Physics Informed Bayesian Machine Learning of Sparse and Imperfect Nuclear Data

This paper presents a physics-informed Bayesian machine learning framework that leverages fission model priors and cumulative yield constraints to accurately evaluate energy-dependent independent fission product yields despite the scarcity and imperfection of experimental nuclear data.

The Gist

The Big Problem: Trying to Cook with a Broken Recipe

Imagine you are a chef trying to create a perfect recipe for a complex dish (nuclear fission yields). You have two major problems:

1. You have very few taste tests: The experimental data (the "taste tests" of how nuclear fuel breaks apart) is extremely scarce, messy, and sometimes contradictory.
2. You have no intuition: If you just use a standard computer program (pure "data-driven" machine learning) to guess the recipe based on those few taste tests, the computer will likely get confused. It might invent flavors that don't exist or miss the subtle spices because it doesn't understand the rules of cooking (physics).

In the world of nuclear physics, this is a huge issue. Scientists need to know exactly how nuclear fuel breaks apart to build better reactors and create medical isotopes, but the data is too thin for computers to learn on their own.

The Solution: A "Smart" Apprentice Chef

The authors of this paper propose a new way to train the computer. Instead of letting the computer start from scratch, they give it a head start using a "Physics-Informed" approach.

Think of it like this:

• The Old Way (Uninformed Learning): You hand a computer a few blurry photos of a cake and ask it to guess the recipe. It might guess wrong because it doesn't know that cakes need flour, eggs, or that they rise in the oven.
• The New Way (Physics-Informed Learning): Before showing the computer the blurry photos, you first teach it a perfect, theoretical textbook on baking (the GEF physics model). The computer reads the whole book and learns the laws of baking (conservation of mass, quantum effects, etc.).
• The Result: Now, when you show the computer the few blurry photos (the real, sparse experimental data), it doesn't start from zero. It uses its knowledge from the textbook to interpret the photos correctly. It knows, "Ah, this blurry spot must be a rising cake because I know how cakes rise."

How They Did It: The Two-Step Training

The researchers used a technique called Bayesian Machine Learning. Here is the process they used, broken down simply:

1. Step 1: The "Textbook" Training (The Prior):
   They took a sophisticated physics model (called GEF) that simulates nuclear fission perfectly based on known laws. They fed this model's generated data into the computer first. This created a "smart prior"—a baseline expectation of what the data should look like.

2. Step 2: The "Real World" Adjustment (The Posterior):
   Then, they showed the computer the actual, sparse, and messy experimental data. Because the computer already knew the "rules of the game" from Step 1, it could adjust its understanding to fit the real data without getting confused or inventing nonsense.

3. Step 3: The "Double-Check" (Constraints):
   They also used a clever trick. They knew that "Independent Yields" (how pieces break apart immediately) and "Cumulative Yields" (how pieces look after they decay over time) are mathematically linked. They used this link as a safety net. If the computer's guess for the immediate break-up didn't match the known rules for the long-term decay, the computer was forced to correct itself.

What They Found: Smarter Predictions

When they tested this new method on Uranium-235 (a common nuclear fuel), the results were impressive:

• Accuracy: The "Smart Apprentice" (Physics-Informed) was much closer to the known "Gold Standard" data than the "Clueless Apprentice" (Uninformed). The error rate dropped from about 5% to less than 1%.
• Understanding the "Fine Print": Nuclear data has tiny wiggles and patterns (like odd and even numbers of particles behaving differently). The old method missed these details. The new method, because it learned the physics rules first, could see and predict these subtle patterns correctly.
• Speed: Because the computer started with a "textbook" education, it learned the real data much faster and with less confusion.

The Bottom Line

This paper demonstrates that you can't just throw data at a computer and expect it to understand nuclear physics. You have to teach the computer the laws of physics first.

By combining a theoretical physics model with real-world data, the researchers created a tool that can fill in the gaps of missing data with high confidence. This is crucial for designing future nuclear energy systems and medical tools, ensuring that the "recipes" for nuclear fuel are accurate, safe, and reliable, even when we don't have enough experimental data to check every single step.

Technical Summary

Technical Summary: Physics Informed Bayesian Machine Learning of Sparse and Imperfect Nuclear Data

Problem Statement
Data-driven machine learning in nuclear physics faces significant hurdles due to the scarcity of experimental data and the inherent imperfections (noise, incompleteness, and discrepancies) of existing datasets. Purely data-driven approaches often fail to exploit the maximum value of scarce data, such as neutron-induced fission product yields, which are critical for advanced nuclear energy applications (e.g., new fuel types, medical isotope production) but are only sparsely available at specific incident energies (thermal, 0.5 MeV, and 14 MeV). Furthermore, while microscopic theories (e.g., TD-DFT, TD-GCM) describe certain aspects of fission, they struggle to reproduce detailed distributions of fission product yields, and semi-empirical models often lack comprehensive energy dependence. The specific challenge addressed in this work is the evaluation of independent fission yields, which have fewer measured data points than cumulative yields but are essential for fuel cycle operations and understanding fission processes.

Methodology
The authors propose a Physics-Informed Bayesian Machine Learning framework that integrates physical models directly into the learning process, specifically through the construction of informative priors. The methodology consists of the following key components:

1. Bayesian Neural Network (BNN) Framework: The core architecture is a BNN with two hidden layers (22 neurons each) using a tanh activation function. Inputs include the fragment mass number ($A$), charge number ($Z$), and neutron incident energy ($E$). The output is the independent fission yield $Y_i$. Uncertainties are quantified via the 95% confidence interval (CI) using Markov Chain Monte Carlo (MCMC) sampling of network weights.
2. Physics-Informed Priors (Transfer Learning): Instead of starting with uninformative priors, the authors use a physics model (the GEF model) to generate a large dataset of energy-dependent independent yields for $^{235}$U. A BNN is first trained on this generated data ($D_{phys}$) to obtain a posterior distribution $P(w_1|D_{phys})$. This posterior is then used as the informative prior $P(w_2)$ for the subsequent evaluation of sparse experimental data ($D_{expt}$). This approach effectively transfers comprehensive physics knowledge (including conservation laws, shell effects, and odd-even effects) into the Bayesian inference.
3. Physical Constraints via Cumulative Yields: To address the scarcity of independent yield data, the framework incorporates heterogeneous data fusion. Cumulative yields, which are more abundant, are linked to independent yields via a conversion matrix (determined by $\beta$-decays). This relationship is enforced as a physical constraint in the cost function ($\chi^2$), penalizing deviations between the predicted independent yields and the measured cumulative yields.
4. Training and Evaluation: The model is trained on a combination of experimental data (from the EXFOR library and JENDL-5 evaluations) and the physics-constrained loss function. The training process utilizes GPU acceleration.

Key Results
The study demonstrates the efficacy of the physics-informed approach through several comparative analyses:

• Improved Accuracy and Convergence: The "informed learning" (using GEF priors) significantly outperforms "uninformed learning" (standard BNN). The normalization deviation for informed learning is approximately 0.22%, compared to 5.3% for uninformed learning. Furthermore, the loss values converge much faster with informed priors, whereas uninformed learning shows slow convergence with large uncertainties.
• Physical Consistency in Energy Dependence: The informed learning correctly reproduces the physical evolution of fission yields with increasing excitation energy. Specifically, it captures the exponential increase of the symmetric fission channel at low energies and the gradual merging of charge peaks. In contrast, uninformed learning fails to capture these trends, showing unreasonable rapid increases in symmetric yields at low energies and incorrect fine structures.
• Fine Structure Reproduction: The informed approach successfully interpolates fine structures, such as the odd-even staggering in charge yields. It correctly predicts the disappearance of odd-even staggering at lower excitation energies (around 6–10 MeV depending on fragment mass) compared to uninformed learning, aligning with theoretical expectations regarding pairing correlations at finite temperatures.
• Constraint Effectiveness: The inclusion of cumulative yield constraints reduces the loss values significantly (e.g., from 0.57 to 0.17 for specific isotopes), demonstrating that the constraints effectively supplement energy dependence information, even though the independent and cumulative yields are largely compatible.
• Uncertainty Quantification: The Bayesian framework provides robust uncertainty estimates. The informed learning yields tighter confidence intervals that better align with experimental data points (e.g., at 3 MeV) compared to the broad uncertainties of uninformed models.

Significance and Claims
The paper claims to demonstrate a "truly Bayesian machine learning" approach that leverages comprehensive physics knowledge to exploit sparse and expensive nuclear data. The primary significance lies in:

1. Bridging the Data Gap: By using physics models to generate informative priors, the method overcomes the limitations of small datasets, enabling reliable evaluations of independent fission yields where experimental data is extremely sparse.
2. Superiority over Pure Data-Driven Methods: The results show that without embedding physical information (via priors or constraints), it is impossible to fully exploit scarce data or reproduce complex physical phenomena like fine structures and non-monotonic energy dependencies.
3. Generalizability: While focused on fission yields, the authors posit that this framework can be extended to other nuclear physics domains with imperfect datasets, such as reaction cross-sections and the equation of state of nuclear matter.
4. Automation and Objectivity: The approach offers a path toward automated, bias-free nuclear data evaluation, reducing reliance on expert-driven manual adjustments in the generation of recommended nuclear data libraries.

The work concludes that incorporating existing nuclear theories and models into machine learning priors is essential for the advancement of nuclear physics data evaluation in the coming years.