The tokamak device, which uses powerful magnetic fields to confine the high-temperature plasma, is one of several types of magnetic confinement facilities being developed to produce controlled thermonuclear fusion power. Estimation of fusion plasma equilibrium parameters from many diagnostic signals is essential for efficient and safe operation of tokamak. To extract the plasma equilibrium parameters such as position, triangularity, internal inductance, and poloidal beta, several schemes have been proposed. The originally used schemes are the first-principles approaches based on a multifilament model or a control surface model for plasma.^[1,2] The currents flowing in the filaments or the current density distribution on the control surface are chosen in such a way as to best fit the measured fields and fluxes. In the development afterward, the plasma is also schematized by means of a set of finite element shape functions.^[3] It has an advantage over the multifilament model since the field singularities are automatically eliminated. However, the calculation amount is also increased. Via using the graphic processing unit (GPU), the fast equilibrium reconstruction has been implemented in Experimental and Advanced Superconducting Tokamak (EAST).^[4]

Recently, a new statistical approach is proposed to make use of a database that contains a large number of equilibria of the plasma and the corresponding measurements. Via applying this dataset statistical method, it is possible to find a function relating the plasma parameters to the external measurements.^[5,6] As one of the big-data-driven statistical methods, the artificial neural network (ANN) has shown its effectiveness in many applications in fusion research. The network estimation approach has no iteration process during prediction once the model has been trained well. We do not have to concern about the computational burden for real-time processing. And the need for the physical interpretation of the measurements will probably be much less important in future reactors.^[7] Under this background, a typical neural network approach has been introduced to establish the required non-linear mapping between the measured diagnostic signals and a finite set of parameters describing the plasma equilibrium. The technique has been successfully applied to single-null diverted discharge equilibria of DIII-D tokamak after having checked its practicability in the simple test case of a circular plasma.^[8] The aim of the present paper is to take a step forward in the application of the neural network approach for the identification of limiter and divertor plasma equilibria. In the remainder of this paper, we investigate the applicability of the new technique. In Section 2, the method of the artificial neural network is introduced, and the database generation procedure is briefly described. In Section 3, the details of the NN-based plasma equilibrium parameter estimation are depicted. And the results analysis and discussion are given in Section 4. In the end, summary and conclusions are drawn in Section 5.

As already pointed out, the estimation process is split into a preliminary classification of the kind of equilibrium (limiter or divertor) and subsequent inference of the equilibrium parameters. Both the classification network and the regression network have the same input parameters, i.e., the above mentioned 38 magnetic probes, 35 flux loops, 14 PF-coil currents, and plasma current. A single hidden layer is proved sufficient for these two tasks. We designed 20 neurons for the hidden layer, and the output layer depends on different tasks.

In the first step, the classification network shows perfect ability to distinguish between the limiter and the divertor configurations on the basis of electromagnetic measurements. The final train performance valued by the binomial cross-entropy function is up to 1.09 × 10⁻⁶. The classification network should give an output equal either to zero or one for limiter or divertor configurations, respectively. In the test dataset, we round the elements of outputs to the nearest integer. All the 498 limiter configurations and 1148 divertor configurations are classified correctly. The results achieved are extremely satisfactory. It has been discussed in Ref. [7] that the ability of ANN to form correct decision regions between the two classes is probably mainly related to the flux loops located near the limiter. Someone can try to use a much smaller number of measurements if there is a demand.

After that, for the subsequent step of inferring the plasma equilibrium parameters, we trained the regression neural network respectively for the limiter and divertor configurations. The R and Z coordinates of the magnetic axis, internal inductance, poloidal beta, major and minor radii, and elongation are chosen as outputs of the limiter plasma parameters inference, and for the divertor parameters inference, we added the safety factor at the magnetic axis, R and Z coordinates of the X point into the model outputs. Table 3 summarizes the global performance of the inference model in terms of MSE, and we list the inference accuracies achieved for the individual plasma parameters in Tables 4 and 5 for the limiter and the divertor cases, respectively. The mean absolute errors between the prediction values and target values are less than 1 mm or 1% for the dimensional and dimensionless parameters, respectively. The max absolute error and relative error also show good inference performance of the regression neural network model. The inference performance of the limiter case is a little bit worse than that of the divertor case, which is probably caused by the stronger non-linear relationship in the plasma ramp-up phase. In these models, all the 88 available electromagnetic measurements are used as the inputs. The prediction parameters are compared with the target values using test datasets.

Table 3

Table 3

Table 3 Summary of the global performance of the inference model in terms of MSE. . Total mean square error Training Testing Regression NN (divertor) 2.30 × 10−6 2.20 × 10−6 Regression NN (limiter) 1.49 × 10−5 1.65 × 10−5

Table 3 Summary of the global performance of the inference model in terms of MSE. .

Table 4

Table 4

Table 4 The inference accuracies achieved for the individual plasma parameters for the divertor case. . Equilibrium parameters (regression NN outputs) Mean absolute error/% Max absolute error/% Relative error/% Z of magnetic axis (Zm)/m 0.04 0.54 0.21 R of magnetic axis (Rm)/m 0.03 0.63 0.12 Safety factor (q0) 0.15 2.30 0.04 Poloidal beta (βp) 0.10 1.41 0.14 Internal inductance (li) 0.08 0.68 0.14 Major radius (R0)/m 0.03 0.51 0.13 Minor radius (a)/m 0.05 0.32 0.27 Elongation (κ) 0.19 2.31 0.19 R of X-point (RX)/m 0.05 0.46 0.70 Z of X-point (ZX)/m 0.04 0.41 0.26

Table 4 The inference accuracies achieved for the individual plasma parameters for the divertor case. .

Table 5

Table 5

Table 5 The inference accuracies achieved for the individual plasma parameters for the limiter case. . Equilibrium parameters (regression NN outputs) Mean absolute error/% Max absolute error/% Relative error/% Z of magnetic axis (Zm)/m 0.13 0.82 0.73 R of magnetic axis (Rm)/m 0.10 0.66 0.39 Poloidal beta (βp) 0.53 4.37 0.74 Internal inductance (li) 0.37 3.27 0.69 Major radius (R0)/m 0.09 1.07 0.37 Minor radius (a)/m 0.1 1.91 0.53 Elongation (κ) 0.13 0.96 0.13

Table 5 The inference accuracies achieved for the individual plasma parameters for the limiter case. .

Figure 3 shows the comparison of divertor plasma equilibrium parameters and their deviations. The limiter inference result is drawn in Fig. 4. For all the inference parameters, the target and prediction values coincide well and the errors are pretty small. Basically, for both the limiter and divertor plasmas, a generally good estimation for the plasma equilibrium parameters can be obtained via the neural network model. Reference [7] investigated the inference model with reduced inputs. The inference performance was found a little decreased but still acceptable when using only half sensors as the inputs. We can also choose fewer measurements for reduced model training according to actual accuracy requirements and available diagnostic signals. Here, we just show the best performance with sufficient inputs. The robustness to noise and possible faults is analyzed and discussed in the next section using the divertor equilibrium parameter inference model with sufficient inputs.

Fig. 3

	Figure Option ViewDownloadNew Window
	Fig. 3 Divertor prediction values and target values, and their deviations.

Fig. 4

	Figure Option ViewDownloadNew Window
	Fig. 4 Limiter prediction values and target values, and their deviations.

In real experiments, the electromagnetic measurements, which are used as the model inputs, may contain noise or faults. In order to assess the robustness of the estimation, we have also attempted to analyze the behavior of the neural network model when the input values are affected by possible faults and measurement noise. The assumed experimental noise level for the EAST electromagnetic measurements is less than 3%. We have trained and tested the regression neural network model with different noise amplitudes from 0 to 3%. As shown in Fig. 5(a), the color bar presents the logarithm value of test MSE performance of a trained model. The top line shows the performance degradation of the noiseless-trained model. We can discover that the model trained with no input noise cannot maintain its performance when there is noise in the test input. In the red frame, when the models are trained with input noise from 0.1% to 0.7%, the noise immunity of these models is improved and keeps the logarithm value of total test MSE performance still smaller than −5. It is the best performance region of the regression neural network model. Adding noise to model training inputs also has a negative influence on the model performance. We can see in Fig. 5(b), the blue line shows the noiseless-test performance degradation when the model is trained with noise from 0 to 3%. The logarithm value of models’ noise-test performance is decreased as the noise in the training inputs is increased. Meanwhile, the yellow line shows the mean values of the test results with 3% noise and the red line denotes the test results with noise from 0.1% to 3%, which indicates that the noise immunity of the model is improved. We need to make a trade between the model total performance and its noise immunity according to the actual application. In Fig. 6, the inference values of R and Z coordinates of the magnetic axis (we only choose a part of divertor equilibrium parameters due to limited space) are compared with the target values. The models, which are trained with noiseless and 0.5% noised inputs, are both tested with 0.5% noised inputs. The red symbols present the good prediction of the noise-trained model while the bad prediction of the noiseless-trained model is denoted with black symbols.

Fig. 5

	Figure Option ViewDownloadNew Window
	Fig. 5 Test performance comparison for the model trained with different noised inputs.

Fig. 6

	Figure Option ViewDownloadNew Window
	Fig. 6 Divertor parameter inference; prediction values and target values of R and Z coordinates of the magnetic axis; noiseless/0.5% noised training, test with noised inputs.

We have also trained a model with three measurements absent while 0.5% percent noise in training inputs and testing inputs is still assumed (green star in Fig. 5(a)). The three absent measurements are randomly chosen in the magnetic probes, flux loops, and PF-coil currents, respectively. It is a primary test for possible faults in measurements. We retrained the model, and the final model performance is tested with input data which has noise and faults. Its global performance and the inference accuracies achieved for the individual plasma parameters are shown in Table 6. The mean absolute error of the dimension parameter is less than 1 mm while the mean absolute error of dimensionless parameter is less than 1%. The max absolute error and relative error also are satisfactory small.

Table 6

Table 6

Table 6 Random faults and noised training, final model performance. . Total mean square error, regression NN (divertor, 3 random faults, 0.5% input noise) Training Testing 1.04 × 10−5 1.16 × 10−5 Equilibrium parameters Mean absolute error/% Max absolute error/% Relative error/% Z of magnetic axis (Zm)/m 0.07 0.46 0.39 R of magnetic axis (Rm)/m 0.06 0.54 0.23 Safety factor (q0) 0.30 3.74 0.07 Poloidal beta (βp) 0.25 1.54 0.35 Internal inductance (li) 0.16 0.80 0.31 Major radius (R0)/m 0.06 0.50 0.24 Minor radius (a)/m 0.07 0.48 0.39 Elongation (κ) 0.61 3.35 0.62 R of X-point (RX)/m 0.06 0.37 0.99 Z of X-point (ZX)/m 0.06 1.05 0.39

Table 6 Random faults and noised training, final model performance. .

In the end, the speed of the inference model based on neural network is discussed as it aims to control application. After the model training has been finished, the topology and parameters of the neural network are settled. For plasma equilibrium parameters inference with presupposed inputs, there are several algebraic calculation processes such as normalization, matrix multiplication, hyperbolic tangent function, and anti-normalization. The total inference time of the neural network model is up to an order of 1 millisecond when running it on an ordinary laptop. Neural network estimation without iteration process is faster than EFIT and some other iterative solvers when they are implemented by traditional CPU computer. With the improvement of computer performance, the elapsed time can be reduced. For the EAST plasma control system, the experimental requirement for shape control is about 1 ms and the neural network model can be promisingly used for it. In case much shorter execution time is required, someone can use a parallel architecture or direct hardware for the implementation of the neural network model.

In this paper, the classification neural network and the regression neural network are both used for estimation of plasma equilibrium parameters. We split the estimation process into a preliminary equilibrium classification and a subsequent inference of the equilibrium parameters. The classification network correctly classifies all the divertor and limiter equilibria in the test dataset. For the parameter inference task, the regression network also shows good performance even with measurement noise and possible faults. The model trained with 0.5% noise is chosen after we trade the global inference performance of the model and its noise immunity. The possible faults in measurements are also tested in the end. The neural network model is proved to be robust with a small number of measurements absent. In conclusion, the method looks very promising especially for the real-time estimation of the plasma equilibrium parameters.