Heat flux closures for two-stream unstable plasmas with nonlinear sparse regression

Balancing accuracy and computational efficiency is a long-standing problem in multiscale modeling of collisionless plasmas, from astrophysical environments to laboratory experiments. Fully kinetic modeling offers high accuracy, but is often also prohibitively expensive computationally. Fluid models, on the other hand, are significantly more efficient, but need to be supplemented with a closure, and finding accurate and robust fluid closures for non-linear systems of interest has proven challenging. Over the past decade, machine learning methods like neural networks and sparse regression have shown a lot of promise for discovering these complex closures from first-principles kinetic simulations. One example of this is the recent discovery of a six-term interpretable model of the combined electron heat flux in a two-stream unstable plasma using sparse regression [Ingelsten et al 2025, J. Plasma Phys. 91, E64]. In this work, we show how the heat flux of the two counter-streaming populations in the setup in question can be modeled separately using a modified version of the same model, resolving the instability without needing to insert it manually through the closure. We also demonstrate how the three most important closure terms can be predicted from box-averaged fluid quantities through nonlinear sparse regression, with accuracy approaching that of a neural network – thus making the closure applicable over a large parameter domain.