# Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure

arXiv: https://arxiv.org/abs/2105.05690 · HTML: https://mlsanity.com/paper/2105.05690/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 62 references that are in the map.
- First posted on arXiv 12 May 2021.

## Builds on (most cited first, 7 of 7)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-equilibrium Flows](https://mlsanity.com/paper/2009.13415/) (2020)
- [Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems](https://mlsanity.com/paper/2007.13977/) (2020)

## Built on by (most cited first, 9 of 9)

- [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)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Hyperbolic Machine Learning Moment Closures for the BGK Equations](https://mlsanity.com/paper/2401.04783/) (2024)
- [Machine learning moment closure models for the radiative transfer equation IV: enforcing symmetrizable hyperbolicity in two dimensions](https://mlsanity.com/paper/2604.20143/) (2026)
- [Structure-preserving neural networks for the regularized entropy-based closure of the Boltzmann moment system](https://mlsanity.com/paper/2404.14312/) (2024)
- [A Hyperbolic Neural Closure for M1 Radiation Transfer](https://mlsanity.com/paper/2607.10364/) (2026)
- [Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs](https://mlsanity.com/paper/2608.18148/) (2026)
- [Learning Closure of Dynamical Systems with Kernel Ridge Regression](https://mlsanity.com/paper/2610.02564/) (2026)

## Abstract (arXiv)

In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. A variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.