Abstract
The following work presents a new method to solve Poisson’s equation and, more generally, sparse linear systems using graph neural networks. We propose a supervised approach to solve the discretized representation of Poisson’s equation at every time step of a simulation. This new method will be applied to plasma physics simulations for Hall-Effect Thruster’s modeling, where the electric potential gradient must be computed to get the electric field necessary to model the plasma’s behavior. Solving Poisson’s equation using classical iterative methods represents a major part of the computational costs in this setting. This is even more critical for unstructured meshes, increasing the problem’s complexity. To accelerate the computational process, we propose a graph neural network to give an initial guess of Poisson’s equation solution. The new method introduced in this article has been designed to handle any meshing structure, including structured and unstructured grids and sparse linear systems. Once trained, the neural network would be used inside a numerical simulation in inference to give an initial guess of the solution for each simulation time step for all right-hand sides of the linear system and all previous time step solutions. In most industrial cases, Hall-Effect thrusters’ modeling requires a large unstructured mesh that one single processor cannot hold regarding memory capacity. We then propose a partitioning strategy to tackle the challenge of solving linear systems on large unstructured grids when they cannot be on a single processor.
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Acknowledgments
This research has been realized under the supervision of Bénédicte Cuenot and Olivier Vermorel (senior researchers at CERFACS). Gabriel Vigot acknowledges the support of Luciano Drozda (senior researcher at CERFACS) in conceiving and validating this present article for the machine learning part. Gabriel Vigot also acknowledges the support of Luc Giraud (senior researcher at INRIA) in conceiving and validating this present article for the part concerning linear algebra.
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The present authors have no competing interests with the participants of this article. Gabriel Vigot acknowledges the financial support from Safran Spacecraft Propulsion, under the supervision of Benjamin Laurent, and the French Space Agency (CNES: Centre National d’Études Spatiales), under the supervision of Ulysse Weller, under the EPIC convention. This work is introduced within the CHEOPS project.
6 Appendix
6 Appendix
(a) 2D map along the simulation time line t in microseconds. From the first to the fourth row is the update \(\mathrm {\Phi }\), the solution of reference \(\mathrm {\Phi _{ref}}\), the absolute difference \(|\mathrm {\Phi - \Phi _{ref}}|\), and \(\textrm{RHS}\) the right-hand side of the Poisson’s equation.
(b) Neural network predictions \(\mathrm {\Phi }\) with the same variable order as in Fig. (6a)
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Vigot, G., Cuenot, B., Vermorel, O. (2024). Solving Sparse Linear Systems on Large Unstructured Grids with Graph Neural Networks: Application to Solve the Poisson’s Equation in Hall-Effect Thrusters Simulations. In: Franco, L., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2024. ICCS 2024. Lecture Notes in Computer Science, vol 14834. Springer, Cham. https://doi.org/10.1007/978-3-031-63759-9_41
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