[Paper Review] Gaussian Process Regression models for the properties of micro-tearing modes in spherical tokamak

W. A. Hornsby, Ander Gray+10 more|arXiv (Cornell University)|Sep 18, 2023

Magnetic confinement fusion researchPhysics and Astronomy3 citations

TL;DR

This paper develops a Gaussian Process Regression (GPR)-based surrogate model for predicting micro-tearing mode (MTM) properties—growth rate, frequency, and normalized electron heat flux—in spherical tokamaks using data from high-fidelity gyrokinetic simulations. By combining GPR with an active learning loop and a Gaussian process classifier, the model achieves high accuracy with only ~5,000 simulations (≈1M CPU hours) across a 7D parameter space, enabling efficient design-space exploration for fusion power plants.

ABSTRACT

Spherical tokamaks (STs) have many desirable features that make them an attractive choice for a future fusion power plant. Power plant viability is intrinsically related to plasma heat and particle confinement and this is often determined by the level of micro-instability driven turbulence. Accurate calculation of the properties of turbulent micro-instabilities is therefore critical for tokamak design, however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities makes repeated use of direct numerical simulations in integrated modelling workflows extremely computationally challenging and creates the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a {\it surrogate model} for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilising Gaussian Process Regression (GPR) and classification; techniques from machine learning. These two components are used in an active learning loop to maximise the efficiency of data acquisition thus minimising computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency and normalised electron heat flux; core components of a quasi-linear transport model. Five-fold cross-validation and direct validation on unseen data is used to ascertain the performance of the resulting surrogate models.

Motivation & Objective

• To develop a fast, accurate, and uncertainty-aware surrogate model for micro-tearing mode (MTM) properties in spherical tokamaks, critical for fusion power plant design.
• To overcome the prohibitive computational cost of repeated high-fidelity gyrokinetic simulations (e.g., GS2) in integrated modeling workflows.
• To enable efficient exploration of a 7D reactor-relevant parameter space using active learning and machine learning techniques.
• To provide a reduced-order model with transparent confidence intervals for use in transport and scenario prediction tools.
• To support future integration into integrated modeling platforms like JINTRAC for full nonlinear turbulence prediction.

Proposed method

• Gaussian Process Regression (GPR) is used to model the linear MTM properties: growth rate, frequency, and normalized electron heat flux, trained on data from the GS2 gyrokinetic code.
• A separate Gaussian process classifier predicts the probability of instability in each point of the 7D parameter space, guiding active learning.
• An active learning loop prioritizes simulations in regions of high uncertainty or high instability probability, minimizing the number of required high-fidelity simulations.
• The model leverages the intrinsic uncertainty estimates of GPs to quantify prediction confidence and guide data acquisition.
• Model performance is validated via cross-validation and generalization on unseen test data, using metrics such as MSE, SMSE, and coverage of credible intervals.
• The framework is designed for extension to higher-dimensional parameter spaces, additional mode types (e.g., KBM, TEM), and full eigenfunction prediction.

Experimental results

Research questions

• RQ1 Can a data-driven surrogate model based on Gaussian Process Regression accurately predict the linear properties of micro-tearing modes in spherical tokamaks with minimal high-fidelity simulation cost?
• RQ2 How effectively can an active learning loop guided by a GP classifier reduce the number of required GS2 simulations in a high-dimensional parameter space?
• RQ3 To what extent does the surrogate model maintain accuracy and reliable uncertainty quantification across unseen regions of the parameter space?
• RQ4 Can the model be extended to cover broader parameter ranges and additional physical quantities, such as flux surface-averaged k⊥² or full eigenfunctions?
• RQ5 How well does the model generalize to other unstable modes (e.g., KBM, TEM) and integrate into existing integrated modeling tools like JINTRAC?

Key findings

• The surrogate model achieved high accuracy with only approximately 5,000 high-fidelity GS2 simulations across a 7D parameter space, requiring roughly 1 million CPU hours.
• The active learning loop significantly improved data acquisition efficiency by focusing simulations on regions of high instability probability and uncertainty.
• The model demonstrated strong generalization performance, with coverage of 95% credible intervals close to the nominal 0.95 level, indicating well-calibrated uncertainty estimates.
• The GP classifier achieved high precision and recall, with F1 scores approaching 1, indicating effective identification of unstable MTM regions.
• The model’s transparent uncertainty quantification enables targeted simulation campaigns to improve accuracy in poorly explored regions of the parameter space.
• The framework is extensible to include additional parameters, broader ranges (e.g., increasing kyρs to 7), and other mode types, supporting future integration into full integrated modeling workflows.