Uniformly accurate structure-preserving neural surrogates for radiative transfer

TL;DR

This paper introduces a neural surrogate model for the radiative transfer equation that maintains accuracy and physical structure preservation uniformly across different scales, ensuring stability and reliability in multiscale simulations.

Contribution

It develops a multiscale parity decomposition framework for neural surrogates that are asymptotic-preserving and structure-preserving, with rigorous uniform error estimates.

Findings

The neural surrogate remains accurate across all parameter regimes.

The model preserves physical structures like parity, conservation, and positivity.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

In this work, we propose a uniformly accurate, structure-preserving neural surrogate for the radiative transfer equation with periodic boundary conditions based on a multiscale parity decomposition framework. The formulation introduces a refined decomposition of the particle distribution into macroscopic, odd, and higher-order even components, leading to an asymptotic-preserving neural network system that remains stable and accurate across all parameter regimes. By constructing key higher-order correction functions, we establish rigorous uniform error estimates with respect to the scale parameter $\varepsilon$, which ensures $\varepsilon$-independent accuracy. Furthermore, the neural architecture is designed to preserve intrinsic physical structures such as parity symmetry, conservation, and positivity through dedicated architectural constraints. The framework extends naturally from one to two dimensions and provides a theoretical foundation for uniformly accurate neural solvers of multiscale kinetic equations. Numerical experiments confirm the effectiveness of our approach.