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Uniformly accurate structure-preserving neural surrogates for radiative transfer

Mengjia Bai, Jingrun Chen, Keke Wu

TL;DR

This work proposes a uniformly accurate, structure-preserving neural surrogate for the radiative transfer equation with periodic boundary conditions based on a multiscale parity decomposition framework, leading to an asymptotic-preserving neural network system that remains stable and accurate across all parameter regimes.

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.

Uniformly accurate structure-preserving neural surrogates for radiative transfer

TL;DR

This work proposes a uniformly accurate, structure-preserving neural surrogate for the radiative transfer equation with periodic boundary conditions based on a multiscale parity decomposition framework, leading to an asymptotic-preserving neural network system that remains stable and accurate across all parameter regimes.

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 , which ensures -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.

Paper Structure

This paper contains 19 sections, 8 theorems, 118 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. One-dimensional method
  5. Multiscale parity decomposition method
  6. Loss function
  7. Convergence of the loss function
  8. Uniform error analysis
  9. Two-dimensional method
  10. Multiscale parity decomposition method
  11. Loss function
  12. Uniform error method
  13. Structure-preserving mechanism
  14. One-dimensional parity symmetry, conservation and positivity
  15. Two-dimensional parity symmetry, conservation and positivity
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $\mathcal{R}_{\mathrm{APNN}}^\varepsilon$ represent the physics-informed loss based on the residual of the multiscale parity-decomposition system, and let $\mathcal{R}_{\mathrm{total}}^\varepsilon$ denote the error between the numerical solution and the exact solution. Then it follows that

Figures (8)

  • Figure 1: Schematic diagram of the asymptotic-preserving neural networks (adapted from jin2023asymptotic). Assume $\mathcal{F^{\varepsilon}}$ is the multi-scale model that depends on the scale parameter $\varepsilon$ and $\mathcal{F}^{0}$ is the corresponding asymptotic limit model as $\varepsilon \to 0$. Define $\mathcal{R}(\mathcal{F^{\varepsilon}})$ as the neural network-based least-squares formulation of the model $\mathcal{F^{\varepsilon}}$. If $\mathcal{R}(\mathcal{F^{\varepsilon}})$ converges to $\mathcal{R}(\mathcal{F}^{0})$ as $\varepsilon \to 0$, and this limit is precisely the least-squares formulation of the limit model $\mathcal{F}^{0}$, then the method is called asymptotic-preserving.
  • Figure 1: Training loss and error evolution for the one-dimensional case ($\varepsilon = 1$).
  • Figure 2: Illustration of uniform convergence of the AP framework.
  • Figure 2: Comparison between reference and neural network solutions at $t = 0.05$ and $t = 0.1$ ($\varepsilon = 1$).
  • Figure 3: Training loss and error evolution for the one-dimensional case ($\varepsilon = 10^{-4}$).
  • ...and 3 more figures

Theorems & Definitions (14)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1: hu2023higher Lemma A.1
  • Theorem 2.2
  • Proof 1
  • Theorem 2.3
  • Proof 2
  • Theorem 2.4
  • Proof 3
  • Theorem 3.1
  • ...and 4 more