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中国物理学会期刊

流体方程约束下的射频容性耦合He和Ar放电的神经网络模拟研究

Parameterized physics-informed neural-network modeling of radio-frequency capacitively coupled He and Ar discharges constrained by fluid equations

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  • 随着人工智能的发展,神经网络在复杂物理系统中的建模、计算和推理能力受到广泛关注。在等离子体模拟领域,对于射频容性耦合等离子体(RF-CCP)这类具有强非线性和多场耦合的放电体系,基于神经网络与流体控制方程相结合的建模研究不足,特别是实现多压强、多射频电压条件下的快速时空场计算推理,尚处于空白。本文构建了用于一维射频容性耦合 He 和 Ar 放电流体模拟的参数化物理信息神经网络(P-PINN)。模型以时空坐标、气体压强 P 和射频电压幅值V0 为输入,并将流体方程组嵌入神经网络训练目标,从而将神经网络的函数逼近能力与流体方程所描述的粒子守恒和能量输运物理约束相结合。结果表明,参数化 PINN 能够在未见压强–电压放电条件下快速和精确地计算出 He 和 Ar 放电中的典型射频容性耦合等离子体各物理的时空结构,包括等离子体区和鞘层区的电势、电子、离子密度和电子能量密度时空分布,揭示了等离子体的输运和等离子体状态变化。相较于纯数据驱动神经网络,参数化 PINN 在整个压强–电压参数空间内表现出更低且更稳定的平均相对 L2 误差,暗示了物理方程残差和边界条件残差能够有效提高未见放电条件下的推理稳定性和物理一致性。此外,参数化物理信息神经网络在 He 和 Ar放电推理计算时间约为毫秒量级。因此,神经网络与 RF-CCP 流体控制方程的融合能够在保持放电物理约束的同时,提供快速的多参数放电条件下等离子体时空场的计算,为等离子体快速模拟、参数扫描和工艺窗口分析提供新的解决方案。

     

    Radio-frequency capacitively coupled plasmas involve strongly coupled particle transport, electron energy transport, sheath dynamics, and self-consistent electric-field evolution. Conventional numerical simulations of the spatiotemporal plasma evolution at different gas pressures and applied radio-frequency voltage amplitudes require the fluid equations coupled with the Poisson equation to be solved separately for each operating condition. Consequently, the total computational cost increases with the number of operating conditions, making large-scale scans of the pressure-voltage parameter space for suitable discharge conditions time-consuming. This work aims to develop a fast surrogate model capable of predicting spatiotemporal plasma characteristics for pressure-voltage combinations not included in training. To this end, a parameterized physics-informed neural network (P-PINN) is constructed for one-dimensional He and Ar discharges between parallel plates separated by 3 cm and driven at 13.56 MHz. The network takes the spatial coordinate, time, gas pressure, and radio-frequency voltage amplitude as inputs and predicts four principal plasma quantities: the electron density, ion density, electric potential, and electron energy density. Its loss function combines sparse fluid-model reference data with the residuals of the electron and ion continuity equations, electron energy equation, Poisson equation, electric field-potential relation, drift-diffusion flux relations, and electrode boundary conditions. Only 1\% of the spatiotemporal reference data from each training condition is used. Separate P-PINN models are trained for the He and Ar discharges and evaluated at pressure-voltage combinations absent from their respective training sets. The predicted spatiotemporal distributions agree closely with the fluid-model reference solutions. The models capture the periodic variations in electric potential, electron density, and electron energy density over a radio-frequency cycle, as well as the main spatial features of the discharge, including the concentration of charged particles in the plasma bulk, the rapid decrease in electron and ion densities near the electrodes, and the potential drops across the sheath regions. They also reproduce the changes in these distributions with gas pressure and radio-frequency voltage amplitude. Compared with neural networks trained using the same sparse data but without the equation and boundary residuals, the P-PINN models generally yield lower relative L_2 errors over the evaluated parameter space, indicating closer agreement with the self-consistent fluid-model solutions. For each gas, ten representative conditions absent from the corresponding training set are evaluated. The mean relative L_2 errors of ion density, electron density, electric potential, and electron energy density are 1.53\%, 1.36\%, 3.44\%, and 1.72\%, respectively, for He, and 1.55\%, 1.21\%, 2.65\%, and 1.60\%, respectively, for Ar. Averaging first over the four quantities and then over the ten conditions gives overall mean errors of 2.01\% for He and 1.75\% for Ar. The electric potential has the largest mean error in both gases, indicating that its reconstruction is the main source of error in the present models. After offline training, the average inference times for one discharge condition are 15.6 ms for He and 14.5 ms for Ar on the computational platform used. For both gases, relatively large errors occur mainly near the boundaries of the sampled parameter domain, particularly under low-pressure and low-voltage conditions. This behavior may be associated with the smaller number of neighboring training conditions near the parameter-space boundaries, which makes accurate inference more challenging. At sufficiently low pressure, the increased electron mean free path also enhances nonlocal electron kinetics, potentially limiting the physical validity of the underlying fluid model and, consequently, that of a surrogate constrained by this model. These results demonstrate that combining sparse fluid-model reference data with the fluid conservation and transport equations, the Poisson equation, and wall boundary conditions enables millisecond-scale prediction of spatiotemporal plasma characteristics. Within the validity range of the underlying fluid model, the proposed method can therefore provide an efficient surrogate for rapid pressure-voltage parameter scans and analysis of discharge operating windows.

     

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