Data-Driven Reconstruction of Spatially Resolved Electron and Ion Energy Distributions from Macroscopic Plasma Quantities with Deep Neural Networks

Low-temperature plasmas (LTPs) are governed by microscopic kinetic properties, such as electron and ion energy distribution functions (EEDFs and IEDFs), which dictate transport, chemical reaction rates, and plasma-surface interactions. While kinetic simulations resolve these distributions directly, experimental measurements are typically invasive, spatially limited, or rely on restrictive assumptions like a Maxwellian shape. However, macroscopic plasma observables such as density and temperature can often be measured non-invasively. This paper investigates whether these readily measurable macroscopic quantities contain sufficient information to reconstruct the underlying spatially resolved kinetic state.

To address this inverse problem, the authors formulate a data-driven supervised learning framework. They generate comprehensive, paired datasets of 2D macroscopic observables and spatially resolved energy distribution functions using high-fidelity 2D-3V Particle-in-Cell Monte-Carlo-Collision (PIC-MCC) simulations across seven distinct physical regimes—incorporating electric fields, magnetic confinement, collisional relaxation, and ionization. Three representative deep learning architectures are evaluated for learning this nonlinear mapping: a convolutional encoder-decoder (U-Net), a Fourier Neural Operator (FNO), and a graph-based MeshGraphNet.

The deep learning models successfully learn the highly nonlinear mapping from macroscopic observables to microscopic energy distributions. All three architectures accurately reproduce both bulk plasma and sheath characteristics when compared against reference PIC-MCC simulation data. Among the tested paradigms, the Fourier Neural Operator (FNO) achieves the best overall performance, capturing complex spatial variations and sharp boundaries near the sheath.

Beyond standard image-based error metrics, the authors perform physics-based validation to verify that the reconstructed energy distribution functions remain physically consistent. The predicted distributions accurately recover the corresponding plasma density, average temperatures, and reaction rate coefficients. This demonstrates that macroscopic plasma observables robustly encode the necessary kinetic information in LTPs, opening new avenues for surrogate kinetic modeling and advanced diagnostics.