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.
Figures & tables
Figure 1: General workflow followed in this study, proceeding from data generation and preprocessing to model development, followed by validation and evaluation.
Case
Description
E field
B field
MCC collisions
1
Electric field only
Yes
Bz=0
No
2
High magnetic field
Yes
Bz=2×10−4 T
No
3
Collisions, low pressure
Yes
Bz=0
Yes ( ng=3×1019m−3 )
4
Collisions, high pressure
Yes
Bz=0
Yes ( ng=3×1020m−3 )
5
Ionization, high pressure
Yes
Bz=0
Yes + ionization ( Eiz=1.43 eV)
6
Low magnetic field with collisions, low pressure
Yes
Bz=2×10−5 T
Yes ( ng=3×1019m−3 )
Table 1: Simulation cases and corresponding physical conditions.
Figure 2: a) Simulation setup for case 1 at 5μs showing plasma density. b) Electron energy plots for various cases averaged along the y direction. c) Electron density plots for various cases averaged along the y direction. d) Ion energy plots averaged along the y direction. e) Space time evolution of X component of electric field exactly at the middle of the domain showcasing the evolution of simulation through the transient phase (case 1).
Figure 3: a) EEDF for full simulation domain. b) EEDF for left sheath. c) IEDF for full simulation domain. d) IEDF for left sheath. (All the EDFs are calculated at time = 5μs )
Figure 4: Schematic representation of dataset construction from two-dimensional PIC simulations. (a) The computational domain is divided into Ny×Nx cells of size Δx and Δy , with multiple simulation particles distributed in each cell. (b) Each particle contributes to its four nearest grid nodes according to its position within the cell, ensuring smooth spatial deposition of physical quantities (c) The resulting node centered mesh ( Gy×Gx ), where macroscopic fields such as densities and temperatures are extracted for use as input data for training.
Figure 5: Schematic representation of the output space as 2D histograms of particle energy discretized in space. In these histograms the vertical axis corresponds to energy bins (E1,E2,…,Emax) and the horizontal axis corresponds to spatial discretization. Four such histograms are constructed, two for electrons and two for ions, where one set resolves the distribution along the x -axis and the other set along the y -axis. (a) Particles with different energies are distributed across a two-dimensional spatial grid. (b) This illustrates the 2D histograms ϵx and ϵy , where particles are grouped into discrete space-energy bins. (c) The corresponding particle count matrices ϵxij and ϵyij represent the 2D histograms, where each element denotes the number of particles in spatial cell i and energy bin j .
Figure 6: Overview of the proposed data-driven framework. (a) Representative input macroscopic observables and corresponding output microscopic energy distribution functions. (b) Workflow illustrating the data preprocessing, neural network training, prediction, and performance evaluation stages adopted in this study.
Figure 7: t-SNE visualization of input channels: (a) electron energy, (b) ion energy, and (c) electron density. Each point represents a data sample projected into a two-dimensional embedding space using t-SNE with perplexity 30. Points are colored by simulation case, showing clear clustering and indicating the presence of distinct physical regimes in the dataset.
Figure 8: Schematic of the U-Net architecture used in this work. The input field X is processed through an encoder path consisting of repeated convolution and max-pooling operations to extract multi-scale feature representations. At the bottleneck, the latent features capture high-level contextual information. The decoder path reconstructs the target output through successive up-convolution and convolution operations. Skip connections concatenate feature maps from corresponding encoder and decoder levels, enabling the recovery of fine spatial details lost during downsampling. A final 1×1 convolution generates the predicted output field Y .
Figure 9: Architecture of the Fourier Neural Operator (FNO). The input field X is lifted to a higher-dimensional latent representation using the operator P and processed through T stacked Fourier layers. Each Fourier layer combines a spectral transformation, consisting of the Fourier transform F , learnable spectral operator R , and inverse Fourier transform F−1 , with a local linear transformation W . The outputs of the spectral and local branches are merged and passed through a nonlinear activation function σ . Finally, the projection operator Q maps the latent representation to the output field Y .
Figure 10: Architecture of the MeshGraphNet model. The input mesh M(V,E) , represented by nodes V and edges E , is first encoded into latent node and edge embeddings using an MLP encoder. The latent graph is then processed through L message-passing layers, where edge features are updated using the edge-update function ϕe and node features are updated using the node-update function ϕv after aggregating information from neighboring nodes. The resulting latent node embeddings are decoded through an MLP decoder to generate predictions on the output mesh M′(V′,E′) .
Model
Parameters
Best Epoch
Inference Time / Sample (ms) ∗
U-Net
31,043,716
74
5.349176
FNO
4,205,652
100
0.309875
MeshGraphNet
251,076
91
25.029857
Table 2: Model complexity and computational performance *Inference time is measured as the average time per sample on Test Data (case 6 and case 7)
Model
Dataset
MSE
SSIM
PSNR
U-Net
Validation
1.134×10−4
0.9636
39.4545
Testing
1.421×10−4
0.9538
38.4751
FNO
Validation
0.332×10−4
0.9811
44.7927
Testing
0.880×10−4
0.9738
40.5544
MeshGraphNet
Validation
0.370×10−4
0.9805
44.3175
Testing
1.284×10−4
0.9708
38.9144
Table 3: Performance of the proposed models on the validation and testing datasets. The reported testing metrics are computed over the combined electron and ion samples from cases 6 and 7.
Figure 11: Comparison of the predicted and ground-truth (GT) EEDFs for an unseen sample from case 6 at time t=1.3425\SIUnitSymbolMicros . The 3×3 panel layout is organized such that the rows correspond to the U-Net, FNO, and MeshGraphNet models, while the columns represent the full computational domain, left sheath, and bottom sheath regions, respectively. The sheath distributions are obtained from a spatial subset of width 10Δx extracted from the 256×256 computational domain. In each subplot, the ground-truth EEDF is shown in red and the corresponding model prediction in blue. The inset table reports the electron density ( ne ) and electron temperature ( Te ) calculated from the ground-truth and predicted (Pred.) distributions, illustrating the ability of the reconstructed EEDFs to recover the corresponding electron density and electron temperature. Electron energy distributions are shown over the range 0-10 eV.
Figure 12: Comparison of the predicted and ground-truth ion energy distribution functions (IEDFs) for the unseen sample from case 6 at time t=1.3425\SIUnitSymbolMicros . The 3×3 panel layout is organized such that the rows correspond to the U-Net, FNO, and MeshGraphNet models, while the columns represent the full computational domain, left sheath, and bottom sheath regions, respectively. The sheath distributions are obtained from a spatial subset of width 10Δx extracted from the 256×256 computational domain. In each subplot, the ground-truth IEDF is shown in red and the corresponding model prediction in blue. The inset table reports the ion density ( ni ) and ion temperature ( Ti ) calculated from the ground-truth (GT) and predicted (Pred.) distributions, illustrating the ability of the reconstructed IEDFs to recover the corresponding electron density and electron temperature. Ion energy distributions are plotted over the range 0-1 eV (zoomed from the full 0-2 eV range) to highlight differences in the low-energy regime.
Model
Case
Electron
Ion
MSE ( ×10−4 )
PSNR
SSIM
JSD ( ×10−3 )
MSE ( ×10−4 )
PSNR
SSIM
JSD ( ×10−3 )
U-Net
6
2.2743
36.6204
0.9118
6.2579
0.6004
42.5335
0.9850
5.2439
6†
2.0390
37.1224
0.9132
5.8342
0.5880
42.5245
0.9858
4.9566
7
1.9895
37.1276
0.9275
2.7291
0.8184
41.2157
0.9911
4.4738
7†
1.8878
37.4030
0.9256
2.8140
0.7583
41.5280
0.9920
4.2864
FNO
6
0.8457
40.9098
0.9547
0.7577
0.1739
47.9712
0.9962
1.3662
Table 4: Performance comparison of U-Net, FNO, and MeshGraphNet on Cases 6 and 7 and their noisy counterparts. Cases marked with † denote inputs corrupted with 10% additive white Gaussian noise (AWGN), while the target distributions remain unchanged. The reported metrics include Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Jensen–Shannon Divergence (JSD), evaluated separately for the electron and ion energy distribution functions (the closer the value is to 0, the better the match).
Energy-based models (EBMs) provide a powerful and flexible way of learning a joint probability distribution over data by constructing an energy surface. This energy surface enables insight extraction and conditional sampling. We apply EBMs to laboratory plasma physics, a domain characterized by highly nonlinear phenomena. These phenomena are studied using plasma diagnostics, which are often difficult to analyze and subject to hardware degradation. In addition, the possible configuration space of a plasma device is sufficiently large that it cannot be efficiently searched using conventional analysis techniques. EBMs address these issues. At the Large Plasma Device (LAPD), a CNN- and attention-based EBM is trained on a set of randomly generated machine conditions and their corresponding diagnostic time series. We demonstrate diagnostic reconstruction using this EBM on real data and show that additional diagnostics improves reconstruction error and generation quality. The energy surface is directly evaluated for an ill-posed inverse problem: inferring probe position from a time-series measurement. This inference illuminates symmetries in the data, potentially leading to a method of inquiry to supplement conventional data analysis. Trends in diagnostic signals are inferred via conditional sampling over machine inputs. In addition, this multimodal EBM is able to unconditionally reproduce all distributional modes, suggesting future potential in anomaly detection on the LAPD. Fundamentally, this work demonstrates the flexibility and efficacy of EBM-based generative modeling of laboratory plasma data, and showcases multiple practical uses of just a single trained EBM in the physical sciences.
Phil Travis, Troy Carter
Ergodic LLC, USA · Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
Reconstructing spatially resolved plasma dynamics from few sensors is essential for diagnostics, reduced-order modelling and control, yet remains difficult because the sparse measurements incompletely constrain multiscale, regime-dependent degrees of freedom. The Shallow Recurrent Decoder (SHRED) partially addresses spatial sparsity by using measurement histories; however, its fully connected decoder provides no explicit mechanism for resolving spatial structure across scales or explicit parametric dependency. We introduce the Recurrent Multiscale Affine-modulated Inference Network (ReMAIN), which preserves SHRED's recurrent temporal encoding but replaces its decoder with a U-Net whose feature hierarchy is conditioned by the recurrent state through feature-wise linear modulation. The temporal representation supplies both a dense prior and scale-specific modulation throughout the U-Net. A parametric extension jointly embeds the operating condition and sensor history, enabling reconstruction to adapt as the governing dynamics change with operating regime. ReMAIN is first benchmarked against SHRED on six one-dimensional nonlinear PDEs representing diverse dynamics. Across all benchmarks, it reduces reconstruction errors on unseen trajectories and more faithfully resolves sharp transitions, localized extrema and fine-scale variations. The parameter-conditioned model is then demonstrated on a collisionless E×B plasma subject to perpendicular axial electric and radial magnetic fields, with the electric-field strength serving as the operating parameter. ReMAIN reconstructs the high-dimensional, multiscale plasma state and recovers its regime-dependent spatiotemporal dynamics at electric-field strengths withheld from training. Together, ReMAIN improves sparse-sensor full-state reconstruction and, through parameter conditioning, generalizes across plasma operating regimes.
Maryam Reza, Farbod Faraji
Independent Researcher · Department of Computing, Huxley Building, Imperial College London, London SW7 2RH, United Kingdom
Real-time control of tokamak plasmas requires the electron density profile. However, in a reactor-relevant device, diagnostic port access is limited and the profile can be observed only partially. In the present work we reconstruct the density profile by inferring the coefficients of a physics-based one-dimensional particle transport equation, using a Fourier Neural Operator (FNO) surrogate of the transport solver as a fast differentiable forward model. The surrogate enables realtime-compatible gradient-based reconstruction and uncertainty quantification from the reflectometry and interferometry measurements. We compare four uncertainty quantification methods - multi-seed ensembles, the Laplace approximation, Metropolis-Hastings, and a Gaussian method independent of the reconstruction fit. The resulting uncertainty is scored with CRPS and coverage calibration, separately on the observed and unobserved regions. We further demonstrate advantage of our approach over standard analytical function fit on synthetic and experimental DIII-D discharges.