Нейросетевые операторы FNO и UNO для расчёта двумерного потока нейтронов

Researchers published a paper on arXiv in July 2026, "Neural Operator Surrogates for Two-Dimensional Neutron Flux Estimation," in which they extended the neural operator method from one-dimensional neutron transport problems to two dimensions and compared three surrogate models based on the 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 now the surrogates operate in two dimensions — a step that brings the method closer to real engineering geometries. The reference ("high-fidelity") solution for each case was computed with a validated discrete ordinates solver, and the quality of the approximation was assessed using 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 numbers to numbers. According to the arXiv abstract, the study involves 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 input data to neutron flux in different ways. The first two tackle the problem head-on, while the third gets a hint from a classical numerical method:

• FNO — maps the material and source fields directly to the 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 on three random initializations (random seeds). This is a key methodological detail: three runs make it possible to distinguish a genuine difference between approaches from the statistical spread that arises from run to run on its own. The single-sweep approximation here is a cheap first iteration of the classical solver, fed to the model as an additional feature.

What this means for radiation shielding

The main practical interest of the work is accuracy in strongly attenuated regions, which are critical for radiation shielding calculations. It is precisely where neutron flux drops by many orders of magnitude that surrogate models usually err the most, and it is exactly the small flux values that matter for assessing biological shielding. The authors test the hypothesis: does training on the logarithm of the flux, rather than on the flux itself, help reconstruct these zones more accurately?

The second research question is whether the additional single-sweep input provides a genuine gain in accuracy compared to direct "field → flux" mappings. The authors state both hypotheses explicitly as the two guiding questions of the research.

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

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

What it means

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