A Filtered MgNet Solver For Radiative Transfer Equations

TL;DR

This paper introduces MgNet, a neural network-based solver for radiative transfer equations that accelerates computations and generalizes well across different medium parameters.

Contribution

It develops a physics-constrained operator learning framework that replaces fixed operators with trainable neural components, improving efficiency and robustness.

Findings

Achieves at least 10x speedup over traditional preconditioners in diffusive regimes.

Demonstrates robust generalization to unseen medium parameters.

Introduces an adaptive angular compression technique to stabilize training.

Abstract

Conventional numerical solvers for the radiative transfer equation (RTE) exhibit severe sensitivity to medium parameters. To address this, we propose an operator learning framework that approximates the RTE solution map as a function of material properties. The core architecture, MgNet, preserves the solution operator framework established by recursive skeleton factorization (RSF) but substitutes its coefficient-specific sub-operators (e.g. smoother, prolongation operator and restriction operator) with learnable neural components. This design transcends the the fixed parametric structure of classical schemes, enabling data-driven sub-operator optimization and learning of their medium-parameter dependence. To mitigate spectral bias in operator learning, we introduce an adaptive angular compression technique within the loss function that dynamically suppresses high-frequency modes responsible for training instability. Comprehensive benchmarks demonstrate that, when deployed as a learned preconditioner, MgNet achieves at least 10 times acceleration over conventional preconditioners in the diffusive regime and maintains robust generalization to unseen parameter configurations. By unifying multilevel factorization structure with deep operator learning, this work establishes a physics-constrained operator-learning paradigm for radiative transport simulations.