Comparing analytical and machine learning heat flux closures

In many plasmas, physical processes of relevance occur over ranges of scales covering many orders of magnitude. Thus, modelling plasmas comes with a trade-off between physical accuracy and computational cost. Fully kinetic models correctly self-consistently describe collisionless plasmas by advancing the velocity distribution functions (VDFs) in time, either directly (Vlasov methods) or sampling it through computational particle (PIC codes). A computationally cheaper but physically less accurate alternative are multi-fluid models. Instead of the VDFs, these models evolve fluid quantities and can approximate kinetic processes of interest by choosing a suitable closure for the hierarchy of fluid moment equations, i. e., an equation for the divergence of the heat flux in the case of ten-moment fluid models. In most heliospheric plasmas, including for example the solar wind, the observed VDFs are non-Maxwellian, which gives rise to many different instabilities that exchange energy between particles and fields. We investigate the use of machine learning models for the discovery of heat flux closures, as an alternative to the typically employed Hammett-Perkins-like analytical closures. As a test case, we use the two-stream instability, which occurs when there is a large velocity drift between two electron populations with respect to their thermal speed, and causes the formation of electron holes and electric field saturation in its nonlinear stage. While the linear stage of the two stream instability is well reproduced by 10-moment models with analytical closures, reproducing electric field evolution at saturation is a challenge for reduced models. In this work, we compare fully kinetic Vlasov simulations against two-fluid 10-moment simulations employing both analytical and ML-driven closures.