Modelling of space plasma from Vlasov to fluid: machine learning applied to the closure problem

In our solar system, the main source of plasma is the Sun, which produces the so-called solar wind by continuously pushing its outermost layer -the corona- into space. The turbulent solar wind impinges on our planet and interacts with its magnetic field, creating a region of space called Earth’s magnetosphere. From its birth to its impact on our planet, the solar wind still harbors numerous unanswered questions. Answering these questions requires the numerical modelling of the plasma itself.

The most physically accurate numerical methods are based on kinetic modeling, which tracks the particles' velocity distribution function. However, these methods are numerically demanding since they involve modeling the complex six-dimensional particle distribution function as it evolves in time. To simplify the problem, such distribution is integrated over the velocity coordinates leading to the (more efficient) three-dimensional fluid plasma framework. Still, the passage to the fluid equations comes with an important caveat. The fluid system of equations needs to be closed by choosing a proper “closure”. The objective of this work is to tackle the closure problem by employing a combination of kinetic simulation and machine learning techniques.

We perform multiple decaying turbulence plasma simulations using a Hybrid-PIC [1] (i.e. kinetic ions, fluid electrons) model. By varying different physical parameters, notably the ion beta, we explore the variability of the solar wind. These kinetic simulations serve as the ground truth to train a machine learning model. The machine's task is to "learn" the best approximation for the closure equation. We focus in particular on the reconstruction of the pressure tensor. We explore various machine learning techniques [2, 3] (CNN, GAN, FNO) that have shown promise in atmospheric science but are new to this specific problem. We show how this reconstructed closure performs better than other analytical approximations [4] (polytropic, CGL, CGL+FLR effects). The final goal is to learn a closure equation that can effectively incorporate complex kinetic physics into a simplified, yet more accurate, fluid simulation. This will significantly increase the fidelity of solar wind models without making them prohibitively expensive to compute.

[1] Behar, Etienne, Shahab Fatemi, Pierre Henri, e Mats Holmström. «Menura: A Code for Simulating the Interaction between a Turbulent Solar Wind and Solar System Bodies». Annales Geophysicae 40, fasc. 3 (2022): 281–97. https://doi.org/10.5194/angeo-40-281-2022.

[2] Kovachki, Nikola, Zongyi Li, Burigede Liu, et al. «Neural Operator: Learning Maps Between Function Spaces». Preprint, 2 maggio 2024. https://doi.org/10.5555/3648699.3648788.

[3] Jeong, Hyun-Jin, Mingyu Jeon, Daeil Kim, et al. «Prediction of the Next Solar Rotation Synoptic Maps Using an Artificial Intelligence–Based Surface Flux Transport Model». The Astrophysical Journal Supplement Series 278, fasc. 1 (2025): 5. https://doi.org/10.3847/1538-4365/adc447.

[4] Hunana, P., A. Tenerani, G. P. Zank, et al. «An Introductory Guide to Fluid Models with Anisotropic Temperatures. Part 1. CGL Description and Collisionless Fluid Hierarchy». Journal of Plasma Physics 85, fasc. 6 (2019): 205850602. https://doi.org/10.1017/S0022377819000801.