Learning nuclear cross sections across the chart of nuclides with graph neural networks

TL;DR

This work introduces a two-stage deep-learning framework to learn nuclear cross sections across the chart of nuclides by combining representation learning (via a variational autoencoder or an implicit neural representation with a hypernetwork) and graph neural networks that exploit the nuclear-chart topology for imputation. The approach demonstrates that cross sections in the fast neutron regime can be encoded into a compact latent space, from which missing values are accurately predicted within a 9×9 region, with VAEs performing best when trained end-to-end and INRs excelling when predictions stay in latent space. Latent-space analyses reveal organized structure with diagonal bands and hints of neutron-magic numbers, supporting the potential to extract covariances and cross-material correlations from data-driven representations. The study suggests a promising path toward unified, data-driven nuclear data libraries and covariance estimation, potentially enhanced by incorporating additional datasets and microscopic theory insights, though it remains contingent on further validation beyond the synthetic TENDL dataset.

Abstract

In this work, we explore the use of deep learning techniques to learn how nuclear cross sections change as we add or remove protons and neutrons. As a proof of principle, we focus on the neutron-induced reactions in the fast energy regime. Our approach follows a two-stage learning framework. First, we apply representation learning to encode cross section data into a latent space using either variational autoencoders (VAEs) or implicit neural representations (INRs). Then, we train graph neural networks (GNNs) on the resulting embeddings to predict missing values across the nuclear chart by leveraging the topological structure of neighboring isotopes. We demonstrate accurate cross section predictions within a 9x9 block of missing nuclei. We also find that the optimal GNN training strategy depends on the type of latent representation used, with VAE embeddings performing best under end-to-end optimization in the original space, while INR embeddings achieve better results when the GNN is trained only in the latent space. Furthermore, using clustering algorithms, we map groups of latent vectors into regions of the nuclear chart and show that VAEs and INRs can discover some of the neutron magic numbers. These findings suggest that deep-learning models based on the representation encoding of cross sections combined with graph neural networks holds significant potential in augmenting nuclear theory models, e.g., by providing reliable estimates of covariances of cross sections, including cross-material covariances.

Figures (14)

• Figure 1: Schematic representation of our framework. The initial data is a (set of) nuclear cross sections represented as a vector indexed by the energy of the incident particle. Panel (a): The data is encoded into a latent representation using either a variational autoencoder (VAE) or an implicit neural representation (INR) network (representation learning). Panel (b): A graph neural network (GNN) then learns how the data in the latent space in nucleus $(Z,N)$ transforms into neighboring nuclei in order to make predictions in nucleus $(Z',N')$. The decoder then transforms the predicted vector in the latent space back into the space of nuclear cross sections (latent space predictions).
• Figure 2: Panel (a): Variational Autoencoder framework illustrating the encoding and decoding process with a latent space bottleneck. Panel (b): Implicit Neural Representation (INR) architecture integrated with a hypernetwork (HyperNet). The hypernetwork generates weights for the INR, informed by a convolutional encoder that processes input data. Note that the INR takes input coordinates, $\{e_{i}\}$ (representing the energy index with a length of $m=256$) and maps them to the corresponding cross sections.
• Figure 3: Comparison of the encoding performance between VAE (top) and INR (bottom) for all four cross sections in the $^{138}$Pm nucleus. Ground-truth results are shown in plain lines and full symbols while reconstructions are in dashed lines and open symbols. Performance is typical of all nuclei in the TENDL dataset.
• Figure 4: Mean prediction errors calculated using Eq. \ref{['eq:test_err']} of the GNN for each nucleus in the TENDL dataset. For better visualization, all plots are normalized between $\epsilon_{\rm min}$ and $0.03\times\epsilon_{\rm max}$, where $\epsilon_{\rm min}$ ($\epsilon_{\rm max}$) is the minimum (maximum) error over all plots. The three panels on the left side are based on VAE encoding while the three on the right on INR encoding. ARMAConv, GatedConv and GraphSAGE refer to the three types of GNNs considered in this work. The red square corresponds to the windowed test data: the GNN was not trained on it.
• Figure 5: Performance of the VAE (left) and INR (right) model for the GraphSAGE GNN architecture in predicting nuclear cross sections within the test set for the central nucleus, $^{102}$Mo and the ones near the lower left and upper right of the window, $^{95}$Y and $^{108}$Ru respectively. Plain curves without symbols correspond to the ground truth while dashed curves with open symbols correspond to GNN predictions. All cross sections are normalized independently in each channel across the entire dataset. The small grid in the upper right corner of the INR figures shows the location of each nucleus within the test set.