基于元学习的气体放电等离子体电子Boltzmann方程数值求解

doi:10.19595/j.cnki.1000-6753.tces.230573

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摘要

在气体放电等离子体中,电子的输运行为可由Boltzmann方程精确描述,该方程的解是许多等离子体仿真模型的基础。物理信息神经网络作为一种求解Boltzmann方程的新型方法,虽克服了传统数值方法网格剖分和方程离散的缺陷,但其参数空间规模大,在求解多任务时训练效率较低。为此,该文构建了一种基于元学习的双循环物理信息神经网络,在内循环中对多个Boltzmann方程求解任务进行优化训练,得到各任务优化后的元损失函数,用于在外循环中进行网络参数更新,从而提高网络在求解新任务时的计算效率。计算结果表明,基于元学习的双循环物理信息神经网络在求解新的Boltzmann方程时,网络损失函数值和L²误差值的下降速度均显著快于普通的物理信息神经网络。此外,该文还研究了网络容量和内循环迭代次数对Boltzmann方程多任务求解效率的影响,结果显示计算效率并不随网络容量的增大而提高,且受内循环迭代次数影响较小。

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关键词 ：气体放电等离子体, Boltzmann方程, 元学习, 物理信息神经网络

Abstract：

The Boltzmann equation is a partial differential equation that describes the variation of particles in a non-equilibrium thermodynamic system. In the field of gas insulation and plasma discharge, the Boltzmann equation can be used to accurately describe the electron transport in gas discharge plasmas, and its solution is the basis of many plasma simulation models. However, the traditional numerical methods for solving the Boltzmann equation all require meshing on the computational domain, and the solution accuracy significantly depends on the quality of the meshing. The physics-informed neural networks (PINNs), as a new method for solving Boltzmann equation, overcomes the shortcomings of traditional numerical methods in mesh generation and equation discretization, but its training is inefficient when dealing with multi-tasks because of huge parameter space of PINNs. To address this issue, this paper proposes a Meta-PINN network with two loops of PINNs based on meta learning. Through the training in inner and outer loops, Meta-PINN solve the Boltzmann equation in multi-tasks accurately and efficiently.
In the Meta-PINN, there are two types of networks, which are the PINN network and the meta network. In the inner loop, the PINN network solve the Boltzmann equation in multi-tasks by minimizing the loss function of PINN. After all multi-tasks are optimized, the sum of the PINN loss function, namely the meta loss function, is obtained. Then, the meta network updates the weights by minimizing the meta loss function in the outer loop. Finally, the updated weights are used to improve the training efficiency when dealing with new tasks of solving Boltzmann equations.
To validate the performance of Meta-PINN, the Boltzmann equations under different reduced electric fields and gas mixing ratios are solved under the framework of Meta-PINN. The results show that, when dealing with new tasks, the loss function values and L² errors of Meta-PINN are reduced faster than that of PINN. Specifically, the minimum and maximum acceleration speeds increase by 1.75 and 23 times respectively, which indicates that the Meta-PINN outperforms the PINN in solution efficiency. Additionally, the effects of network capacities and inner steps on the computational efficiency are investigated in multi-tasks of solving Boltzmann equation. The result reveals that the computational efficiency does not improve with the increase of network capacities and is less affected by the inner steps.
The following conclusions could be drawn from the numerical experiments: (1) Meta-PINN can improve the efficiency of solving Boltzmann equation describing electron transport in gas discharge plasmas. Moreover, in some cases, the loss function value and L² error of Meta-PINN can be reduced to a lower order of magnitude than PINN, indicating that solution accuracy of Meta-PINN is also better than that of PINN. (2) In solving the Boltzmann equation, the computation efficiency is not improved with the increase of network capacity. The most suitable network for solving the Boltzmann equation under different reduced electric fields in argon plasma is the neural network with 3 hidden layers and 300 neurons per layer. (3) The computation efficiency is either not improved with the increase of inner steps. The most suitable inner steps for solving the Boltzmann equation under different reduced electric fields in argon plasma is 5 steps. Generally, the Meta-PINN can be extended easily to other numerical solution of plasma governing equations.

Key words： Gas discharge plasma Boltzmann equation meta learning physics-informed neural network

收稿日期: 2023-04-28

PACS:

TM11

基金资助:

国家自然科学基金（92066106）、江苏省科协青年科技人才托举工程（2021031）、东南大学“至善青年学者”支持计划（中央高校基本科研业务费）（2242022R40022）资助项目

通讯作者: 仲林林男,1990年生,副研究员,博士生导师,研究方向为高电压技术、放电等离子体技术、人工智能技术。E-mail：linlin@seu.edu.cn

作者简介: 吴冰钰女,1998年生,硕士研究生,研究方向为放电等离子体技术E-mail：wubingyu@seu.edu.cn

引用本文:

仲林林, 吴冰钰, 吴奇. 基于元学习的气体放电等离子体电子Boltzmann方程数值求解[J]. 电工技术学报, 0, (): 239621-239621. Zhong Linlin, Wu Bingyu, Wu Qi. Numerical Solution of Electron Boltzmann Equation in Gas Discharge Plasmas Based on Meta Learning. Transactions of China Electrotechnical Society, 0, (): 239621-239621.

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