Non-linear control variate in 𝛿𝑓 particle-in-cell methods using symplectic neural networks

Victor Fournet, Martin Campos Pinto, Emmanuel Franck, Victor Michel-Dansac

2026 · Elsevier BV · Elsevier BV · 10.2139/ssrn.7070225

Abstract

We present a novel 𝛿𝑓 particle-in-cell (PIC) method for the kinetic simulation of electrostatic plasmas in which the bulk density, acting as a control variate, is evolved using symplectic neural networks (SympNets). The SympNets are used as an approximation of the backward flow and trained using the particle trajectories. We introduce a periodic variant of the SympNet architecture that encodes the spatial periodicity of the problem into the network itself. We validate the approach with numerical results in 1D1V and 3D3V for the Vlasov-Poisson system.

@article{Fournet_2026,
 title={Non-linear control variate in 𝛿𝑓 particle-in-cell methods using symplectic neural networks},
 url={http://dx.doi.org/10.2139/ssrn.7070225},
 DOI={10.2139/ssrn.7070225},
 publisher={Elsevier BV},
 author={Fournet, Victor and Campos Pinto, Martin and Franck, Emmanuel and Michel-Dansac, Victor},
 year={2026} }