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

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 ($\omega_{T_e}$), the ratio of normalized electron temperature and density radial gradients ($\eta_e$), and the electron-to-ion temperature ratio ($\tau$). 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.

• The paper presents a machine-learning framework that develops reduced models to capture electron temperature gradient (ETG) turbulence in the W7-X stellarator.
• It leverages sparse grid initialization and active learning to efficiently sample high-fidelity gyrokinetic simulation data, achieving low prediction errors.
• The approach provides uncertainty-aware predictions with robust confidence intervals and parameter regression, outperforming generic models in localized regimes.

Machine Learning-Driven Reduced Models for Electron-Scale Turbulence in W7-X

Introduction and Motivation

The paper establishes a data-driven framework to construct and validate reduced models for electron temperature gradient (ETG) turbulence-driven transport in the Wendelstein 7-X (W7-X) stellarator. High-fidelity simulations, typically based on first-principles gyrokinetic codes, are foundational for studying turbulent transport but are computationally prohibitive for many-query tasks such as uncertainty quantification, extensive parameter scans, and iterative design optimization of fusion devices. Reduced models offer the computational efficiency needed for these applications, provided they maintain fidelity with the underlying physics.

A central innovation in the work is the development of ETG transport models grounded in three critical plasma parameters: normalized electron temperature gradient ($\omega_{T_e}$), the ratio of temperature and density gradients ($\eta_e$), and the electron-to-ion temperature ratio ($\tau$), leveraging systematic regression and active learning strategies over sparse and high-fidelity simulation data. The approach is generalizable and enables uncertainty quantification, advancing the predictive capabilities required for stellarator optimization.

Modeling Approach and Methodology

The authors propose a scaling-law-based reduced model for electron heat flux, expressed as:

$$Q_e/Q_{\mathrm{GB}} = c_0 \, \omega_{T_e}^{p_1} \left(\eta_e - 1\right)^{p_2} \tau^{p_3}$$

where $Q_{\mathrm{GB}}$ is the gyro-Bohm normalized flux. Coefficients $\{c_0, p_1, p_2, p_3\}$ are inferred for seven radial locations $\hat{\rho} \in \{0.2, ..., 0.8\}$ via regression in logarithmic space.

The training process is initialized using low-cardinality, structure-exploiting sparse grids. These grids leverage anisotropic influence of input parameters, targeting directions in parameter space most influential on turbulence properties. Initial models are thus efficiently fit on limited data.

Subsequent model refinement is driven by active learning, selecting new training points from a high-fidelity simulation database generated via the Gene-KNOSOS-Tango framework (Figure 1).

Figure 1: Gene-KNOSOS-Tango framework iteratively evolves plasma profiles and generates simulation data for model building.

Uncertainty in model predictions is estimated via nonparametric bootstrap, yielding $95\%$ prediction intervals. At each active learning step, the most informative input triplet is selected by maximizing the relative deviation of the prediction interval, independent of the flux magnitude. This point is added to the training set, and the model is retrained. The process iterates until both prediction uncertainty and error on the newly added data fall below prescribed thresholds.

Model Formulation and Parameter Trends

Initial fitting demonstrates a strong dependence on $\omega_{T_e}$, moderate dependence on $\eta_e$, and weaker (often negative) dependence on $\tau$. After active learning-based refinement on the pre-existing database, final fitted coefficients exhibit smooth radial dependency:

• $p_1$ decreases with radius, consistent with reduced TEM drive in the edge.
• $p_2$ (exponent of $(\eta_e-1)$) remains nearly constant.
• $p_3$ (exponent of $\tau$) is increasingly negative near the core but weakens toward the edge.

The authors provide explicit confidence intervals for all four coefficients at all seven locations, calculated via $10^4$ bootstrap replicates, demonstrating robustness and low estimator bias.

Validation and Generalization

Validation of each position-dependent model is performed on large out-of-sample datasets (typically $> 393$ points per location). For all positions, deterministic relative errors ($\epsilon_{\rm pred}$) are consistently below $0.18$, and the outlier rate—where prediction errors exceed $100\%$—is negligible.

Figure 2: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.2$.

Figure 3: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.3$.

Figure 4: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.4$.

Figure 5: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.5$.

Figure 6: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.6$.

Figure 7: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.7$.

Figure 8: Out-of-sample prediction and uncertainty intervals for $\hat{\rho} = 0.8$.

Prediction intervals are generally narrow, capturing the majority of the reference values. This performance exceeds previously reported results for ETG models in tokamak pedestals.

The work further explores model generalization to untrained radial positions ($\hat{\rho} \in \{0.1, 0.55, 0.75\}$) via:

1. Coefficient regression: Fitting explicit functions for each coefficient as a function of radius and evaluating at arbitrary $\hat{\rho}$.
2. Generic (position-independent) model: Fitting a single model over the aggregated data, with fixed coefficients.

   Figure 9: Generic reduced model performance across all seven positions.

   

   Figure 10: Comparison of generic and coefficient-regression model predictions at three additional radial positions.

Coefficient-regression-based models yield superior accuracy at new locations (errors $< 0.17$), while the generic model can underpredict in regions with distinct local turbulence characteristics.

Computational and Practical Considerations

The active learning framework is well suited for scenarios where simulation evaluation is costly and high-dimensional. By using uncertainty-aware selection, the model reaches saturation accuracy with $73$–$138$ training points per location. Bootstrapping enables robust confidence interval estimation without distributional assumptions.

The pre-existing simulation database, generated using the integrated Gene-KNOSOS-Tango workflow, was crucial for efficient data reuse and iterative model improvement. However, the approach can be extended to online settings where simulation runs are performed dynamically as new points are selected.

Resource requirements are modest for regression and bootstrapping, scaling linearly with the number of training points and bootstrap replicates. The framework is adaptable to larger parameter spaces as long as anisotropic input influence allows effective sparse grid exploration.

Theoretical and Practical Implications

This work represents one of the first systematic validations of machine-learning-driven ETG reduced turbulence models in stellarator geometry. Key implications are:

• The scaling-law model form is robust and physically interpretable; exponents capture the core drivers and stabilizers of ETG turbulence.
• Rigorous prediction intervals empower uncertainty-aware deployment in critical applications, facilitating reliable many-query use cases.
• Active learning strategies leveraging uncertainty estimates are essential for efficient surrogate construction in underexplored regimes such as stellarators.
• Position-dependent model coefficients are vital for high-fidelity flux prediction; generic models may introduce unacceptable bias in regions with distinct physical conditions.

Future Directions

The methodological framework is directly extensible to ion-scale turbulence and multichannel flux prediction (electron/ion heat and particle fluxes). Further, expanding database size, including more diverse magnetic configurations and heating regimes, would enable generalized models applicable to multiple stellarator devices.

Integration of convolutional neural networks for exploiting spatial structure, and formal analyses of model transferability across magnetic topologies, are promising future avenues (as highlighted by recent literature on machine learning in fusion turbulence modeling [landreman2025doesiontemperaturegradient]).

Conclusion

This work demonstrates that regression-driven, active learning methods can yield accurate, robust, and uncertainty-aware reduced models for ETG turbulent transport in W7-X. The systematic use of bootstrapped uncertainty for model refinement and validation sets a high standard for deploying data-driven surrogates in fusion modeling. Extrapolation via parameter regression outperforms generic approaches, emphasizing the need for localized physics-aware modeling in high-performance stellarators. The approach is primed for rapid adoption in coupled transport workflows, facilitating large-scale optimization and real-time profile prediction in experimental and design contexts.