Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Machine Learning for Electron-Scale Turbulence Modeling in W7-X

Ionut-Gabriel Farcas, Don Lawrence Carl Agapito Fernando, Alejandro Banon Navarro, Gabriele Merlo, Frank Jenko

TL;DR

This work develops ML-driven reduced surrogates for electron-temperature-gradient turbulence in the W7-X stellarator by mapping three key plasma parameters $(\omega_{T_e}, \eta_e, \uptau)$ to the normalized ETG heat flux $Q_e/Q_{GB}$ via a scaling law with four coefficients $(c_0,p_1,p_2,p_3)$. The authors combine sparse-grid initialization with active-learning-based iterative refinement and bootstrapping to quantify predictive uncertainty, achieving accurate predictions across seven radial locations and robust generalization to three additional radii. They explore two generalization strategies—coefficient regression for radius-dependent coefficients and a radius-independent generic model—finding that radius-specific coefficients yield superior accuracy, while the generic model performs well in some regions but not others. The results demonstrate reliable, uncertainty-aware reduced models that approach the accuracy of high-fidelity simulations, with potential applicability to rapid design studies and multi-query analyses in stellarator turbulence research.

Abstract

Constructing reduced models for turbulent transport is essential for accelerating profile predictions and enabling many-query tasks such as uncertainty quantification, parameter scans, and design optimization. This paper presents machine-learning-driven reduced models for Electron Temperature Gradient (ETG) turbulence in the Wendelstein 7-X (W7-X) stellarator. Each model predicts the ETG heat flux as a function of three plasma parameters: the normalized electron temperature radial gradient ($ω_{T_e}$), the ratio of normalized electron temperature and density radial gradients ($η_e$), and the electron-to-ion temperature ratio ($τ$). We first construct models across seven radial locations using regression and an active machine-learning-based procedure. This process initializes models using low-cardinality sparse-grid training data and then iteratively refines their training sets by selecting the most informative points from a pre-existing simulation database. We evaluate the prediction capabilities of our models using out-of-sample datasets with over $393$ points per location, and $95\%$ prediction intervals are estimated via bootstrapping to assess prediction uncertainty. We then investigate the construction of generalized reduced models, including a generic, position-independent model, and assess their heat flux prediction capabilities at three additional locations. Our models demonstrate robust performance and predictive accuracy comparable to the original reference simulations, even when applied beyond the training domain.

Machine Learning for Electron-Scale Turbulence Modeling in W7-X

TL;DR

This work develops ML-driven reduced surrogates for electron-temperature-gradient turbulence in the W7-X stellarator by mapping three key plasma parameters to the normalized ETG heat flux via a scaling law with four coefficients . The authors combine sparse-grid initialization with active-learning-based iterative refinement and bootstrapping to quantify predictive uncertainty, achieving accurate predictions across seven radial locations and robust generalization to three additional radii. They explore two generalization strategies—coefficient regression for radius-dependent coefficients and a radius-independent generic model—finding that radius-specific coefficients yield superior accuracy, while the generic model performs well in some regions but not others. The results demonstrate reliable, uncertainty-aware reduced models that approach the accuracy of high-fidelity simulations, with potential applicability to rapid design studies and multi-query analyses in stellarator turbulence research.

Abstract

Constructing reduced models for turbulent transport is essential for accelerating profile predictions and enabling many-query tasks such as uncertainty quantification, parameter scans, and design optimization. This paper presents machine-learning-driven reduced models for Electron Temperature Gradient (ETG) turbulence in the Wendelstein 7-X (W7-X) stellarator. Each model predicts the ETG heat flux as a function of three plasma parameters: the normalized electron temperature radial gradient (), the ratio of normalized electron temperature and density radial gradients (), and the electron-to-ion temperature ratio (). We first construct models across seven radial locations using regression and an active machine-learning-based procedure. This process initializes models using low-cardinality sparse-grid training data and then iteratively refines their training sets by selecting the most informative points from a pre-existing simulation database. We evaluate the prediction capabilities of our models using out-of-sample datasets with over points per location, and prediction intervals are estimated via bootstrapping to assess prediction uncertainty. We then investigate the construction of generalized reduced models, including a generic, position-independent model, and assess their heat flux prediction capabilities at three additional locations. Our models demonstrate robust performance and predictive accuracy comparable to the original reference simulations, even when applied beyond the training domain.

Paper Structure

This paper contains 12 sections, 8 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Target reduced models for ETG turbulence in W7-X
  3. Datasets for reduced model development and numerical validation
  4. Low-cardinality sparse grid datasets for constructing the initial reduced models
  5. Database for iterative model refinement and numerical validation at seven radial locations
  6. Database for three additional radial locations
  7. Active-learning-based iterative reduced model refinement
  8. Predictions using the proposed models
  9. Model generalization and predictions for three additional radial locations
  10. Summary and discussion
  11. Role of sparse grid data in constructing the active learning base models
  12. Regression formulae for the scaling law coefficients

Figures (11)

  • Figure 1: The Gene-KNOSOS-Tango framework evolves the plasma density ($n_e$) and pressure ($p_e,\;p_i$) profiles until good agreement between the fluxes and their respective sources (energy or particle) are obtained. A simulation database is created from the input parameters, profiles, and fluxes generated at each iteration $l$.
  • Figure 2: Evolution of reduced model predictions for four heat fluxes from the validation set as a function of $N_{\mathrm{train}}$ in the active learning procedure for $\hat{\rho} \in \{0.2, 0.5, 0.8\}$. The shaded regions represent the 95% prediction intervals computed using bootstrapping. For improved visualization, the very large prediction intervals from the first four iterations have been excluded.
  • Figure 3: Numerical validation of the reduced model \ref{['eq:red_model_x0_0_2']} at $\hat{\rho} = 0.2$. The left plot compares its deterministic predictions with all $N_{\mathrm{pred}}=547$ reference values. The right plot shows the predictions plus their $95\%$ prediction intervals computed via bootstrapping for a randomly selected subset of $40$ points for an easier visualization.
  • Figure 4: Numerical validation of the reduced model \ref{['eq:red_model_x0_0_3']} at $\hat{\rho} = 0.3$. The left plot compares its deterministic predictions with all $N_{\mathrm{pred}}=617$ reference values. The right plot shows the predictions plus their $95\%$ prediction intervals computed via bootstrapping for a randomly selected subset of $40$ points for an easier visualization.
  • Figure 5: Numerical validation of the reduced model \ref{['eq:red_model_x0_0_4']} at $\hat{\rho} = 0.4$. The left plot compares its deterministic predictions with all $N_{\mathrm{pred}}=506$ reference values. The right plot shows the predictions plus their $95\%$ prediction intervals computed via bootstrapping for a randomly selected subset of $40$ points for an easier visualization.
  • ...and 6 more figures