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

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.