Surrogate Modeling for Neutron Transport: A Neural Operator Approach

Technical Deep Dive: Surrogate Modeling for Neutron Transport

Overview

This work introduces neural operator based surrogate modeling frameworks for neutron transport computation. Two architectures, the Deep Operator Network (DeepONet) and the Fourier Neural Operator (FNO), were trained to learn the mapping from anisotropic neutron sources to the corresponding angular fluxes in a one-dimensional slab geometry. The models were evaluated on a wide range of unseen source configurations and demonstrated significant speedups compared to traditional transport solvers, while maintaining high predictive accuracy. The neural operator surrogates were further integrated into k-eigenvalue neutron transport solvers, enabling rapid and reliable evaluations.

Methodology

Fixed Source Problem

The steady-state one-group neutron transport equation (NTE) was considered in a 1D slab geometry, with a prescribed internal source term Q(x, μ) and boundary conditions. The goal was to develop neural operator surrogates to efficiently map Q(x, μ) to the resulting angular flux ψ(x, μ).

The DeepONet architecture encodes the source term into a latent vector using a BranchNet, which is then combined with spatial and angular coordinates via a TrunkNet to predict the angular flux. The FNO approach lifts the source term into a higher-dimensional latent space, which is then propagated through a sequence of Fourier layers to obtain the angular flux prediction.

Eigenvalue Problem

The neural operator surrogates were also integrated into the eigenvalue formulation of the NTE, replacing the computationally intensive inner transport sweep loop. This approach enables rapid and reliable evaluations of the k-eigenvalue and associated scalar flux distributions.

Results

Fixed Source Problem

• Both DeepONet and FNO models demonstrated high accuracy in predicting the angular flux for a wide range of unseen anisotropic source configurations, with FNO generally achieving lower prediction errors.
• In terms of computational efficiency, the surrogate models required less than 0.3% of the runtime of a traditional S₂ solver.
• The models maintained good performance across different scattering regimes, from absorption-dominated to purely scattering transport.

Eigenvalue Problem

• When integrated into the k-eigenvalue solver, the neural operator models reproduced reference eigenvalues with deviations up to 135 pcm for DeepONet and 112 pcm for FNO.
• The computational cost of the eigenvalue calculations was reduced to less than 0.1% of the traditional S₂ solver on fine spatial and angular grids.
• Both the DeepONet and FNO based solvers achieved scalar flux errors of less than 1% compared to the reference S₂ solution.

Interpretation

The results demonstrate the strong potential of neural operator frameworks as accurate, efficient, and generalizable surrogates for neutron transport problems. By directly learning the underlying transport solution operator, the models can provide rapid and reliable evaluations, enabling their use in applications requiring repeated analyses, such as design optimization, uncertainty quantification, and digital twin frameworks.

Limitations and Uncertainties

• The study was limited to a 1D slab geometry. Future work should extend the methodology to higher-dimensional geometries.
• The models were trained and evaluated on fixed macroscopic cross sections. Generalizing the approach to handle a broader range of cross section values remains an area for further investigation.
• The performance of the neural operator surrogates was assessed using mean squared error and mean relative error metrics. Additional validation against other reference quantities (e.g., reaction rates, reaction rate ratios) would strengthen the findings.

Future Work

• Extend the neural operator methodology to higher-dimensional neutron transport problems, including 2D and 3D geometries.
• Develop approaches for learning transport solution operators that can generalize across a broad range of macroscopic cross section values, enabling more realistic reactor physics applications.
• Investigate the integration of the fast neural operator surrogates into reduced-order modeling frameworks, such as proper generalized decomposition (PGD), to further accelerate multiphysics simulations.