Multi-task deep neural network for predicting both nuclear fission yields and their experimental errors in peak-shaped data

This paper introduces a multi-task deep neural network equipped with a novel loss function and odd-even effect incorporation to more accurately predict both nuclear fission product yields and their experimental errors in peak-shaped data compared to conventional independent learning methods.

The Gist

Imagine you are trying to predict the weather. You have a lot of data about sunny days, rainy days, and stormy days. But you need to predict a very specific, rare event: a sudden, violent thunderstorm that happens only in a tiny, jagged valley.

This is exactly the challenge nuclear physicists face when studying Nuclear Fission.

The Problem: The "Jagged Mountain"

When a heavy atom (like Uranium) splits, it doesn't just break into two random pieces. It breaks into a specific set of smaller atoms called Fission Products. If you graph how often each type of atom is produced, you don't get a smooth hill. You get a double-humped mountain range with very sharp, jagged peaks and deep valleys.

• The Peaks: These are the most common results.
• The Valleys: These are rare results.
• The Problem: Existing computer models are great at drawing smooth hills, but they struggle to predict the sharp, jagged peaks accurately. They also struggle to tell you how confident they are in their prediction (the "error").

The Solution: A "Twin-Brain" AI

The researchers in this paper built a new kind of Artificial Intelligence (AI) to solve this. Instead of using a standard AI, they used a Multi-Task Deep Neural Network.

Think of this AI as a twin-brain system:

1. Brain A is tasked with predicting the amount of fission products (the height of the mountain).
2. Brain B is tasked with predicting the uncertainty or error (how shaky the ground is under that mountain).

In the past, scientists trained two separate AIs: one for the mountain and one for the shaky ground. But the researchers realized these two tasks are deeply connected. If you know the mountain is very high, you can guess the ground is more stable. By training them together, they help each other learn, much like how a pianist's left and right hands improve each other when practicing a duet.

The Secret Sauce: Two New Tricks

To make this twin-brain system work perfectly, the researchers added two special ingredients:

1. The "Spotlight" Loss Function (Weighted Loss)

Standard AI training is like a teacher who grades a student's test by giving every question the same weight. But in nuclear fission, the "peak" questions (the jagged mountains) are the most important. If the AI gets those wrong, the whole prediction fails.

The researchers invented a Weighted Loss Function. Imagine a teacher who puts a giant spotlight on the most important questions.

• If the AI gets a "valley" question wrong, the teacher gives a small penalty.
• If the AI gets a "peak" question wrong, the teacher screams, "This is critical! Try again!"
  This forces the AI to obsess over getting the sharp peaks right, rather than just averaging out the whole graph.

2. The "Odd-Even" Clue

Nature has a quirky habit: atoms with an even number of particles are generally more stable and common than those with an odd number. This creates a "jagged" pattern in the data (like a sawtooth wave).

The researchers gave the AI a cheat sheet: a simple Odd-Even switch.

• If the number is even, the switch flips to "1" (High Stability).
• If the number is odd, the switch flips to "0" (Low Stability).
  This helps the AI understand the underlying rhythm of the jagged peaks, allowing it to draw the sawtooth pattern much more accurately.

The Results: A Masterpiece of Prediction

When they tested this new system against older methods (like standard AI or Bayesian Networks):

• Accuracy: It drew the jagged peaks much closer to the real experimental data.
• Confidence: It didn't just guess the height of the mountain; it also gave a very accurate estimate of how much that height might vary (the error).
• Versatility: It worked well not just for known energy levels, but also predicted what would happen at new, untested energy levels by learning the patterns from the data it did have.

Why Does This Matter?

Nuclear energy is the future of clean power, but it requires precise safety calculations.

• Reactor Design: Engineers need to know exactly how much radioactive waste will be produced and what kind.
• Safety: If you don't know the "error" (the uncertainty), you can't design safe containment systems.

This new AI method acts like a super-accurate crystal ball. It tells engineers not only what will happen when a nuclear reactor runs, but also how sure we can be about that prediction. This leads to safer, more efficient nuclear power plants and better waste management strategies.

In short: The researchers taught a computer to stop guessing the shape of a jagged mountain and start mapping it with a spotlight and a rhythm guide, giving us a much clearer picture of the future of nuclear energy.

Technical Summary

Title

Multi-task deep neural network for predicting both nuclear fission yields and their experimental errors in peak-shaped data

1. Problem Statement

Fission Product Yields (FPY) are critical for nuclear energy applications, including reactor design, safety analysis, and fuel cycle management. However, existing nuclear data libraries (e.g., JENDL-5, ENDF) often lack FPY data for specific incident neutron energies, particularly for fast reactors.

• The Challenge: FPY data exhibits a characteristic peak-shaped distribution (double-humped structure) with jagged, non-smooth variations due to the odd–even effect.
• Limitations of Existing Methods:
  • Phenomenological/Semi-microscopic models: Fit existing data well but struggle to predict unobservable FPY at different energy levels.
  • Fully microscopic models (e.g., TDHFB, TDGCM): Computationally prohibitive.
  • Bayesian Neural Networks (BNNs): While excellent for uncertainty quantification, BNNs struggle to accurately reproduce complex, jagged peak structures because they approximate data distributions using Gaussian confidence intervals, which tend to smooth out sharp peaks.
  • Standard Deep Learning: Standard Mean Squared Error (MSE) loss functions often lead to underfitting of the peak regions because the model prioritizes minimizing error across the entire dataset, neglecting the high-value peak data points.

2. Methodology

The authors propose a Multi-task Deep Neural Network (MT-DNN) based on the Multi-gate Mixture-of-Experts (MMoE) architecture to simultaneously predict FPY values and their associated experimental errors.

A. Core Architecture: MMoE

• Structure: The model utilizes a gating mechanism to assign weights to multiple feedforward neural networks (experts).
• Tasks: It performs two simultaneous tasks:
  1. Predicting the FPY value ($\hat{y}_1$).
  2. Predicting the FPY experimental error ($\hat{y}_2$).
• Rationale: FPY values and their errors are strongly correlated (nuclides with higher yields often have lower relative errors). Joint learning exploits this correlation to improve the accuracy of both tasks.

B. Novel Loss Function (Weighted Loss)

To address the underfitting of peak data caused by standard MSE, the authors introduced a Weighted Loss Function specifically for the FPY prediction task:

• Mechanism: The loss assigns higher weights to data points with larger normalized FPY values (the "peaks").
• Definition:
  • If $normalize(y_i) < r$ (non-peak): Loss is scaled by a small factor $\beta$.
  • If $normalize(y_i) \ge r$ (peak): Loss is scaled by $w(y_i) = 1 + normalize(y_i)$.
• Goal: This forces the model to prioritize learning the high-yield peak structures, which are critical for accurate physical representation.

C. Input Features & Odd-Even Effect

• Input Vector: Includes Charge number ($Z$), Neutron number ($N$), Mass number ($A$), and Excitation energy ($E$).
• Odd-Even Effect: An auxiliary binary indicator ($O_i$) is added to the input.
  • $O_i = 1$ if $A$ is even (higher stability/yield).
  • $O_i = 0$ if $A$ is odd.
  • $O_i = 0.5$ for non-peak data to reduce contrast where the effect is less pronounced.
• Data Source: Training data is derived from JENDL-5, which provides evaluated FPY values and their uncertainties (derived via Generalized Least-Squares methods).

D. Data Augmentation

To predict FPY at unmeasured energies, the authors generated supplementary data using a Gaussian model (Katakura model) representing the mass-yield distribution. These data points were added to the training set to bridge gaps in energy coverage.

3. Key Contributions

1. Simultaneous Prediction: First application of an MMoE-based multi-task framework to jointly predict FPY values and their experimental errors, leveraging their intrinsic correlation.
2. Weighted Loss Function: Development of a novel loss function that dynamically weights training data to specifically target and improve the prediction of peak-shaped distributions, overcoming the limitations of standard MSE.
3. Integration of Physical Priors: Successful incorporation of the odd–even effect as an explicit input feature to capture the jagged nature of fission yields.
4. Energy Extrapolation: Demonstration of the model's ability to predict FPY at intermediate and high energies (e.g., 14 MeV) where experimental data is sparse, using Gaussian-model-generated priors.

4. Results

The proposed method was evaluated against Bayesian Neural Networks (BNN) and standard Deep Neural Networks (DNN) using $\chi^2$ error metrics across various actinides ($^{235}\text{U}$, $^{238}\text{U}$, $^{239}\text{Pu}$, $^{241}\text{Pu}$) and energies (Thermal, 0.5 MeV, 14 MeV).

• Peak Prediction Accuracy: The Multi-task DNN with Weighted Loss + Odd-Even effect consistently achieved the lowest $\chi^2$ errors for peak data compared to all other methods. It significantly outperformed BNNs in reproducing the jagged peak structures.
• FPY Error Prediction: The multi-task approach also yielded lower errors for predicting FPY uncertainties compared to single-task DNNs and BNNs, validating the benefit of joint learning.
• Physical Consistency: The model successfully reproduced physical trends, such as:
  • The widening of the double-humped structure at higher energies.
  • The smoothing of fine structures as temperature increases.
  • The shift of fission fragment peaks due to prompt neutron emission.
• Negative Predictions: The proposed method drastically reduced the rate of physically impossible negative FPY predictions (8.29%) compared to BNNs (17.52%).
• Data Augmentation: Using 6 divisions of Gaussian-generated supplementary data (535 points) provided the optimal balance for predicting FPY at unmeasured energies.

5. Significance and Conclusion

This research demonstrates that Multi-task learning combined with physics-informed loss functions is a superior approach for nuclear data prediction compared to traditional statistical models or standard deep learning.

• Practical Impact: The ability to accurately predict FPY and their uncertainties at unmeasured energies is crucial for the design and safety assessment of next-generation fast reactors and for nuclear waste management (reprocessing and geological disposal).
• Methodological Shift: The study highlights that while BNNs are powerful for uncertainty quantification, they may be suboptimal for capturing complex, non-Gaussian physical structures like fission peaks. The proposed deterministic multi-task approach offers a more robust solution for structural prediction while still providing error estimates.
• Future Work: The authors plan to extend validation to a broader range of nuclides and develop simulation-based frameworks to estimate FPY errors for supplementary data where experimental covariance information is currently unavailable.