A comprehensive review published on arXiv details the growing application of machine learning (ML) techniques to address the critical need for accurate closure relations in fluid plasma simulations. These simulations are essential for modeling both astrophysical and laboratory plasmas, but fluid models require closure relations to account for higher-order plasma moments, which are often computationally prohibitive to resolve kinetically. The review categorizes ML approaches into two main families: neural network surrogates and equation discovery methods, highlighting their diverse architectures and applications.

Neural network surrogates, ranging from multilayer perceptrons to Fourier Neural Operators (FNOs), are being trained to approximate complex closure terms. FNOs, in particular, have demonstrated the ability to reproduce linear and non-linear Landau damping in real-time within fluid solvers, a significant step towards integrating kinetic effects into macroscopic simulations. Equation discovery methods, such as sparse regression, are also employed to identify governing equations for closure terms directly from data, offering a data-driven path to improved theoretical models.

Neural network surrogates, ranging from multilayer perceptrons to Fourier Neural Operators (FNOs), are being trained to approximate complex closure terms.

The research surveyed examines whether these ML-based closures are tested offline against existing datasets or integrated online into time-evolving fluid solvers. Offline validation provides a benchmark for accuracy, while online integration is crucial for their practical utility in large-scale simulations. The review emphasizes the challenges inherent in this approach, including ensuring the accuracy of off-diagonal pressure tensor components and achieving robust generalization beyond the specific training data distributions.

Despite these challenges, the integration of ML into plasma closure modeling holds substantial promise for advancing fusion energy research. More accurate fluid simulations could reduce the computational burden of designing and optimizing fusion devices like tokamaks and stellarators, potentially accelerating the path to net energy gain. The development of reliable ML closures could also enhance the predictive capabilities of simulations for plasma instabilities and transport phenomena, crucial for maintaining plasma confinement and achieving sustained fusion reactions.

Future research directions identified in the review focus on improving the stability and generalizability of ML closures for large-scale simulations. Addressing issues such as off-diagonal pressure tensor accuracy and ensuring stable numerical integration are key to unlocking the full potential of these techniques. Continued development in this area could lead to more efficient and accurate plasma simulations, supporting the design and operation of future fusion power plants and furthering our understanding of fundamental plasma physics.