# Investigating the Physics of Tokamak Global Stability with Interpretable Machine Learning Tools

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## Abstract

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## Featured Application

**Machine-learning-based techniques have been applied to disruption prediction in Tokamaks and, by symbolic regression via genetic programming, physically meaningful equations have been extracted from the machine learning models, helping to investigate disruption physics and to transfer present-day knowledge to future devices.**

## Abstract

## 1. Disruptions in Tokamaks: An Operational Perspective

_{95}the percentage of disruptions can easily exceed 25%. To put this value in perspective, on DEMO even, one unmitigated disruption could severely damage the reactor [2].

_{95}, where q

_{95}is the safety factor at 95% of the plasma radius. The density limit is typically based on the Murakami factor n

_{e}R/B

_{T}, where n

_{e}is the mean electron density, R the plasma major radius and B

_{T}the toroidal field. For a large JET database with the ILW (see Section 4), the Hugill diagram is reported in Figure 1. Unfortunately, such a plot has very poor predictive and interpretative capability, since the disruptive and safe regions overlap almost completely in this space.

_{*}l

_{i}/(aB

_{T}), where I is the plasma current, l

_{i}the internal inductance and a the minor radius. The β-limit plot for various campaigns of JET with the ILW is reported in Figure 2. Again, inspection of the plot reveals that, in this space, it is practically impossible to separate the disruptive from the safe operational regions. Therefore, from a practical point of view, these representations have poor predictive and interpretative capability, at least for JET with the ILW.

- Training the machine learning tools for classification, i.e., to discriminate between disruptive and non-disruptive examples;
- Identifying a sufficient number of points on the boundary between safe and disruptive regions of the operational space on the basis of the models derived by the machine learning tools;
- Deploying Symbolic Regression via Genetic Programming to express the equation of the boundary in a physically meaningful form, using the points identified in the previous step;
- Converge on the final model with the help of the Pareto Frontier and non-linear fitting.

## 2. Classifiers and Symbolic Regression via Genetic Programming

#### 2.1. Support Vector Machines

**x**as disruptive or non-disruptive depends on the sign of this distance to the hyperplane [35].

#### 2.2. Classification and Regression Trees

#### 2.3. Symbolic Regression via Genetic Programming

## 3. Identification of the Boundary Equation between Safe and Disruptive Regions of the Operational Space

#### 3.1. Combining SVM and Symbolic Regression

#### 3.2. Combining CART Ensembles and Symbolic Regression

#### 3.3. Computational Requirements

^{4}× 51 takes of the order of 3 h. The GP calculation, including the building of the Pareto Frontier, is typically about one order of magnitude longer. Therefore, the computationally more demanding task is SR via GP step. In this respect, one should consider that genetic programs, of the type deployed in this application, are easy to parallelise. Consequently, very significant reductions in computational time can be expected if a more sophisticated version of the routines is implemented.

## 4. Database of JET with the ITER-Like Wall

## 5. A Data-Driven Model for JET with the ILW

_{i}amplitudes, the posterior probabilities have then been calculated as indicated in Section 2. The outputs of the ML classifiers have been investigated for a whole range of threshold probabilities. It turns out that the probability value, which provides the best performance for both classifiers, is around 80% for both techniques; this has been confirmed by a systematic scan of the classifiers developed for the present study. Therefore, the model trained with this threshold is the one whose results have been reported in the paper. It is worth mentioning that, for the interval (50%, 80%) of threshold probability, all the models give very similar results [27]. So, the choice of the threshold is not too critical for the purpose of the present paper, the identification of a manageable formula to describe the boundary between safe and disruptive regions of the operational space.

^{−4}Tesla, l

_{i}the internal inductance and the coefficients assume the values:

## 6. Deployment of the Proposed Approach in Support to Model Building

_{c}is the distance between the plasma centre and the location of the magnetic loops measuring the amplitude of the locked mode.

## 7. Conclusions and Future Prospects

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Definition of an initial function for the boundary;
- Generating samples of the two classes from the function;
- Training the SVM for classification;
- Building an appropriate mesh on the domain;
- Determining a sufficient number of points on the hypersurface identified by the SVM;
- Deploying symbolic regression to obtain the equation of the hypersurface from the points previously generated.

Steps | Values |
---|---|

Initial Function | y = sin(x_{1} + x_{2}) − 0.5 x_{3} x_{4} |

Ranges of Variables | −1.5 < x_{1} < 1.5 and −2 < x_{2} < 20 < x _{3} < 2 and 2 < x_{4} < 4 |

Number of Points for Each Class | 2000 |

Offset | 10% of y domain |

Classification Noise | ~5% |

**Table A2.**The accuracies obtained by the SVM for the train and test sets, by classifying the synthetic database with the best sigma that equals to 0.6.

Database Type | Classification Accuracy in Percent |
---|---|

Train Data | 96.1337 |

Test data | 96.0422 |

_{1}+ x

_{2}))−0.5010 x

_{3}x

_{4}

**Table A3.**The accuracies obtained for the train and test sets, by classifying the synthetic database with the expression obtained via GP, Equation (A1).

Database Type | Classification Accuracy in Percent |
---|---|

Train Data | 96.1060 |

Test data | 96.3061 |

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**Figure 1.**Hugill plot for a large database of JET with the ITER-like Wall (ILW) covering campaigns C29–C30. The disruptive and safe discharges overlap completely in this space. For the non-disruptive discharges, 14 samples during the flat top are shown. For the disruptive discharges, 14 samples, taken during the 210 ms before the beginning of the current quench, are reported.

**Figure 2.**The beta limit plot for a large database of JET with the ILW covering campaigns C29–C30. In addition, in this space the disruptive and safe discharges overlap completely. For the non-disruptive discharges, 14 samples during the flat top are shown. For the disruptive discharges, 14 samples, taken during the 210 ms before the beginning of the current quench, are reported.

**Figure 3.**Hypersurface points for a synthetic data set. The black and red points belong to the two classes. The blue points are the ones lying of the hypersurface, obtained with the method explained in the text and illustrated pictorially in the insert.

**Figure 4.**Overview of the database. Green triangles: disruptive examples. Black asterisks: safe discharges. The vertical line indicates the beginning of C31, the campaign used as test setT.

**Figure 5.**Plot of the safe and disruptive regions of the operational space in JET with the ILW according to the Support Vector Machine (SVM). The colour code represents the posterior probability of the classifier. The black circles are all the non-disruptive shots (10 random time slices for each shot). The red asterisks are the data of the disruptive shots at the time slice when the predictor triggers the alarm. The light blue squares are the false alarms. In red is the 80% disruption probability.

**Figure 6.**Plot of the safe and disruptive regions of the operational space in JET with the ILW according to the SVM. The colour code represents the posterior probability of the classifier. The black circles are all the non-disruptive shots (10 random time slices for each shot). The red asterisks are the data of the disruptive shots at the time slice when the predictor triggers the alarm. The light blue squares are the false alarms. In red is the 80% disruption probability.

**Figure 7.**Plots of the critical value of the locked mode, predicted by Equation (3), versus the various quantities used in the regression for the examples of campaigns C29–C30. Red circles: critical values of the locked mode at the moment of the alarm. Black crosses: critical values of the locked mode 15 ms before the beginning of the current quench. Green circles: critical values of the locked mode at the flat top of the same discharges. Before the beginning of the current quench, the overlap between the values of the locked mode is almost complete.

**Figure 8.**The boundary between the safe and disruptive regions of the operational space for campaigns C29 and C30. Top left: in the three dimensions LM, li and q

_{95}. Top right and bottom: the two projections. Blue safe points, red disruptive points. The hypersurface obtained with symbolic regression and non-linearly fitted to the data is shown in green. Only the first disruptive point of each discharge has been reported to help with visualizing the behaviour of the data.

**Table 1.**The figures of merit obtained using SVM (threshold probability 80%) and Equation (1) for campaigns C29–C30.

Method | Success Rate | Missed | Early | Tardy | False | Mean (ms) | Std (ms) |
---|---|---|---|---|---|---|---|

SVM | 97.9% (183/187) | 0% (0/187) | 0% (0/187) | 2.1% (4/187) | 2.8% (29/1020) | 335 | 344 |

CART Ensemble | 94.7% (177/187) | 0% (2/187) | 0% (0/187) | 4.3% (8/187) | 2.3% (23/1020) | 336 | 345 |

SR via GP Equation (1) | 97.9% (183/187) | 0% (0/187) | 0% (0/187) | 2.1% (4/187) | 2.7% (28/1020) | 336 | 345 |

**Table 2.**The figures of merit obtained with Equation (1) for campaign C31. These statistics have been obtained by applying the final model directly to campaign C31.

Method | Success Rate | Missed | Early | Tardy | False | Mean (ms) | Std (ms) |
---|---|---|---|---|---|---|---|

SR via GP Equation (1) | 92.21 (142/154) | 0.65 (1/154) | 1.30 (2/154) | 5.84 (9/154) | 7.34 (32/436) | 406 | 474 |

**Table 3.**The traditional figures of merit to assess the performance of predictors for the case of Equation (3) applied to campaigns C29–C30.

Method | Success Rate | Missed | Early | Tardy | False | Mean (ms) | Std (ms) |
---|---|---|---|---|---|---|---|

Equation (3) | 52.35% (89/170) | 24.12% (41/170) | 0% (0/170) | 23.53% (40/170) | 1.72% (17/987) | 175 | 328 |

Method | Success Rate | Missed | Early | Tardy | False | Mean (ms) | Std (ms) |
---|---|---|---|---|---|---|---|

Equation (4) | 91.7% (156/170) | 1.2% (2/170) | 0% (0/170) | 7.1% (12/170) | 2.3% (23/987) | 338 | 357 |

**Table 5.**Performance of Equation (4) for campaign C31. The models obtained at the end of C30 have been applied to the examples of C31 without any retraining.

Method | Success Rate | Missed | Early | Tardy | False | Mean (ms) | Std (ms) |
---|---|---|---|---|---|---|---|

Equation (4) | 92.21% (142/154) | 3.25% (5/154) | 0.65% (1/154) | 3.9% (6/154) | 11.93% (52/436) | 424 | 484 |

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**MDPI and ACS Style**

Murari, A.; Peluso, E.; Lungaroni, M.; Rossi, R.; Gelfusa, M.; JET Contributors. Investigating the Physics of Tokamak Global Stability with Interpretable Machine Learning Tools. *Appl. Sci.* **2020**, *10*, 6683.
https://doi.org/10.3390/app10196683

**AMA Style**

Murari A, Peluso E, Lungaroni M, Rossi R, Gelfusa M, JET Contributors. Investigating the Physics of Tokamak Global Stability with Interpretable Machine Learning Tools. *Applied Sciences*. 2020; 10(19):6683.
https://doi.org/10.3390/app10196683

**Chicago/Turabian Style**

Murari, Andrea, Emmanuele Peluso, Michele Lungaroni, Riccardo Rossi, Michela Gelfusa, and JET Contributors. 2020. "Investigating the Physics of Tokamak Global Stability with Interpretable Machine Learning Tools" *Applied Sciences* 10, no. 19: 6683.
https://doi.org/10.3390/app10196683