Source-linked AI summary
Physics-Informed Neural Networks to Infer the Perpendicular Energy Conductivity in the Scrape-Off Layer of Stellarator Devices
J. Gallego, P. Protopapas, A. Bustos, A. Alonso, S. Barquero, A. Baciero, I. Rivera, J. A. Moríñigo, R. Mayo-García
TL;DR
The paper addresses how to infer the SOL perpendicular conductivity dependence on density and temperature from sparse plasma profiles. It uses an inverse PINN constrained by a reduced transport equation, recovering synthetic conductivity with below-10% error in data-constrained regions and producing an exploratory TJ-II estimate. The results support inverse PINNs as a framework for estimating effective SOL transport coefficients within model and data-coverage limits.
Problem
Perpendicular SOL conductivity is difficult to derive from first principles, motivating inference of κ_perp(n,T) from sparse measurements.
Method
An inverse PINN jointly reconstructs temperature and density profiles and represents κ_perp(n,T), using profile data and a reduced 1D transport-equation residual.
Results
Errors were below 10% for synthetic κ_perp(n,T) recovery in the data-constrained region, while the TJ-II application achieved S_f = 0.82 and inferred κ_perp of 10^18–10^20 m^-1s^-1.
Takeaways & Limitations
Inverse PINNs can estimate effective SOL transport coefficients from sparse experimental profiles within the reduced-model and data-covered domain.
Takeaways & Limitations
The inferred conductivity should be interpreted mainly within the reduced 1D model assumptions and the region of (T,n) covered by data.
Abstract
from arXiv · showhide
In this work, we develop an inverse Physics-Informed Neural Network (PINN) framework to infer the dependence of the scrape-off layer (SOL) perpendicular heat conductivity on plasma density and temperature, $κ_\perp(n,T)$. The method combines radial profile measurements of electron density and temperature with the residual of a reduced one-dimensional SOL transport equation, so that the inferred conductivity is constrained by both the measurements and the underlying transport model. Three neural networks are trained simultaneously: two reconstruct the temperature and density profiles as functions of the radial coordinate and transported power, while a third represents the effective conductivity as a function of the local density and temperature. The framework is first validated using synthetic data generated from a prescribed conductivity function, allowing the inferred $κ_\perp(n,T)$ to be compared directly with the ground truth. The model recovers the imposed functional dependence with errors below $10~\%$ in the data-constrained region. Bootstrap resampling is shown to provide a practical indicator of prediction reliability and consistency. A scan in the number of plasma profiles used for training and the number of radial measurement positions per profile identifies a practical trade-off between reconstruction accuracy and data availability. Finally, the method is applied to an experimental dataset from the TJ-II stellarator obtained with the helium-beam diagnostic. This exploratory application provides an initial estimate of the effective SOL conductivity and illustrates the potential of inverse PINNs for extracting transport information from plasma edge measurements.
1 Introduction
The paper motivates inverse PINNs for inferring stellarator SOL perpendicular conductivity from sparse plasma-profile measurements. It targets the functional dependence κ_perp(n,T), combining measurements with a reduced 1D transport model.
- PINNs constrain neural-network solutions with governing physical laws in addition to available data, addressing weaknesses of purely data-driven approaches under sparse or unrepresentative measurements.
- Inverse PINNs recover unknown equation quantities from sparse solution observations, extending parameter inference from scalar coefficients to functional relations.
- Perpendicular and parallel conductivities and their dependencies are important for predicting target heat loads and wetted area because they determine SOL transport-channel width.
- κ_perp is difficult to derive from first principles because perpendicular SOL transport is largely governed by turbulence and may differ between stellarators and tokamaks.
- The work infers κ_perp(n,T) in stellarator SOLs using sparse density and temperature profiles and the residual of a reduced 1D transport model.
- The study validates the framework on synthetic data and applies it to a small TJ-II experimental dataset after examining robustness, profile counts, and radial measurement positions.
2 Background
The background develops a reduced sheath-limited SOL transport model and formulates inverse PINNs for recovering conductivity parameters and functions. It also emphasizes the simplifying assumptions and identifiability constraints underlying the approach.
- Reduced SOL transport model: The model assumes equal ion and electron temperatures, negligible ion parallel conduction, and negligible parallel electron conduction for the predominantly sheath-limited TJ-II SOL.
- Reduced SOL transport model: The SOL is straightened into perpendicular x and parallel y directions because its width is much smaller than the plasma minor radius.
- Reduced SOL transport model: In the reduced 1D description, parallel convective losses act as a local thermal-energy sink while perpendicular heat flux decreases radially toward the targets.
- Reduced SOL transport model: The parallel particle-flux loss is approximated using upstream and target quantities, with the connection length Lc setting the upstream-to-target scale.
- Reduced SOL transport model: The reduced equation is a strongly simplified SOL heat-transport description, potentially suited to small devices such as TJ-II but adaptable to more sophisticated models.
- Perpendicular transport scalings: Reference scalings include classical, Bohm, and gyro-Bohm forms, but core gyro-Bohm-like behavior is not clearly transferable to the SOL because open field lines and collisionality can alter transport.
- Inverse PINN formulation: Inverse PINNs combine sparse measurements with differential-equation residuals to learn unknown coefficients, while functional conductivity inference can require additional constraints to avoid non-uniqueness.
3 PINN model
The inverse PINN reconstructs temperature and density profiles while inferring κ⊥(n, T) through a third network constrained by measurements and a reduced 1D transport equation. Its architecture incorporates transported power to represent multiple plasma profiles, while weighted and adaptively balanced losses jointly guide training.
- 3.1 Proposed PINN architecture: The model simultaneously reconstructs temperature and density profiles and infers κ⊥(n, T), enforcing agreement with measurements and the reduced 1D transport equation.Three neural networks are trained together so the reconstructed profiles fit labelled data while the conductivity makes them consistent with the transport model.
- 3.1 Proposed PINN architecture: Including transported power Ptr as an input allows one PINN to represent profiles from different heating conditions rather than a single plasma scenario.The approach supports simultaneous training on several discharges when the magnetic configuration is fixed.
- 3.2 Network optimization: The total objective combines physics, temperature-data, density-data, and conductivity-boundary losses through explicit partial-loss weights.The conductivity reference values are imposed at the LCFS, where the heat-flux relation connects κ⊥ to transported power.
- 3.1 Proposed PINN architecture: The temperature, density, and conductivity components use ResNet networks, with automatic differentiation supplying derivatives for the transport residual and parameter updates.The profile networks take x and Ptr, while the conductivity network receives the predicted temperature and density profiles.
- 3.2 Network optimization: The partial-loss weights are initially normalized and then adaptively updated so their backpropagated gradients remain comparable during training.Exponential averaging of the adaptive factors reduces fluctuations and helps prevent noisy updates from destabilizing optimization.
4 Validation with synthetic data
Synthetic validation tests whether the inverse PINN can reconstruct temperature and density profiles and recover a prescribed perpendicular conductivity from noisy, sparse measurements. The conductivity error stays below 10% in the data-constrained region, while bootstrap variability tracks extrapolation uncertainty and additional profiles or positions improve accuracy until saturation.
- Synthetic data generation: The synthetic benchmark prescribes density and conductivity, then numerically solves the transport equation to generate temperature profiles for direct ground-truth comparison.The imposed conductivity is physically reasonable but is not intended as a definitive SOL transport scaling.
- Synthetic data generation: Neural networks reconstruct smooth temperature and density profiles from noisy measurements sampled at discrete radial positions and transported powers.The representative example uses three transported power values and eight radial positions.
- Conductivity reconstruction: Below 10% relative conductivity error is achieved in the region constrained by the synthetic training dataset.The inferred conductivity is compared pointwise with the known ground truth.
- Uncertainty and consistency: Bootstrap variability is lowest within the measurement-covered region and larger in extrapolated low-density, low-temperature regions, where relative errors also tend to increase.This agreement supports bootstrap resampling as a practical indicator of sampling sensitivity.
- Data sufficiency scan: Increasing the number of profiles and radial positions generally reduces conductivity MAPE, but accuracy saturates around NP = 5 and Nx = 6.Beyond this combination, additional data provide only marginal accuracy gains.
5 Application to experimental data
The TJ-II application tests the inverse PINN on sparse helium-beam measurements and produces an exploratory estimate of κ⊥(T,n), with uncertainty and physics-consistency diagnostics.
- Experimental scope: The limited TJ-II dataset is used to test real-data applicability and obtain an initial conductivity estimate, not to systematically characterize κ⊥ dependencies.The diagnostic provides Nx = 4 radial positions and NR = 2 measurements per discharge, limiting systematic experimental analysis.
- Diagnostic setup: The helium-beam diagnostic reconstructs spatially resolved SOL electron temperature and density profiles for the PINN analysis.It injects neutral helium and interprets line emission with a collisional-radiative model.
- Experimental reconstruction: The reconstructed profiles reproduce the measurements reasonably well, with bootstrap variations of ±2 eV in temperature and ±0.5 10^17 m−3 in density.For x ≳ 8 mm, density becomes weakly varying and low, providing an estimate of the measured SOL extent.
- Physics consistency: The physics-consistency score is 0.814, below the synthetic value of 0.998, indicating that the experimental transport may not be fully captured by the reduced equation.The authors interpret this difference cautiously because the experimental dataset is limited.
- Inferred conductivity: The inferred conductivity is approximately 10^18–10^20 m−1s−1 and is largest at high temperature and density near the LCFS.It decreases outward as temperature and density fall, while the low-density temperature dependence appears weaker and may reflect uncertainty or omitted physics.
- Cross-machine comparison: The normalized TJ-II conductivity is roughly eight times larger than reported W7-X values, motivating broader multi-machine validation.The comparison may partly reflect differing SOL transport regimes.
6 Conclusions
The inverse PINN framework infers SOL perpendicular heat conductivity from sparse profiles while enforcing a reduced 1D transport model. Synthetic validation and exploratory TJ-II application support its usefulness, with interpretation bounded by model assumptions and data coverage.
- The framework combines sparse temperature and density profiles with a reduced 1D transport equation, using three neural networks to reconstruct profiles and infer κ⊥(n, T).
- Synthetic validation recovered the prescribed conductivity dependence with errors below 10% in the data-constrained region, while bootstrap variability increased in weakly sampled areas.The reconstruction error saturated around NP = 5 and Nx = 6 under the tested synthetic conditions.
- The TJ-II exploratory application reproduced measurements reasonably well with Sf = 0.82 and inferred conductivity of 10^18–10^20 m−1s−1, corresponding to κ⊥/κgB of order 800.
- TJ-II’s normalized conductivity was roughly eight times larger than the reported W7-X value, possibly reflecting differences in SOL transport regimes and motivating broader multi-device analysis.
- The inferred conductivity should be interpreted mainly within the data-covered (T, n) domain and under the assumptions of the reduced 1D model.The paper identifies Bayesian uncertainty quantification, broader datasets, and refined geometry treatment as future directions.