# The Machine Learning Approach to Moment Closure Relations for Plasma: A Review

arXiv: https://arxiv.org/abs/2511.22486 · HTML: https://mlsanity.com/paper/2511.22486/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 106 references that are in the map.
- First posted on arXiv 27 Nov 2025.

## Builds on (most cited first, 22 of 22)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [Identification of high order closure terms from fully kinetic simulations using machine learning](https://mlsanity.com/paper/2110.09916/) (2021)
- [Machine learning of hidden variables in multiscale fluid simulation](https://mlsanity.com/paper/2306.10709/) (2023)
- [Learning Physically Interpretable Atmospheric Models from Data with WSINDy](https://mlsanity.com/paper/2501.00738/) (2025)
- [Data-driven, multi-moment fluid modeling of Landau damping](https://mlsanity.com/paper/2209.04726/) (2022)
- [Electron neural closure for turbulent magnetosheath simulations: energy channels](https://mlsanity.com/paper/2510.00282/) (2025)

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

- [Learning Closure of Dynamical Systems with Kernel Ridge Regression](https://mlsanity.com/paper/2610.02564/) (2026)

## Abstract (arXiv)

The requirement for large-scale global simulations of plasma is an ongoing challenge in both space and laboratory plasma physics. Any simulation based on a fluid model inherently requires a closure relation for the high order plasma moments. This review compiles and analyses the recent surge of machine learning approaches developing improved plasma closure models capable of capturing kinetic phenomena within plasma fluid models. We survey two methodological families: neural-network surrogates (from multilayer perceptrons to Fourier neural operators, the latter recently reproducing both linear and non-linear Landau damping online within a fluid solver) and equation-discovery methods such as sparse regression; and organise the studies by whether they are tested offline against reference data or online within a time-evolving solver. We outline the challenges associated with machine-learning closures, including off-diagonal pressure-tensor accuracy, generalisation beyond the training distribution, and stable integration into large-scale simulations, and the directions future research might take to address them.