Discovering heat flux closures using machine learning methods

In computational plasma physics kinetic models are used to simulate plasma phenomena where small scale physics is expected to be of importance. These models contain the full information of the particle velocity distribution function but are computationally expensive. Therefore, computationally cheaper models are utilized, which can then be deployed to larger scales e. g. 10-moment fluid models or magnetohydrodynamics (MHD). However, the large scale behavior is critically influenced by small scale behavior. For example, solar wind observations show that ion and electron scale instabilities constrain the solar wind temperature anisotropy over the entire heliosphere (Berčič et al., 2019; Matteini et al., 2013)  and in our group we have recently demonstrated via fully kinetic numerical simulations the non-trivial link between the small and the large scales in heat flux regulation in the solar wind (Micera et al., 2021; Micera et al., 2025). Thus, models are required that can include kinetic processes, in reduced form, into large scale simulations. At the moment, analytical closures are used to close the hierarchy of fluid equations, but these closures are strictly valid only in certain regimes. For example, Landau fluid closures (Hammett & Perkins, 1990; Hunana et al., 2019) assume that the plasma is close to Local Thermodynamic Equilibrium, which is not the case for most space plasmas. Finding suitable closure equations is an ongoing research topic that gets increasingly more difficult in complex systems. In this study, we try to improve fluid models by learning a suitable symbolic closure for the heat flux by applying machine learning methods (Alves & Fiuza, 2022; Long et al., 2019) to data from kinetic simulations.
At first, these methods were tested by learning the lower moment equations using simulation data of the two stream instability.
In the long term, closure equations for more complex systems will be addressed.