An Interesting Poster to look at from the Tri Alpha Energy Team in California

High Time Resolution Reconstruction of Electron Temperature Profiles with a Neural Network in C-2U
Gabriel Player, Richard Magee, Erik Trask, Sergey Korepanov, Ryan Clary and the TAE Team
Abstract
One of the most important parameters governing fast ion dynamics in a plasma is the electron
temperature, as the fast ion-electron collision rate goes as 𝝂 ~ 𝑻𝟑/𝟐. Unfortunately, the electron 𝒆𝒊 𝒆
temperature is difficult to directly measure—methods relying on high-powered laser pulses or fragile probes lead to limited time resolution or measurements restricted to the edge. In order to improve the time resolution of core electron temperature measurements in the C-2U database, a type of learning algorithm, specifically a neural network, was implemented. This network uses 3 hidden layers to correlate information from nearly 250 signals, including magnetic probes, interferometers, and several arrays of bolometers, with Thomson scattering data over the entire C-2U database, totalling nearly 20,000 samples. The network uses the Levenberg-Marquardt algorithm with Bayesian regularization to learn from the large number of samples and inputs how to accurately reconstruct the entire electron temperature time history at a resolution of 500 kHz, a huge improvement over the 2 time points obtainable by Thomson alone. These results can be used in many different types of analysis and plasma characterization—in this work, we use the network to quantify electron heating.
Motivation
 The project began as an attempt to determine neutral beam heating in the FRC core using beam termination and beam power scan experiments in C-2U
 Diagnostic coverage of Thomson scattering became an issue – it can take 100+ shots to construct a time-resolved Te signal
 In order to resolve this issue, a neural network was developed to reconstruct Te from other diagnostic signals
 The network was based on previous work by E. Trask, et. al. in [1]
Experimental Validation
 A comparison between Thomson measurements and the network output of ~40 similar C-2U shots shows excellent correlation with expected values
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TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
Network Structure & Performance The neural network connects input signals Thomson data
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Preliminary Application of Neural Net Data
Electron temperature response to neutral beam injection can be examined using two sets of experiments: beam terminations and beam power scans
In beam terminations, the electron heating by the neutral beams can be found as follows:
with a network of hidden layers
The C-2U network utilizes ~250 signals to determine plasma parameters, including:
Input Layer
Hidden Layers
Output Layer
 Magnetics: Axial magnetic probes, Mirnov probes  CO Interferometer
 𝑯 = 𝒅𝑻𝒆 𝒕<𝒕𝒃𝒕 − 𝒅𝑻𝒆 𝒕>𝒕𝒃𝒕, where tbt is the time 𝒅𝒕 𝒅𝒕
2
 Radial Bolometer arrays
of beam termination
The network was trained using Levenberg-Marquardt backpropagation with Bayesian regularization to prevent overfitting
High leverage inputs were determined through performance tests
After determination of a high-leverage input set, a genetic algorithm was implemented to finalize the inputs for each impact parameter
The final network utilizes Thomson error bars as additional targets, which (shown right) had a highly positive impact on performance
Similarly, beam power scans can determine the heating:
𝚫 𝐝𝐓𝐞
𝐝𝐭 ∗ 𝐈𝐛𝐞𝐚𝐦, where Ibeam is the beam current 𝚫𝐈𝐛𝐞𝐚𝐦
These heating values H, in eV/s, represent the electron heating due to neutral beams
Estimates of neutral beam electron heating tend to be lower than results from 0D calculations and Monte Carlo/MHD codes. Possible reasons for this discrepancy are:
 Underestimation of warm neutral density
 Compressional heating
 Neglecting beam heating of edge plasma
 Neural net bias towards high-power beam shots
Summary
An attempt to experimentally determine electron heating due to neutral beam injection highlighted a diagnostic gap – lack of time resolution on single-shot Thomson measurements
The proposed solution was to implement a neural network (similar to C-2 [1]) which could use learn the connections between time-resolved diagnostics with Te dependence and the Thomson measurements
The network was successfully implemented, and reconstructs Te traces with a time resolution of 10 μs
To display the utility of the network, the original problem, neutral beam heating of electrons, was revisited, to great effect – there are multiple ways to use the neural network in this analysis
Future work:
 C-2W network – update the network to run on C-2W diagnostics
 Continue neutral beam heating analysis using C-2U network output
 𝑯 = in kA
Network Results
 The left examines the relationship between Te and the number of active beams (or, equivalently, kA of beam current)
 The right displays the effects of beam termination at different times
Both plots also show a double hump Te distribution, which is expected due to fast ion orbits from
neutral beam injection [2]
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contours with a time resolution of 10 μs, and a spatial resolution The two plots below were both generated using output from the neural net:
The neural network produces T equivalent to that of the Thomson arrays
e
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References
[1] E. Trask, et. al. Electron Temperature Estimate in C-2 FRC Using Neural Network. Bull. Am. Phys. Soc. Vol. 58, (2013)
[2] M. Beall, et. al. Improved density profile measurements in the C-2U advanced beam-driven FRC plasma. Bull. Am. Phys. Soc. Vol. 60, (2015)