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Data-Driven Reconstruction of Spatially Resolved Electron and Ion Energy Distributions from Macroscopic Plasma Quantities with Deep Neural Networks
Libin Varghese, Kaushik Prajapati, Bhaskar Chaudhury
TL;DR
Reconstructing spatially resolved electron and ion energy distributions from macroscopic plasma observables remains challenging because the inverse mapping is nonlinear and largely unexplored. This work learns that mapping from paired PIC-MCC simulation data, finding that all three models reconstruct the distributions, with FNO performing best overall.
Problem
Whether spatially resolved macroscopic plasma observables contain sufficient information to reconstruct underlying kinetic distributions remains largely unexplored, despite the importance of spatially resolved EEDFs and IEDFs.
Method
The study trains data-driven models on paired spatially resolved macroscopic observables and EEDFs/IEDFs generated by high-fidelity PIC-MCC simulations, evaluating reconstruction with image-based and physics-based measures.
Results
All three architectures approximate the macro-to-micro mapping, with the FNO achieving the best overall accuracy and generalization across the unseen test cases.
Takeaways & Limitations
Within the represented plasma conditions, macroscopic observables retain sufficient information for approximate kinetic-state reconstruction, establishing a proof of concept for future diagnostics and reduced-order modeling.
Takeaways & Limitations
Transferability beyond the simulation models, collision cross sections, boundary conditions, parameter space, and predominantly Maxwellian-like distributions remains to be established.
Abstract
from arXiv · showhide
Spatially resolved EEDFs/IEDFs provide essential kinetic information about low-temperature plasmas (LTPs) and play a central role in determining transport, chemical reaction rates, and plasma surface interactions. While kinetic simulations directly resolve these distributions, experimental measurements remain challenging and are often invasive, spatially limited, or require assumptions regarding the distribution shape such as a Maxwellian. However, several macroscopic plasma observables can be measured non-invasively using advanced diagnostic techniques, providing spatially resolved information about the plasma state. An important inverse problem is therefore whether readily measurable macroscopic plasma quantities contain sufficient information to reconstruct the underlying kinetic state. In this work, we investigate this problem by learning a nonlinear mapping from spatially resolved macroscopic plasma observables to the corresponding spatially resolved EEDFs/IEDFs using a deep learning framework. Paired datasets comprising 2D macroscopic observables and spatially resolved EDFs are generated using 2D-3V PIC-MCC simulations. Three representative learning paradigms, a U-Net, a FNO, and a MeshGraphNet, are employed in this study to learn this inverse mapping. The predicted EDFs reproduce both bulk plasma and sheath characteristics with good agreement to the PIC-MCC reference data, with the FNO providing the best overall performance. Beyond conventional metrics, physics-based validation demonstrates that the reconstructed EDFs accurately recover the corresponding density and temperature, and rate coefficients. These results demonstrate that macroscopic plasma observables encode sufficient information to infer important kinetic properties in LTPs, providing a potential foundation for surrogate kinetic modeling and next-generation plasma diagnostics.
1 Introduction
Spatially resolved electron and ion energy distributions are essential to describe nonequilibrium plasma kinetics and spatially varying processing behavior, yet recovering them from macroscopic observables is a difficult inverse problem. The section motivates learning this inverse mapping from paired kinetic-simulation data using deep-learning architectures.
- Kinetic importance: Electron and ion energy distributions govern reaction rates, transport, ionization, excitation, and plasma-surface interactions in low-temperature plasmas.The EEDF controls electron-impact processes, while the IEDF directly influences ion-assisted plasma-surface processes.
- Spatial resolution: Spatially resolved kinetic information is essential because reactor geometry, sheath dynamics, and non-uniform power deposition create substantial energy-distribution variations.Bulk or spatially averaged measurements can therefore be insufficient for accurately describing local etch and deposition behavior.
- Data-driven reconstruction: PIC simulations provide paired macroscopic plasma quantities and underlying particle energy distributions, enabling investigation of whether microscopic states can be inferred from macroscopic observables.The simulations resolve particle dynamics and fields while providing quantities such as density, temperature, and potential.
- Deep-learning framework: Convolution-based, graph-based, and neural-operator frameworks offer distinct inductive biases for learning complex mappings between high-dimensional spatial fields.Recent deep-learning advances make these architectures relevant to spatially resolved plasma-state reconstruction.
- Inverse problem: The inverse mapping is challenging because velocity-space moments compress much microscopic information into a limited set of observable quantities.A statistically meaningful approximation learned from representative kinetic simulations would test whether macroscopic observables retain recoverable kinetic information.
2 Data-driven Methodology
The methodology learns a nonlinear mapping from spatially resolved macroscopic plasma quantities to corresponding EEDFs and IEDFs using data-driven statistical learning. Reconstructed distributions are evaluated with image-based metrics and physics-based validation for reconstruction accuracy and physical consistency.
- The framework learns a nonlinear mapping between spatially resolved plasma quantities and corresponding spatially resolved energy distributions.
- Reconstructed EEDFs and IEDFs are assessed using image-based performance metrics and physics-based validation for reconstruction accuracy and physical consistency.
- The workflow proceeds from 2D PIC-MCC data generation and preprocessing to model development, validation, and evaluation.
- Model parameters θ ∗ are estimated by minimizing a loss functional L measuring discrepancies between predicted and reference distributions.
- The inverse problem reconstructs microscopic distributions from macroscopic observables without relying on simplifying analytical assumptions.
3 Data Generation
The dataset is generated with 2D PIC-MCC simulations spanning controlled physical cases, then organized into spatially resolved macroscopic inputs and electron/ion energy-distribution outputs. Sheath-specific distributions use a common normalization based on an average sheath index of k = 10.
- PIC-MCC methodology: 2D PIC-MCC iteratively couples particle dynamics and electrostatic fields by depositing charge, solving Poisson’s equation, interpolating fields, and advancing particles with the Boris pusher.The simulation cycle self-consistently updates grid fields and particle trajectories at each time step.
- Simulation setup: All cases use a 1 × 1 m domain with 256 cells per direction, ne = ni = 5×10^11 m^-3, Te = 1 eV, Ti = 300 K, tf = 5 µs, and ∆t = 2.5 × 10^-9 s.Grounded absorbing walls and a uniform quasineutral hydrogen plasma define the common initialization and boundary conditions.
- Input construction: Macroscopic inputs are stored on a node-centered Cartesian grid of size (Ny + 1) × (Nx + 1), formed by distributing each particle’s contribution among four nearest vertices.The resulting entries represent averaged physical quantities over surrounding spatial regions.
- Output construction: Each output is an Me × Ms matrix whose entries count particles within a spatial strip and energy interval, producing spatially resolved electron and ion energy distributions.Energy and spatial axes are discretized with resolutions ∆E and ∆s over [Emin,Emax] and [Smin,Smax].
- Sheath analysis: k = 10 is used across all cases for sheath-specific analysis after applying the Brinkmann integral criterion to density profiles near the wall.The sheath edge is located at x = k∆x, and the same normalization procedure is applied with region-specific summation limits.
4 Data Preprocessing
The framework preprocesses diverse multi-case PIC-MCC data by analyzing distributions and applying channel-wise normalization before jointly reconstructing electron and ion energy distributions. This preparation addresses scale disparities while supporting learning across distinct physical regimes.
- Data preprocessing: The preprocessing workflow includes data normalization and distribution analysis across simulation cases to verify coverage of diverse physical regimes.The overall framework also includes model training and evaluation.
- Data distribution analysis: Two hundred samples are randomly selected from each simulation case to assess distributional differences across distinct operating conditions.The cases come from multiple PIC-MCC simulations representing different physical conditions.
- Normalization: Channel-wise min-max normalization is applied independently to input macroscopic quantities and output energy-distribution histograms because their magnitudes span several orders and can bias optimization.Normalization is required to improve conditioning and prevent higher-magnitude features from dominating learning.
5 Deep Learning Architectures for EDF Reconstruction
The study compares three representation paradigms for reconstructing spatially resolved EDFs from macroscopic plasma fields: convolutional, operator-based, and graph-based models. U-Net targets structured field-to-field mappings, FNO learns resolution-invariant function-space operators, and MeshGraphNet uses mesh-based message passing to model spatial interactions.
- Architectures: Three architectures are evaluated: U-Net, Fourier Neural Operator (FNO), and MeshGraphNet, representing convolutional, neural operator-based, and graph-based paradigms.The comparison is designed to assess how architectural representation affects the macro-to-micro learning task.
- U-Net: U-Net is a natural baseline because macroscopic quantities and target EDFs share the same spatial grid, enabling structured field-to-field prediction.Its encoder-decoder structure captures contextual information across spatial regions, while skip connections support localized spatial features.
- Implementations: The implemented models use a four-level U-Net with 64 base channels, a two-dimensional FNO with four Fourier layers and 32 latent channels, and MeshGraphNet with six message-passing blocks and width 64.The FNO retains 16 Fourier modes per spatial dimension, while MeshGraphNet applies two-layer MLP encoders and decoder with ReLU activations.
- FNO: FNO is well suited to the inverse mapping because it learns nonlinear operators between continuous function spaces and can, in principle, operate at resolutions different from training.Fourier layers model global spatial dependencies, while local transformations capture localized variations.
- MeshGraphNet: MeshGraphNet represents grid locations as nodes and physical relationships as edges, using message passing to model local states and neighborhood interactions.Its encode-process-decode framework iteratively updates edge and node representations, expanding the effective receptive field.
6 Model Evaluation and Results
The three neural architectures are evaluated for distributional accuracy, physical consistency, and robustness, with FNO showing the strongest overall reconstruction performance. Physics-based checks recover key macroscopic quantities and rate coefficients, while noisy-input tests indicate limited sensitivity within the simulated conditions.
- Conventional metrics: FNO achieves the lowest MSE and highest PSNR and SSIM across validation and testing datasets, indicating the strongest reconstruction accuracy and structural fidelity.The testing metrics combine electron and ion samples from cases 6 and 7, with consistent validation–testing trends indicating stable generalization.
- Qualitative reconstruction: Qualitative comparisons evaluate U-Net, FNO, and MeshGraphNet predictions across the full domain and left and bottom sheath regions against PIC-MCC ground truth.The comparisons use unseen test samples from case 6 for both EEDFs and IEDFs.
- Physics-based validation: Moment-based validation shows that reconstructed EEDFs and IEDFs retain the density and effective-temperature properties of the corresponding simulation distributions across all three architectures.Density is obtained from the zeroth EDF moment, while mean particle energy is computed from the first moment.
- Physics-based validation: For test case 6 at t = 1.3425 µs, FNO and ground-truth momentum-transfer rate coefficients are 1.32×10−13 and 1.31×10−13 m3,s−1, respectively, with 0.16% relative error.The coefficients are evaluated from the model-predicted and ground-truth EEDFs using the momentum-transfer collision process.
- Distributional similarity: FNO achieves the lowest Jensen-Shannon Divergence values for both electron and ion distributions, indicating the closest agreement with the reference kinetic distributions.JSD complements pixel-wise metrics by measuring probabilistic similarity between reconstructed and reference EEDFs and IEDFs.
- Noise robustness: With 10% signal-magnitude noise added independently to each input channel without retraining, all models show minor metric changes, while FNO is most robust.The noisy-input evaluation covers cases 6 and 7 and reports marginal changes in MSE, PSNR, SSIM, and JSD.
- Scope and limitations: The reconstruction is established only within the PIC-generated plasma-condition range and remains conditioned on its physical models, collision cross sections, boundaries, parameter space, and predominantly Maxwellian-like distributions.Transferability beyond these assumptions remains to be established.
7 Conclusion
Within seven physically distinct 2D-3V PIC-MCC simulation conditions, macroscopic plasma observables supported approximate reconstruction of spatially resolved EEDFs and IEDFs. U-Net, FNO, and MeshGraphNet all learned the macro-to-micro mapping, with FNO performing best overall and remaining robust to moderate measurement noise.
- Paired datasets from high-fidelity electrostatic 2D-3V PIC-MCC simulations covered seven physically distinct conditions for reconstructing spatially resolved EEDFs and IEDFs.
- U-Net, FNO, and MeshGraphNet all approximated the macro-to-micro mapping, while FNO achieved the best overall accuracy and generalization on unseen test cases.
- Controlled perturbation tests showed robustness to moderate measurement noise, indicating limited sensitivity to uncertainties typical of LTP diagnostics.
- The results provide evidence that readily measurable macroscopic quantities retain sufficient information for approximate reconstruction of the underlying kinetic state.
- The study establishes a proof of concept and provides a potential foundation for future integration.