FNO and UNO Neural Operators for Calculating Two-Dimensional Neutron Flux

Researchers published a paper on arXiv in July 2026 titled "Neural Operator Surrogates for Two-Dimensional Neutron Flux Estimation." It extends neural operators from one-dimensional neutron transport problems to two-dimensional ones and compares three surrogate models built on FNO and UNO architectures.

What the researchers did

The researchers trained neural operators to predict scalar neutron flux in a one-group transport problem with isotropic scattering. The work directly continues their earlier one-dimensional experiments with single-sweep operators, but the surrogates now operate in two dimensions, a step that brings the method closer to real engineering geometries. The reference ("high-fidelity") solution for each case came from a verified discrete ordinates solver, and approximation quality was measured with the mean relative L2 error norm.

A neural operator is a model that learns to map one continuous function (the material and source field) to another (the flux field), rather than simply mapping a number to a number. According to the arXiv abstract, the study uses two types of such architectures: Fourier neural operators (FNO) and U-shaped neural operators (UNO).

Three surrogate models

The paper compares three surrogates that map inputs to neutron flux in different ways. The first two tackle the problem head-on; the third gets a hint from a classical numerical method:

• FNO — maps material and source fields directly to scalar flux
• UNO — the same direct problem, but on a U-shaped neural operator architecture
• FNO with single-sweep input — additionally receives the flux after one source iteration (single-sweep approximation)

Each of the three models was trained with three random seeds. This is a key detail of the methodology: three runs make it possible to tell a real difference between approaches from the statistical scatter that arises between runs on its own. The single-sweep approximation here is a cheap first iteration of the classical solver, fed to the model as an extra feature.

What this means for radiation shielding

The main practical interest of the work is accuracy in heavily attenuated regions, which are critical for radiation shielding calculations. Surrogate models typically make their largest errors exactly where neutron flux drops by many orders of magnitude, and small flux values are what matter for assessing biological shielding. The authors test a hypothesis: does training on the logarithm of the flux, rather than on the flux itself, help recover these regions more accurately?

The second research question is whether the extra single-sweep input gives a real accuracy gain over direct "field → flux" mappings. The authors state both hypotheses explicitly as the two guiding questions of the study.

"Two questions guide the study: does the single-sweep input improve

accuracy compared with direct mappings, and does training on the logarithm of the flux improve accuracy in heavily attenuated regions relevant to shielding," the paper's arXiv abstract states.

What this means

Neural operators are gradually moving into computational physics: if a surrogate reproduces the neutron field accurately enough, the expensive discrete ordinates calculation can be partly replaced by fast model inference. The shift from one-dimensional to two-dimensional problems brings the approach closer to engineering radiation shielding calculations, where full-scale simulations are especially time-consuming and the number of scenarios to sweep is large.