Poisson's equation plays an important role in modeling many physical systems. In electrostatic self-consistent low-temperature plasma (LTP) simulations, Poisson's equation is solved at each simulation time step, which can amount to a significant computational cost for the entire simulation. In this paper, we describe the development of a generic machine-learned Poisson solver specifically designed for the requirements of LTP simulations in complex 2D reactor geometries on structured Cartesian grids. Here, the reactor geometries can consist of inner electrodes and dielectric materials as often found in LTP simulations. The approach leverages a hybrid CNN-transformer network architecture in combination with a weighted multiterm loss function. We train the network using highly-randomized synthetic data to ensure the generalizability of the learned solver to unseen reactor geometries. The results demonstrate that the learned solver is able to produce quantitatively and qualitatively accurate solutions. Furthermore, it generalizes well on new reactor geometries such as reference geometries found in the literature. To increase the numerical accuracy of the solutions required in LTP simulations, we employ a conventional iterative solver to refine the raw predictions, especially to recover the high-frequency features not resolved by the initial prediction. With this, the proposed learned Poisson solver provides the required accuracy and is potentially faster than a pure GPU-based conventional iterative solver. This opens up new possibilities for developing a generic and high-performing learned Poisson solver for LTP systems in complex geometries.

翻译：泊松方程在众多物理系统的建模中扮演着重要角色。在静电自洽低温等离子体（LTP）模拟中，每个仿真时间步均需求解泊松方程，这可能导致整个模拟产生显著的计算开销。本文描述了一种通用型机器学习泊松求解器的开发过程，该求解器专为满足结构化笛卡尔网格上复杂二维反应器几何构型的LTP模拟需求而设计。此类反应器几何构型可包含内部电极及介电材料——这在LTP模拟中颇为常见。该方法采用混合CNN-Transformer网络架构，并结合加权多任务损失函数。我们使用高度随机化的合成数据训练网络，以确保所学求解器对未见反应器几何构型的泛化能力。结果表明，该求解器能够产生定量和定性均准确的解。此外，它在新的反应器几何构型（如文献中的参考几何构型）上表现出良好的泛化性能。为提升LTP模拟所需解的数值精度，我们采用传统迭代求解器对原始预测值进行精化，特别是恢复初始预测未能解析的高频特征。由此，所提出的泊松求解器在满足精度要求的同时，可能比纯基于GPU的传统迭代求解器具有更快的速度。这为开发适用于复杂几何体LTP系统的高性能通用型泊松求解器开辟了新的可能性。