Work

A hierarchical adaptive Fourier-feature physics-informed neural network for radio-frequency capacitively coupled plasma discharges

Year: 2026

Type: article

Abstract: Physics-informed neural networks (PINNs) are difficult to train for radio-frequency capacitively coupled plasma (RF-CCP) fluid--Poisson systems because missing initial conditions, steep sheath gradients, and strongly imbalanced residuals can lead to unstable or nonunique reconstructions. We propose a Hierarchical Adaptive Fourier-feature PINN (HA-FPINN) for such coupled plasma models. The method combines Fourier feature mapping to improve the representation of small-scale sheath structures, sparse physical anchors to provide interior constraints, and a two-level adaptive weighting scheme to balance the PDE, boundary, and anchor losses as well as individual equation residuals. The framework is tested on an 8-equation helium discharge model and a 23-equation electronegative $\mathrm{Ar}/\mathrm{O}_2$ model. In the helium case, HA-FPINN reconstructs the space--time evolution of the main physical variables with $1\%$ anchors and converges for all tested network architectures, with convergence times of $110$--$246$ s. In the $\mathrm{Ar}/\mathrm{O}_2$ case, HA-FPINN achieves stable reconstruction with $5\%$ anchors, whereas the non-adaptive Fourier-feature PINN and standard PINN fail to converge under the same setting. The best performance in both systems is obtained at the tested Fourier frequency parameter $\sigma=2.20$. Noise-injection tests show that HA-FPINN maintains the lowest reconstruction error in the helium case and preserves the main space--time structures in the $\mathrm{Ar}/\mathrm{O}_2$ case up to $15\%$ anchor noise. These results show that HA-FPINN provides a robust PINN formulation for strongly coupled plasma fluid models with missing initial conditions.

Source: Machine Learning Science and Technology

DOI: 10.1088/2632-2153/aeba37