Addressing the closure problem using supervised Machine Learning

When deriving fluid equations from the Vlasov equation for collisionless plasmas, one runs into the so-called closure problem: each equation for the temporal evolution of one particle moment (density, current, pressure, heat flux, …) includes terms depending on the next order moment. Therefore, when choosing to truncate the description at the nth order, one must approximate the terms related to the (n+1)th order moment included in the evolution equation for the nth order moment. The order at which the hierarchy is closed and the assumption behind the approximations used determine how accurately a fluid description can reproduce kinetic processes.

In this work, we aim at reconstructing specific particle moments from kinetic simulations, using as input the electric and magnetic field and the lower moments. We use fully kinetic Particle In Cell simulations, where all physical information is available, as the ground truth. The approach we present here uses supervised machine learning to enable a neural network to learn how to reconstruct higher moments from lower moments and fields.

Starting from the work of Laperre et al., 2022 we built a framework which makes it possible to train feedforward multilayer perceptrons on kinetic simulations to learn to predict the higher moments of the Vlasov equation from the lower moments, which would also be available in fluid simulations. We train on simulations of magnetic reconnection in a double Harris current sheet with varying background guide field obtained with the semi-implicit Particle-in-Cell code iPiC3D (Markidis et al, 2010). We test the influence of data preprocessing techniques, of (hyper-)parameter variations and of different architectures of the neural networks on the quality of the predictions that are produced. Furthermore, we investigate which metrics are most useful to evaluate the quality of the outcome.