Imputation of faulty magnetic sensors with coupled Bayesian and Gaussian processes to reconstruct the magnetic equilibrium in real time

Abstract

A Bayesian with GP(Gaussian Process)-based numerical method to impute a few missing magnetic signals caused by impaired magnetic probes during tokamak operations is developed such that the real-time reconstruction of magnetic equilibria, whose performance strongly depends on the measured magnetic signals and their intactness, are affected minimally. Likelihood of the Bayesian model constructed with the Maxwell's equations, specifically Gauss's law for magnetism and Ampère's law, results in infinite number of solutions if two or more magnetic signals are missing. This undesirable characteristic of the Bayesian model is remediated by coupling the model with the Gaussian process. Our proposed numerical method infers nine non-consecutive missing magnetic signals correctly in less than 1 msec suitable for real-time reconstruction of magnetic equilibria during tokamak operations.

Single Gaussian Process Method for Arbitrary Tokamak Regimes with a Statistical Analysis

“…Here, rather than use raw experimental data as in prior work [33][34][35][36][37], we create synthetic data to enable a more accurate understanding of the errors associated with different GPR methods. Specifically, for L-mode profiles we use…”

“…The specific technique used in this paper will be Gaussian Process Regression, a technique that is becoming widely used in the uncertainty quantification and machine learning [32]. It's introduction into the fusion community was by Svensson [33] for soft X-ray tomography, but has since grown to include other diagnostics [33][34][35][36][37]. It's use within the fusion community has broadened to be used for verification and validation [38][39][40][41], and in equilibrium reconstruction as well [42,43].…”

“…For H-mode profiles, the short length scale of the pedestal region motivates the use of non-stationary kernels; e.g., as used in Chilenski [38]. Similarly, inferring magnetic data can be challenging because of rapid changes in the data requiring non-stationary kernels [35].…”

Deep neural network Grad–Shafranov solver constrained with measured magnetic signals

“…Since inferring the missing values is better than the null replacement [49], we resolve the issue by using the recently proposed imputation method [50] based on Gaussian processes [52] and Bayesian inference [51], where the likelihood is constructed based on Maxwell's equations. The imputation method infers the missing values fast enough, i.e.…”

“…This number becomes large rapidly, and it becomes formidable, if not impossible, to train the networks with reasonable computational resources. Since the magnetic pick-up coils are susceptible to damage, we have developed our networks to be capable of inferring a few missing signals of the magnetic pick-up coils in real time by invoking an imputation scheme [50] based on Bayesian probability [51] and Gaussian processes [52].…”