⚛️ nuclear theory

Extracting Barrier Distributions from Fusion Cross Sections

This paper demonstrates that Bayesian neural networks outperform Gaussian processes in extracting fusion barrier distributions from experimental cross-section data by providing more faithful reconstructions with quantified uncertainties, while also identifying key regions for future experimental impact and highlighting that the fidelity of all methods depends primarily on the magnitude of experimental uncertainties.

Original authors: Aaron Philip

Published 2026-07-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Aaron Philip

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

The Big Picture: Smashing Atoms to See Inside

Imagine you are trying to figure out what a locked box looks like on the inside, but you can't open it. The only way to learn about it is to throw other objects at it and see how they bounce off or stick to it.

In nuclear physics, scientists do this by smashing heavy atomic nuclei together. Sometimes they fuse (stick together), and sometimes they bounce apart. The "fusion cross section" is just a fancy way of measuring how often they stick together at different speeds (energies).

For a long time, scientists thought these collisions were like rolling a ball over a single, smooth hill. If the ball has enough energy to get over the hill, it rolls down the other side. But real atoms are messy. They have internal parts that wiggle and vibrate. This means the "hill" isn't a single smooth line; it's actually a messy, bumpy landscape with many different heights and shapes.

The goal of this thesis is to take the messy data from these collisions and reconstruct what that "bumpy landscape" (called the Barrier Distribution) actually looks like.

The Problem: The "Blurry Photo" Effect

To see the shape of this landscape, scientists have to do a specific math trick: they take the data and calculate the second derivative.

Think of it like this:

The Data: A slightly blurry photo of a mountain range.
The Math Trick: Trying to draw the exact outline of every peak and valley by looking at the photo.

The problem is that this math trick is extremely sensitive to "noise" (imperfections in the photo). If the data has even a tiny bit of error, the math gets confused and starts drawing fake mountains and valleys that don't exist. It's like trying to trace a jagged line on a shaky hand; the result looks like a scribble rather than a mountain.

The Old Solution: The "Smooth Painter" (Gaussian Processes)

In a previous study, a scientist named Godbey tried to fix this by using a tool called Gaussian Processes (GP).

The Analogy: Imagine you have a very flexible, stretchy rubber sheet. You pin it down at the points where you have data. The sheet naturally smooths out between the pins. You can then trace the shape of the sheet to see the landscape.

The Thesis Findings:

What worked: When the data was very clean and precise, this rubber sheet worked great. It gave a smooth, accurate picture of the main hill.
What failed: When the data was a bit noisy (which happens a lot in real experiments), the rubber sheet started to wiggle wildly at the edges. It created fake peaks (ghost mountains) that weren't real. This is called "aliasing." It's like when a spinning wheel in a movie looks like it's spinning backward; the math trick creates an illusion.

The New Solution: The "Smart Detective" (Bayesian Neural Networks)

The author of this thesis, Aaron Philip, decided to try a newer, more powerful tool called Bayesian Neural Networks (BNNs), specifically using a software tool called AutoBNN.

The Analogy: Instead of one rubber sheet, imagine hiring a team of detectives.

Each detective looks at the same blurry photo.
They all try to draw the landscape, but they are allowed to have different opinions.
Because they are "Bayesian," they don't just give you one drawing; they give you a drawing plus a confidence level. They can say, "I'm 90% sure there's a hill here, but I'm only 50% sure about that weird bump over there."
The "AutoBNN" tool is like a manager who automatically tries out different styles of drawing (different "architectures") to see which detective team works best.

The Thesis Findings:

Better Vision: The detective team (specifically a type called "Changepoint" models) was much better at ignoring the noise. They didn't get tricked into drawing fake mountains.
Honest Uncertainty: When the data was bad or missing, the detectives were honest. They drew a wide, fuzzy area and said, "We don't know what's here yet." The old rubber sheet (GP) would often draw a sharp, confident line even when it was wrong.
Logarithmic Space: The author found that doing the math in a "logarithmic" way (like looking at a map where distances are compressed) helped the detectives see the shape much more clearly, especially when the data was messy.

The Results: What Did We Learn?

The author tested these new methods on real data from past experiments:

The "Ghost" Peaks: In some experiments, the old method (GP) claimed there was a second, smaller hill in the landscape. The new method (BNN) said, "Actually, we aren't sure. The data is too noisy to tell if that hill is real or just a trick of the light."
Where to Look Next: Because the new method knows exactly where it is confused, it can tell scientists: "Hey, if you want to know the truth, go measure the collision at this specific speed." It acts like a GPS for future experiments.
No Data? No Problem: The author also figured out how to use these tools even when scientists didn't write down how "noisy" their measurements were. The new method can guess the noise level and still give a good answer.

The Bottom Line

This thesis is about building better tools to clean up noisy scientific data.

Old Tool: Good for clean data, but creates fake details when data is messy.
New Tool: A team of smart, probabilistic detectives that can handle messy data, admit when they are unsure, and tell you exactly where you need to look next to get the truth.

The author has released the code for these new tools so other scientists can use them to stop guessing and start seeing the true shape of the atomic world.

Technical Summary: Extracting Barrier Distributions from Fusion Cross Sections

Problem Statement
The fusion of heavy ions is a complex process where the interaction between nuclei cannot be adequately described by a simple one-dimensional tunneling model through a single barrier. Instead, couplings to collective states and other channels induce a continuous "distribution of barriers," $D(E)$ . Extracting this distribution from experimental fusion cross-section data ( $\sigma(E)$ ) is theoretically possible via the second derivative of the energy-weighted cross section, $D(E) \propto \frac{d^2(E\sigma)}{dE^2}$ . However, in practice, this extraction is numerically unstable. Standard point-difference schemes are hypersensitive to experimental noise, data resolution, and irregularity, often yielding unreliable results, particularly at energies above the primary barrier. While recent work (Godbey 2024) utilized Gaussian Processes (GPs) to smooth data and provide uncertainty estimates, limitations remain regarding the fidelity of the extracted distribution at higher energies and the reliability of uncertainty quantification.

Methodology
This thesis employs a multi-stage approach to benchmark and improve the extraction of barrier distributions:

Benchmarking with Toy Models: The study utilizes the Wong formula (Wong 1973), an analytic model for fusion cross sections, to generate ground-truth data. This data is corrupted with realistic noise levels ( $\eta = \delta\sigma/\sigma$ ), varying resolutions ( $\Delta E$ ), and irregular spacing to simulate experimental conditions.
Gaussian Process Regression (GPR): The author extends the method from Godbey (2024) by training GPs with Radial Basis Function (RBF) kernels. A critical methodological shift involves training in logarithmic space (modeling multiplicative errors) rather than linear space to stabilize the covariance matrix and better reflect the statistical nature of fusion data. The study also introduces a technique to infer unknown relative uncertainties by optimizing a global variance parameter via marginal likelihood maximization when experimental errors are not quoted.
Bayesian Neural Networks (BNNs): To overcome the limitations of GPs, the thesis implements AutoBNN (Carroll et al. 2024), a tool for training interpretable Bayesian Neural Networks. These networks are trained using Markov Chain Monte Carlo (MCMC) via the No-U-Turn Sampler (NUTS) algorithm. The study explores specific architectures, including "Sum of Changepoints" and "Changepoint of Products," which allow the model to adapt its statistical behavior across different energy regions.
Regularization: To address numerical instabilities in the second derivative (specifically at changepoint locations), a prior penalizing the magnitude of the third derivative is implemented during BNN training.

Key Results

Gaussian Process Limitations:
- Aliasing: GPs exhibit "aliasing" (false oscillatory peaks) at energies higher than the primary barrier. This is attributed to the ill-conditioning of the covariance matrix caused by large cancellations in the linear combination of kernel functions.
- Noise Sensitivity: While GPs are robust to data irregularity and sparsity if the noise level is low ( $\eta \leq 10^{-2}$ ), they fail to capture the barrier location and structure when noise levels are high ( $\eta \approx 10^{-1}$ ).
- Uncertainty Underestimation: GP-quoted uncertainties are often underestimated, potentially leading to overconfident conclusions about nuclear structure.
- Log-Space Training: Training in log-space improves stability and primary peak recovery but does not eliminate high-energy aliasing.
Bayesian Neural Network Performance:
- Superior High-Energy Recovery: Specific AutoBNN architectures, particularly those with "Changepoint" capabilities, successfully recover the barrier distribution across both sub-barrier and above-barrier regions without the spurious aliasing observed in GPs.
- Uncertainty Quantification: BNNs provide more reliable and expansive uncertainty bands, correctly identifying regions where the data is insufficient to constrain the model (e.g., distinguishing between a plateau and a secondary peak).
- Log-Space Necessity: Similar to GPs, training BNNs in logarithmic space is essential for effective regularization and performance on cross-section data.
Application to Experimental Data:
- The methods were applied to real datasets (e.g., $^{16}\text{O} + ^{208}\text{Pb}$ , $^{40}\text{Ca} + ^{40}\text{Ca}$ ).
- In cases with quoted uncertainties, the BNNs identified tertiary rises and secondary structures that discrete 3-point formulas missed.
- In cases without quoted uncertainties, the inferred relative uncertainty method allowed for the extraction of barrier distributions, though the BNNs correctly flagged high uncertainty in regions where models disagreed (e.g., around 55 MeV in $^{48}\text{Ca} + ^{48}\text{Ca}$ ), suggesting a need for further experimental measurements.

Significance and Contributions
The thesis contributes to the "theory-experiment cycle" in nuclear physics by providing robust, statistically sound tools for extracting barrier distributions.

Methodological Advancement: It demonstrates that while GPs are useful, they suffer from specific numerical artifacts (aliasing) at high energies. The introduction of AutoBNN architectures offers a more faithful recovery of the barrier distribution with well-quantified uncertainties.
Experimental Guidance: By leveraging the continuous posterior and ensemble disagreement of models, the work identifies specific energy ranges where additional experimental data would be maximally impactful, moving beyond simple data fitting to active experimental design.
Open-Source Implementation: The author releases a user-friendly, open-source implementation of the analysis pipeline, including the modified AutoBNN code and scripts for handling irregular experimental data, to encourage adoption by the experimental community.
Handling Missing Data: The thesis provides a validated strategy for analyzing experimental datasets where measurement uncertainties are not explicitly quoted, a common scenario in nuclear physics literature.

Limitations and Future Work
The author notes that the choice of the third-derivative penalty strength ( $\lambda$ ) requires expert tuning and that the "Sum of Changepoint" and "Changepoint of Products" models can yield conflicting predictions without a clear metric for reliability. Furthermore, while MCMC training yields superior results, it is computationally expensive. The thesis proposes future work to translate the finite-width AutoBNN models back into infinite-width Gaussian Process kernels to accelerate training while retaining the learned structural insights, though this translation faces challenges regarding the preservation of multi-modality in the function distribution.