# Machine learning moment closure models for the radiative transfer equation IV: enforcing symmetrizable hyperbolicity in two dimensions

arXiv: https://arxiv.org/abs/2604.20143 · HTML: https://mlsanity.com/paper/2604.20143/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 22 Apr 2026.

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

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure](https://mlsanity.com/paper/2105.05690/) (2021)
- [Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-equilibrium Flows](https://mlsanity.com/paper/2009.13415/) (2020)
- [Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures](https://mlsanity.com/paper/2105.14410/) (2021)
- [Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds](https://mlsanity.com/paper/2109.00700/) (2021)
- [Hyperbolic Machine Learning Moment Closures for the BGK Equations](https://mlsanity.com/paper/2401.04783/) (2024)
- [Data-driven construction of a generalized kinetic collision operator from molecular dynamics](https://mlsanity.com/paper/2503.24208/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Hyperbolic Neural Closure for M1 Radiation Transfer](https://mlsanity.com/paper/2607.10364/) (2026)

## Abstract (arXiv)

This is our fourth work in the series on machine learning (ML) moment closure models for the radiative transfer equation (RTE). In the first three papers of this series, we considered the RTE in slab geometry in 1D1V (i.e. one dimension in physical space and one dimension in angular space), and introduced a gradient-based ML moment closure [1], then enforced the hyperbolicity through a symmetrizer [2], or together with physical characteristic speeds by learning the eigenvalues of the Jacobian matrix [3]. Here, we extend our framework to the RTE in 2D2V (i.e. two dimensions in physical space and two dimensions in angular space). The main idea is to preserve the leading part of the classical PN model and modify only the highest-order block row. By analyzing the structural properties of the PN model, we show that its coefficient matrices are symmetric and admit a block-tridiagonal structure. Then we use this property to introduce a block-diagonal symmetrizer for the ML moment model and derive explicit algebraic conditions on the closure blocks which guarantee the symmetrizable hyperbolicity of the resulting ML system. These conditions lead to a natural parametrization of the closure in terms of a symmetric positive definite matrix together with symmetric closure blocks, which can be learned from data while automatically enforcing symmetrizable hyperbolicity by construction. The numerical results show that the proposed framework improves upon the classical PN model while maintaining hyperbolicity.