# Using neural networks to solve the 2D Poisson equation for electric field computation in plasma fluid simulations

arXiv: https://arxiv.org/abs/2109.13076 · HTML: https://mlsanity.com/paper/2109.13076/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 27 Sep 2021.

## Builds on (most cited first, 13 of 13)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Network](https://mlsanity.com/paper/2109.09363/) (2021)
- [Learning Incompressible Fluid Dynamics from Scratch -- Towards Fast, Differentiable Fluid Models that Generalize](https://mlsanity.com/paper/2006.08762/) (2020)

## Built on by (most cited first, 10 of 10)

- [Invariant preservation in machine learned PDE solvers via error correction](https://mlsanity.com/paper/2303.16110/) (2023)
- [Principled Acceleration of Iterative Numerical Methods Using Machine Learning](https://mlsanity.com/paper/2206.08594/) (2022)
- [Physics-Preserving AI-Accelerated Simulations of Plasma Turbulence](https://mlsanity.com/paper/2309.16400/) (2023)
- [Deep Learning Architecture Based Approach For 2D-Simulation of Microwave Plasma Interaction](https://mlsanity.com/paper/2206.01263/) (2022)
- [An Implicit GNN Solver for Poisson-like problems](https://mlsanity.com/paper/2302.10891/) (2023)
- [Physics-Informed Boundary Integral Networks (PIBI-Nets): A Data-Driven Approach for Solving Partial Differential Equations](https://mlsanity.com/paper/2308.09571/) (2023)
- [A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions](https://mlsanity.com/paper/2310.00177/) (2023)
- [Towards a Machine-Learned Poisson Solver for Low-Temperature Plasma Simulations in Complex Geometries](https://mlsanity.com/paper/2306.07604/) (2023)
- [SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling](https://mlsanity.com/paper/2505.04738/) (2025)
- [Machine learning for advancing low-temperature plasma modeling and simulation](https://mlsanity.com/paper/2307.00131/) (2023)

## Abstract (arXiv)

The Poisson equation is critical to get a self-consistent solution in plasma fluid simulations used for Hall effect thrusters and streamer discharges, since the Poisson solution appears as a source term of the unsteady nonlinear flow equations. As a first step, solving the 2D Poisson equation with zero Dirichlet boundary conditions using a deep neural network is investigated using multiple-scale architectures, defined in terms of number of branches, depth and receptive field. One key objective is to better understand how neural networks learn the Poisson solutions and provide guidelines to achieve optimal network configurations, especially when coupled to the time-varying Euler equations with plasma source terms. Here, the Receptive Field is found critical to correctly capture large topological structures of the field. The investigation of multiple architectures, losses, and hyperparameters provides an optimal network to solve accurately the steady Poisson problem. The performance of the optimal neural network solver, called PlasmaNet, is then monitored on meshes with increasing number of nodes, and compared with classical parallel linear solvers. Next, PlasmaNet is coupled with an unsteady Euler plasma fluid equations solver in the context of the electron plasma oscillation test case. In this time-evolving problem, a physical loss is necessary to produce a stable simulation. PlasmaNet is finally tested on a more complex case of discharge propagation involving chemistry and advection. The guidelines established in previous sections are applied to build the CNN to solve the same Poisson equation in cylindrical coordinates with different boundary conditions. Results reveal good CNN predictions and pave the way to new computational strategies using modern GPU-based hardware to predict unsteady problems involving a Poisson equation.