Reconstruction of Unobserved Core Electron Density in Tokamak Plasmas with a Neural Operator Transport Surrogate and Uncertainty Quantification

D. Sorokin; A. Granovskiy; M. Stokolesov; E. Adishchev; M. Nurgaliev; G. Subbotin

physics.plasm-phOct 17, 2026

Reconstruction of Unobserved Core Electron Density in Tokamak Plasmas with a Neural Operator Transport Surrogate and Uncertainty Quantification

Authors: D. Sorokin, A. Granovskiy, M. Stokolesov, E. Adishchev, M. Nurgaliev, G. Subbotin

Organizations: Next Step Fusion, Bertrange, L-8070, Luxembourg

Abstract

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.

Figures & tables

Synthetic
Method	axis	core	full
FNO	$\mathbf{0.0565\pm 0.0450}$	$0.0596\pm 0.0334$	$0.0657\pm 0.0353$
mtanh	$0.080\pm 0.083$	$\mathbf{0.055\pm 0.051}$	$\mathbf{0.056\pm 0.058}$
DIII-D
FNO	$\mathbf{0.103\pm 0.076}$	$0.102\pm 0.035$	$0.106\pm 0.033$
mtanh	$0.161\pm 0.097$	$\mathbf{0.089\pm 0.054}$	$\mathbf{0.082\pm 0.053}$

Table 1: MAP relative

L^{2}

error, for the realistic-noise synthetic dataset with hidden plasma core (100 discharges) and for 13 experimental DIII-D discharges. core =

\rho<0.7

(unobserved); axis =

\rho<0.05

(unobserved).

Panel of Figure 1 — Figure 1: Reconstruction of the unobserved core over time, for synthetic discharge (left) and DIII-D discharge (right). Solid: true profile; dashed: FNO MAP reconstruction; shaded bands: 1 $\sigma$ /2 $\sigma$ posterior spread (multi-seed ensemble); color encodes time within the reconstructed window. Shaded $\rho$ region: observed edge ( $\rho\geq 0.7$ ).

FNO ( $n=100$ )
	CRPS		cov. $1\sigma$ (0.68)		cov. $2\sigma$ (0.95)
Method	core	edge	core	edge	core	edge
Multi-seed	$0.087\pm 0.108$	$0.091\pm 0.119$	$0.235\pm 0.258$	$0.212\pm 0.304$	$0.398\pm 0.296$	$0.334\pm 0.341$
Laplace	$0.091\pm 0.110$	$0.082\pm 0.124$	$0.262\pm 0.274$	$0.244\pm 0.298$	$0.561\pm 0.312$	$0.619\pm 0.354$
MCMC	$0.101\pm 0.109$	$0.105\pm 0.158$	$0.239\pm 0.181$	$0.128\pm 0.232$	$0.467\pm 0.273$	$0.259\pm 0.324$
Gaussian control	$\mathbf{0.082\pm 0.095}$	$\mathbf{0.075\pm 0.116}$	$\mathbf{0.660\pm 0.281}$	$\mathbf{0.336\pm 0.303}$	$\mathbf{0.921\pm 0.220}$	$\mathbf{0.780\pm 0.307}$

Table 2: Reconstruction and UQ quality, median

\pm

std over 100 synthetic discharges (top) and 13 DIII-D discharges (bottom), FNO only. Core = unobserved

\rho<0.7

; axis =

\rho<0.05

, the deepest extrapolation from the observed window; edge = observed

\rho\geq 0.7

. Best per column in bold. Both MCMC and Laplace are underconfident in the unobserved core at

1\sigma

. Reconstruction accuracy (rel. err) is identical across UQ methods within each model (shared MAP fit) and is reported in Table 1 .

Figure 2: Gaussian-method empirical vs. nominal coverage at the 50/70/90% levels, as a function of $\rho$ (median per level, binned into 20 $\rho$ intervals; shaded region marks the observed edge, $\rho\geq 0.7$ ), comparing the 100 synthetic discharges (left) against the 13 DIII-D discharges (right). IQR bands are omitted for readability – per-shot spread is wide and reported numerically as the $\pm$ std in Table 2 .

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Group	Parameter	Range
diffusion	$\rho_{\mathrm{ped}}$	$[0.7,0.9]$
	$D_{\mathrm{core}}$	$[0.3,3.0]$
	$D_{\mathrm{ped}}$	$[0.3,2.0]$
	$D_{\mathrm{edge}}$	$[0.05,0.5]$
pinch	$v_{\mathrm{core}}$	$0$ (fixed)
	$v_{\mathrm{ped}}$	$[-0.5,0.5]$

Appendix

Table 3: Prior ranges for the transport/source coefficients

\kappa

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Open paper AskPDF arXiv

The boundary and divertor plasma govern how a tokamak exhausts power and particles, setting heat fluxes, target conditions, and the onset of detachment. Predicting these quantities is essential for operating current and future devices, but edge simulations that resolve them are too slow for parameter scans, optimization, or real-time control. Machine-learning surrogates offer a fast alternative, yet most are forward-only: they cannot recover input parameters from observations or assess the reliability of their predictions. We introduce a cycle-consistent neural surrogate for edge plasmas, combining a conditional U-Net forward model with an optimization-based inverse method built on the frozen forward network. The forward model maps five control parameters to two-dimensional plasma-state fields on the SOLPS-ITER mesh; the inverse method enforces consistency between forward and inverse predictions, a self-supervised quality check needing no ground-truth labels at inference. An ensemble of multilayer perceptrons also predicts electron temperature and density profiles at the outboard midplane and divertor targets, with uncertainty estimates that flag where more simulations are needed. The forward model achieves normalized root-mean-square errors below 2.6% and Pearson correlations above 0.95 for all fields. Cycle-consistency regularization raises the average cyclical

R^2

from 0.59 to 0.99 without degrading forward accuracy and enables recovery of the core fueling rate; all five control parameters are recovered with Pearson

r\ge0.97

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k

-d tree warm start yields a database completion rate above 95%, versus roughly 30% outright failures when cold-started. With about

4\times10^6

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Abstract

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