⚛️ nuclear theory

Bayesian Inference for Extracting Barrier Distributions from Fusion Excitation Functions

This paper introduces a Bayesian inference framework using AutoBNN to robustly extract barrier distributions with well-calibrated uncertainties from sparse fusion excitation function data, outperforming Gaussian process regression in benchmarks and successfully mitigating spurious structures in experimental heavy-ion fusion reactions.

Original authors: Aaron Philip, Pablo Giuliani, Kyle Godbey

Published 2026-07-17

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Aaron Philip, Pablo Giuliani, Kyle Godbey

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 the atomic nucleus not as a solid marble, but as a squishy, vibrating balloon of protons and neutrons. When two of these balloons crash into each other at high speeds, they sometimes stick together in a process called nuclear fusion. This isn't just a cool party trick; it's the engine that powers the stars and the key to unlocking clean energy on Earth. But to make them stick, you have to push them hard enough to overcome a giant invisible wall of repulsion called the "Coulomb barrier."

Here's the tricky part: because these nuclear balloons are wiggly and complex, they don't just face one single wall. Instead, the quantum mechanics of the collision create a whole "distribution" of barriers—some lower, some higher, some bumpy, some smooth. Scientists call this the "barrier distribution." It's like a fingerprint of the nucleus's internal structure. If you can read this fingerprint, you learn how the nucleus is built and how it behaves. The problem is, the data we get from experiments is messy, like a blurry photo of a fast-moving car. To see the fingerprint clearly, you have to do some intense math to find the "second derivative" of the data, which is a fancy way of saying "how fast the rate of change is changing." Unfortunately, doing this math on messy, sparse data often creates fake patterns and wild guesses, making the fingerprint look like static on a TV screen.

This paper introduces a new, smarter way to clean up that static. The authors, Aaron Philip, Pablo Giuliani, and Kyle Godbey, treated the problem like a detective story: instead of just crunching numbers, they used a Bayesian machine learning tool called AutoBNN. Think of this tool as a super-smart, cautious artist who doesn't just draw what it sees, but also draws a "confidence zone" around its sketch to show where it's guessing and where it's sure. They tested this new artist against an old, popular method (Gaussian processes) and a standard math trick (the 3-point difference formula) using thousands of simulated experiments.

The results were clear: the old methods often got it wrong, creating fake bumps and peaks in the barrier distribution that weren't really there, and they were way too confident about their mistakes. The AutoBNN artist, however, faithfully recovered the true "fingerprint" in the simulations, even when the data was noisy or sparse, and it gave honest, well-calibrated estimates of its own uncertainty. When the team applied this new method to real-world heavy-ion fusion experiments, it smoothed out the weird, spurious structures that other methods had reported above the energy barrier. Essentially, they built a better lens to see the hidden structure of the atomic nucleus, helping scientists trust their data more and understand the secrets of how stars burn and how we might one day harness fusion energy.

Technical Summary: Bayesian Inference for Extracting Barrier Distributions from Fusion Excitation Functions

Problem Statement
Barrier distributions ( $D_f(E)$ ) provide critical insights into the structure and dynamics of fusing nuclei, encoding information about channel couplings, deformations, and transfer processes. These distributions are theoretically linked to experimental fusion excitation functions, $\sigma(E)$ , via the second derivative relation $D_f(E) = \frac{d^2}{dE^2}(E\sigma)$ . However, extracting $D_f(E)$ from sparse, noisy experimental data is numerically unstable. Traditional methods, such as the 3-point difference formula, amplify measurement noise and are limited to discrete energy points. While Gaussian Process (GP) regression offers a continuous alternative, previous studies indicate that standard GP approaches (specifically those using Radial Basis Function kernels) can struggle to faithfully recover the underlying barrier distribution under realistic experimental conditions, often introducing spurious oscillations and providing poorly calibrated uncertainty estimates.

Methodology
The authors frame the extraction of barrier distributions as a Bayesian inference problem. They propose a novel approach utilizing AutoBNN, an interpretable Bayesian machine learning framework originally developed for time series forecasting.

Bayesian Framework: The problem is formulated to find the posterior distribution $p(\sigma|D_\sigma)$ of the excitation function given experimental data $D_\sigma = (E, y, \delta y)$ . To handle the wide dynamic range of cross-sections, the authors work in logarithmic space, defining $g(E) = \log(\sigma(E)/\bar{y})$ and assuming an additive error model in this transformed space.
AutoBNN Implementation: Unlike standard GPs, AutoBNN employs Bayesian Neural Networks (BNNs) as surrogates for Gaussian processes. The authors utilize the "Sum of Changepoint" architecture, which concatenates piecewise kernel combinations. This allows the model to adapt to different statistical structures across various energy ranges.
Regularization: To prevent unphysical discontinuities in the derived barrier distribution, a modified prior is introduced that penalizes large values of the third derivative of the latent function $g_\theta(E)$ .
Differentiation: A key technical advantage is the use of forward-mode automatic differentiation (via JAX) to compute the second derivative required for $D_f(E)$ , avoiding the finite difference errors inherent in discrete methods.
Benchmarking: The method is benchmarked against standard GP regression (using RBF kernels) and the 3-point difference formula. Simulated data is generated using the Wong formula across 16 distinct datasets varying in energy spacing irregularity ( $\kappa$ ), average resolution ( $\Delta E$ ), and measurement noise ( $\eta$ ).

Key Results

Recovery Accuracy: In all 16 simulated scenarios, AutoBNN models outperformed GP models in recovering the true underlying barrier distribution. While GPs occasionally recovered peak height and width more accurately, AutoBNN excelled at capturing the peak location and the overall shape, particularly in datasets with high irregularity, low resolution, or high noise.
Uncertainty Calibration: Empirical coverage plots demonstrated that AutoBNN provides well-calibrated uncertainty estimates, with coverage rates closely matching nominal credible intervals. In contrast, GP models exhibited overconfident predictions, with coverage rates significantly below the diagonal, indicating their uncertainty estimates were unreliable.
Application to Experimental Data: The authors applied the AutoBNN framework to four heavy-ion fusion reactions ( $^{16}\text{O} + ^{208}\text{Pb}$ $^{16} O +^{208} Pb$ , $^{40}\text{Ca} + ^{40}\text{Ca}$ $^{40} Ca +^{40} Ca$ , $^{48}\text{Ca} + ^{40}\text{Ca}$ $^{48} Ca +^{40} Ca$ , and $^{48}\text{Ca} + ^{48}\text{Ca}$ $^{48} Ca +^{48} Ca$ ).
- For $^{16}\text{O} + ^{208}\text{Pb}$ , AutoBNN provided greater insight into above-barrier structures while constraining uncertainties reported by the 3-point formula.
- For $^{40}\text{Ca} + ^{40}\text{Ca}$ , the model supported the presence of a secondary bump but indicated that the primary peak's location was not rigidly constrained by the data.
- For the Calcium-48 reactions, AutoBNN predictions diverged from the 3-point formula above the barrier, suggesting a smoother secondary plateau rather than the sharp secondary rise predicted by the difference formula, effectively mitigating spurious above-barrier structures.

Significance and Claims
The paper claims that the AutoBNN-based approach offers a robust, model-agnostic statistical method for analyzing fusion excitation functions. Its primary significance lies in its ability to:

Faithfully recover underlying barrier distributions from sparse and noisy data where traditional finite difference and standard GP methods fail.
Provide well-calibrated uncertainty estimates, which are crucial for interpreting nuclear structure and testing theoretical models (e.g., Coupled-channels or Time-Dependent Density Functional Theory).
Mitigate the introduction of spurious structures in the above-barrier region, a common artifact in previous extraction methods.

The authors emphasize that they have developed a user-friendly software implementation to facilitate the application of this method to future heavy and light-ion fusion experiments, aiming to improve the reliability of barrier distribution extraction for the nuclear science community.

Technical Summary: Bayesian Inference for Extracting Barrier Distributions from Fusion Excitation Functions

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