The Two Models: The Baseline vs. The Physics-Expert

The Results: A Clear Winner

The Bottom Line

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