# Neural Operator Surrogates for Two-Dimensional Neutron Flux Estimation

arXiv: https://arxiv.org/abs/2607.19388 · HTML: https://mlsanity.com/paper/2607.19388/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 2 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Solver Recommendation For Transport Problems in Slabs Using Machine Learning](https://mlsanity.com/paper/1906.08259/) (2019)
- [Evolutionary Discovery of Sequence Acceleration Methods for Slab Geometry Neutron Transport](https://mlsanity.com/paper/2512.24559/) (2025)
- [Surrogate Modeling for Neutron Transport: A Neural Operator Approach](https://mlsanity.com/paper/2602.15890/) (2026)

## Abstract (arXiv)

This work extends our one-dimensional single-sweep neural-operator studies to two dimensions. We consider one-group transport with isotropic scattering. As in the one-dimensional work, we use Fourier neural operators (FNOs) to approximate the high-fidelity scalar flux. Additionally, we also investigate U-shaped neural operators (UNOs) in this study. We consider three surrogates. The first two map the material and source fields directly to the flux, one using an FNO and one using a UNO. The third is an FNO that additionally takes the scalar flux after one source iteration, the single-sweep approximation, as an input. Each case is solved to high fidelity with a verified discrete-ordinates solver, and an average relative L_2 error norm is used to characterize the quality of the inferred maps. We train every surrogate over three random seeds so that differences between them can be assessed against run-to-run variability. Two questions guide the study: whether the single-sweep input improves accuracy over the direct maps, and whether training on the logarithm of the flux improves accuracy in the strongly attenuated regions relevant to shielding.