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

We develop neural networks that not only output a value of $\psi$ but also satisfy equation (1), the GS equation. With a total of 94 input nodes (91 for the plasma current and magnetic signals, two for the R and Z position, and one for the bias) and one output node for a value of $\psi$, each network has three fully connected hidden layers with an additional bias node at each hidden layer. Each layer contains 61 nodes including the bias node. The structure of our networks is selected by examining several different structures by trial and error.

Denoting the value of $\psi$ calculated by the networks as $\psi^{\rm NN}$, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:nn_structure} \nonumber &\psi^{\rm NN} = s_{0} + \sum_{l=1}^{60} s_{l} \nonumber \\ &\times f \left(u_{l0} + \sum_{k=1}^{60} u_{lk} f \left(v_{k0} + \sum_{j=1}^{60} v_{kj} f \left(w_{j0} + \sum_{i=1}^{93} w_{ji} x_{i} \right) \right) \right), \nonumber \end{align} \tag{ 2 }$$

where x_i is the ith input value with $i=1,\dots,93$, i.e. 91 measured values with various magnetic diagnostics and two for the R and Z positions. w_ji is an element in a $61\times 94$ matrix, whereas $v_{kj}$ and u_lk are elements in $61\times 61$ matrices. s_l connects the $l^{\rm th}$ node of the third (last) hidden layer to the output node. $w, v, u$ and s are the weighting factors that need to be trained to achieve our goal of obtaining accurate $\psi$. $w_{j0}, v_{k0}, u_{l0}$ and s₀ are the weighting factors connecting the biases, where values of all the biases are fixed to unity. We use a hyperbolic tangent function as the activation function f  giving the network non-linearity [54]:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:NN-actfcn} f \left(t \right) = \tanh(t) = \frac{2}{1 + {\rm e}^{-2t}} - 1. \nonumber \end{align} \tag{ 3 }$$

The weighting factors are initialized as described in [55] so that a good training can be achieved. They are randomly selected from a normal distribution whose mean is zero with the variance set to be an inverse of the total number of connecting nodes. For instance, our weighting factor w connects the input layer (94 nodes with bias) and the first hidden layer (61 nodes with bias), therefore the variance is set to be $1/(94+61)$. Likewise, the variances for $v$, u and s are $1/(61+61)$, $1/(61+61)$ and $1/(61+1)$, respectively.

With the aforementioned network structure, training (or optimizing) the weighting factors to predict the correct value of $\psi$ highly depends on the choice of cost function. A typical choice of such cost function would be:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:costfcn} \epsilon = \frac{1}{N} \sum_{i=1}^{N} \left(\psi_{i}^{\rm NN} - \psi_{i}^{\rm Target} \right)^{2}, \nonumber \end{align} \tag{ 4 }$$

where $\psi^{\rm Target}$ is the target value, i.e. the value of $\psi$ from the off-line EFIT results in our case, and N the number of data sets.

As will be shown shortly, minimizing the cost function $\newcommand{\e}{{\rm e}} \epsilon$ does not guarantee to satisfy the GS equation (equation (1)) even if $\psi^{\rm NN}$ and $\psi^{\rm Target}$ match well, i.e. the network is well trained with the given optimization rule. Since $\Delta^*\psi$ provides information on the toroidal current density directly, it is important that $\Delta^*\psi^{\rm NN}$ matches $\Delta^*\psi^{\rm Target}$ as well. We have an analytic form representing $\psi^{\rm NN}$ as in equation (2); therefore, we can analytically differentiate $\psi^{\rm NN}$ with respect to R and Z, meaning that we can calculate $\Delta^*\psi^{\rm NN}$ during the training stage. Thus, we introduce another cost function:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:costfcn_plusdiff} \nonumber \epsilon^{\rm new}& = \frac{1}{N} \sum_{i=1}^{N} \left(\psi_{i}^{\rm NN} - \psi_{i}^{\rm Target} \right)^{2} \nonumber \\ &\quad+ \frac{1}{N} \sum_{i=1}^{N} \left(\Delta^*\psi_{i}^{\rm NN} - \Delta^*\psi_{i}^{\rm Target} \right)^{2}, \nonumber \end{align} \tag{ 5 }$$

where we obtain the value of $\Delta^*\psi^{\rm Target}$ from the off-line EFIT results as well.

To acknowledge the difference between the two cost functions $\newcommand{\e}{{\rm e}} \epsilon$ and $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$, we first discuss the results. Figure 3 shows the outputs of the two trained networks with the cost function (a) $\newcommand{\e}{{\rm e}} \epsilon$ and (b) $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$. It is evident that in both cases the network output $\psi^{\rm NN}$ (red dashed line) reproduces the off-line EFIT $\psi^{\rm Target}$ (black line). However, only the network trained with the cost function $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$ reproduces the off-line EFIT $\Delta^*\psi^{\rm Target}$. Both networks are trained well, but the network with the cost function $\newcommand{\e}{{\rm e}} \epsilon$ does not achieve our goal of correctly predicting $\psi^{\rm Target}$ and $\Delta^*\psi^{\rm Target}$.

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Since our goal is to develop a neural network that solves the GS equation, we choose the cost function to be $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$ to train the networks. We determine the weighting factors found by minimizing the cost function $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$ with Adam optimizer [56] which is one of the gradient-based optimization algorithms. With 90% and 10% of the total data samples for training and validation of the networks, respectively, we stop training the networks with a fixed number of iterations that is large enough but not too large such that the validation errors do not increase, i.e. to avoid overfitting problems. The whole workflow is carried out with Python and Tensorflow [57].

With the selected cost function we create three different networks that differ only by the training data sets. NN₂₀₁₇, NN₂₀₁₈ and NN_{2017, 2018} refer to the three networks trained with the data sets from only 2017 (744 discharges), from only 2018 (374 discharges) and from both 2017 and 2018 (744  +  374 discharges) campaigns, respectively.

The descending feature of the cost function $\newcommand{\e}{{\rm e}} \epsilon^{\rm new}$ as a function of the training iteration for the NN_2017,2018 network is shown in figure 4. Both the training errors (blue line) and validation errors (red dashed line) decrease together with similar values which means that the network is well generalized. Furthermore, since the validation errors do not increase, the network does not have an overfitting problem. Note that fluctuations in the errors, i.e. standard deviation of the errors, are represented as shaded areas.

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Small undulations repeated over the iterations in figure 4 are due to mini-batch learning. Contrary to batch learning, i.e. optimizing the network with the entire training set in one iteration, mini-batch learning divides the training set into a number of small subsets (1000 subsets in our case) to optimize the networks sequentially. One cycle that goes through all the subsets once is called an epoch. Mini-batch learning helps to escape from local minima in the weighting factor space [58] via the stochastic gradient descent scheme [59].

Figure 5 show the benchmark results of the NN_2017,2018 network, i.e. network trained with the data sets from both 2017 and 2018 campaigns. (a) and (b) show the results with the test discharges from the 2017 campaign, while (c) and (d) present the results with the test discharges from the 2018 campaign. Histograms of (a)(c) $\psi^{\rm NN}$ versus $\psi^{\rm Target}$ and (b)(d) $\Delta^*\psi^{\rm NN}$ versus $\Delta^*\psi^{\rm Target}$ are shown with colors representing the number of counts. For instance, there is a yellow colored point in figure 5(a) around $(-0.1, -0.1)\pm\varepsilon$, where $\varepsilon$ is a bin size for the histogram. Since yellow represents about $2\times 10^5$ counts, there are approximately $2\times 10^5$ data whose neural network values and EFIT values are $-0.1\pm\varepsilon$ simultaneously within our test data set. Note that each KSTAR discharge contains numerous time slices whose number depends on the actual pulse length of a discharge, and each time slice generates a total of $22\times 13=286$ data points. The values of $\psi^{\rm Target}$ and $\Delta^*\psi^{\rm Target}$ are obtained from the off-line EFIT results. It is clear that the network predicts the target values well.

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As a figure of merit, we introduce the R² metric (coefficient of determination) defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm R}^2 = 1 - \frac{\sum\nolimits_{i=1}^{L} \left(y_i^{\rm Target} - y_i^{\rm NN} \right)^2}{\sum\nolimits_{i=1}^{L} \left(y_i^{\rm Target} - \frac{1}{L}\sum\nolimits_{j=1}^{L} y_{j}^{\rm Target} \right)^2}, \nonumber \end{align} \tag{ 6 }$$

where y  takes either $\psi$ or $\Delta^*\psi$, and L is the number of test data sets. The calculated values are written in figure 5, and they are indeed close to unity, implying the existence of very strong linear correlations between the predicted (from the network) and target (from the off-line EFIT) values. Note that R²  =  1 means the perfect prediction. The red dashed lines in the figures are the y   =  x lines.

Figure 6 is an example of reconstructed magnetic equilibria using KSTAR shot #18057 from the 2017 campaign. (a) shows the evolution of the plasma current. The vertical dashed lines indicate the time points where we show and compare the equilibria obtained from the network (red) and the off-line EFIT (black) which is our target. (b) and (c) are taken during the ramp-up phase, (d) and (e) during the flat-top phase, and (f ) and (g) during the ramp-down phase. In each sub-figure from (b) to (g), the left panels compare $\psi$, and the right panels are for $\Delta^*\psi$. We mention that the equilibria in figure 6 are reconstructed with $65 \times 65$ grid points even though the network is trained with $22 \times 13$ grid points, demonstrating how the spatial resolution is flexible in our networks.

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For a quantitative assessment of the network, we use an image-relevant figure of merit that is the PSNR [60] (see appendix B), originally developed to estimate the degree of artifacts due to image compression compared to an original image. The typical PSNR range for a JPEG image, which preserves the original quality with a reasonable degree, is generally 30–50 dB [34, 61]. For our case, the network errors relative to the off-line EFIT results can be treated as artifacts. As listed in figures 6(b)–(g), the PSNR for $\psi$ is very good, while we achieve acceptable values for $\Delta^*\psi$.

Similar to shown in figure 5, we show the benchmark results of NN₂₀₁₇ and NN₂₀₁₈ in figures 7 and 8, respectively. The R² metric is also provided in the figures. Again, the overall performance of the networks is good.

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The NN₂₀₁₇ and NN₂₀₁₈ networks are trained with only in-campaign data sets, e.g. NN₂₀₁₈ with data sets from only the 2018 campaign, and we find slightly poorer results, but still good, in predicting cross-campaign magnetic equilibria, e.g. NN₂₀₁₈ predicting equilibria for the 2017 campaign. Notice that NN₂₀₁₇ seems to predict cross-campaign equilibria better than the in-campaign ones by comparing figures 7(a) and (c), which contradicts our intuition. Although the histogram in figure 7(c) seems tightly aligned with the y   =  x line (red dashed line), close inspection reveals that the NN₂₀₁₇ network, in general, underestimates the off-line EFIT results from the 2018 campaign marginally. This will be evident when we compare image quality.

MSSIM [62] (see appendix B) is another image relevant figure of merit used to estimate perceptual similarity (or perceived differences) between the true and reproduced images based on the inter-dependence of adjacent spatial pixels in the images. The MSSIM ranges from zero to one, where the closer to unity the better the reproduced image is.

Together with the PSNR, figure 9 shows the MSSIM for (a) NN₂₀₁₇, (b) NN₂₀₁₈ and (c) NN_2017,2018 where the off-line EFIT results are used as a reference. Notice that counts in all the histograms of MSSIM and PSNR in this work correspond to the number of reconstructed magnetic equilibria (or a number of time slices) since we obtain a single value of MSSIM and PSNR from one equilibrium, whereas counts in figures 5, 7 and 8 are much bigger since $286(=22\times 13)$ data points are generated from each time slice. The red (green) line indicates the test results for the data sets from the 2017 (2018) campaign. In general, whether the test data sets are in-campaign or cross-campaign, the image reproducibility of all three networks, i.e. predicting the off-line EFIT results, is good as attested by the fact that the MSSIM is quite close to unity and the PSNR for $\psi$ ($\Delta^*\psi$) ranges approximately from 40 to 60 (20 to 40). It is easily discernible that the in-campaign results are better for both NN₂₀₁₇ and NN₂₀₁₈ unlike what we noted in figures 7(a) and (c). Not necessarily guaranteed, we find that the NN_2017,2018 network works equally well for both campaigns as shown in figure 9(c).

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It is widely recognized that rt-EFIT results and off-line results are different from each other. If we allow the off-line EFIT results used to train the networks to be accurate ones, then the reconstruction of equilibria with the neural networks (nn-EFIT) must satisfy the following criterion: nn-EFIT results must be more similar to the off-line EFIT results than the rt-EFIT results are to the off-line EFIT, as mentioned in section 1. Once this criterion is satisfied, then we can always improve the nn-EFIT as genuinely more accurate EFIT results are collected. For this reason, we make comparisons among the nn-EFIT, rt-EFIT and off-line EFIT results.

Figure 10 shows an example of the reconstructed magnetic equilibria for (a) rt-EFIT versus off-line EFIT and (b) nn-EFIT (the NN_2017,2018 network) versus the off-line EFIT for KSTAR shot #17975 at 0.7 s with $\psi$ (left panel) and $\Delta^*\psi$ (right panel). The green, red and black lines indicate rt-EFIT, nn-EFIT and off-line EFIT results, respectively. This simple example shows that the nn-EFIT is more similar to the off-line EFIT than the rt-EFIT is to the off-line EFIT, satisfying the aforementioned criterion.

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To validate the criterion statistically, we generate histograms of the MSSIM and PSNR for the nn-EFIT and the rt-EFIT with reference to the off-line EFIT. This is shown in figure 11 as histograms, where the MSSIM (left panel) and PSNR (right panel) of $\psi$ (top) and $\Delta^*\psi$ (bottom) are compared between the nn-EFIT (black) and the rt-EFIT (green). Here, the nn-EFIT results are obtained with the NN_2017,2018 network on the test data sets. We confirm that the criterion is satisfied with the NN_2017,2018 network as the histograms in figure 11 are in favor of the nn-EFIT, i.e. larger MSSIM and PSNR are obtained by the nn-EFIT. This is more conspicuous for $\Delta^*\psi$ than $\psi$.

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We perform similar statistical analyses for the other two networks, NN₂₀₁₇ and NN₂₀₁₈, which are shown in figures 12 and 13. Since these two networks are trained with the data sets from only one campaign, we show the results where the test data sets are prepared from the (a) 2017 campaign and (b) 2018 campaign so that in-campaign and cross-campaign effects can be assessed separately. We find that the criterion is fulfilled for both $\psi$ and $\Delta^*\psi$ with both the in- or cross-campaign data.

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If one or more magnetic pick-up coils forming part of the input to the nn-EFIT are impaired, then we will have to re-train the network without the damaged ones, or hope that the network will reconstruct equilibria correctly by padding a fixed value, e.g. zero-padding, to the broken ones. Of course, one can anticipate training the network by considering possible combinations of impaired magnetic pick-up coils. With a total number of 68 signals from the magnetic pick-up coils being input to the network in our case, we immediately find that the number of possible combinations increases too quickly to consider it as a solution.

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. less than 1 msec to infer up to nine missing values on a typical personal computer; thus, we can apply the method during plasma discharge by replacing the missing values with the real-time inferred values.

We have applied the imputation method to KSTAR shot #20341 at 2.1 s for the normal ($B_{\rm n}$) and tangential ($B_{\rm t}$) components of the magnetic pick-up coils as an example. We have randomly chosen nine signals from the 32 $B_{\rm n}$ measurements and another nine from the 36 $B_{\rm t}$ measurements and pretended that all of them (9  +  9) are missing simultaneously. Figure 14 shows the measured (blue open circles) and the inferred (red crosses with their uncertainties) values for (a) $B_{\rm n}$ and (b) $B_{\rm t}$. Probe # on the horizontal axis is used as an identification index of the magnetic pick-up coils. Table 2 provides the actual values of the measured and inferred ones for better comparisons. We find that the imputation method infers the correct (measured) values very well except probe #37 of $B_{\rm n}$. The inferred (missing) probes are probes #3, 4, 6, 14, 18, 24, 30, 35, 37 for $B_{\rm n}$ and probes #4, 6, 8, 11, 17, 29, 30, 32, 35 for $B_{\rm t}$. Here, we provide all the probe #'s used for the neural network: $B_{\rm n}$ probe #[2, ..., 6, 8, 9, 11, ..., 15, 17, ..., 20, 23, ..., 26, 28, ..., 32, 34, 35, 37, ..., 41] (a total of 32); and $B_{\rm t}$ probe #[2, ..., 6, 8, 9, 11, ..., 32, 34, 35, 37, ..., 41] (a total of 36).

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Table 2. The imputation results shown in figure 14 with KSTAR shot #20341 at 2.1 s.

Bn (T) $\times 10^{-2}$			Bt (T) $\times 10^{-2}$
No.	Measured	Inferred	No.	Measured	Inferred
3	−1.45	−1.88 $\pm$ 0.22	4	−14.69	−13.97 $\pm$ 0.47
4	−1.72	−2.31 $\pm$ 0.24	6	−12.38	−11.42 $\pm$ 0.97
6	4.62	4.45 $\pm$ 0.65	8	−7.82	−7.88 $\pm$ 0.67
14	6.13	6.36 $\pm$ 0.27	11	−3.15	−3.22 $\pm$ 0.65
18	−8.27	−8.11 $\pm$ 0.48	17	0.10	0.30 $\pm$ 0.52
24	1.86	1.65 $\pm$ 0.30	29	3.84	2.65 $\pm$ 0.64
30	−7.52	−7.19 $\pm$ 0.18	30	1.15	0.49 $\pm$ 0.61
35	−7.93	−7.08 $\pm$ 0.65	32	−2.65	−2.11 $\pm$ 0.62
37	−4.27	−1.41 $\pm$ 0.93	35	−8.07	−8.87 $\pm$ 0.55

Comparisons between the nn-EFIT without any missing values, which we treat as reference values, and the nn-EFIT with the imputation method or with the zero-padding method are made. Here, the nn-EFIT results are obtained using the NN_2017,2018 network. The top panel of figure 15 shows $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \psi\lp R, Z\rp$ obtained from the nn-EFIT without any missing values (black line) and from the nn-EFIT with the two missing values replaced with the inferred values (green line), i.e. the imputation method, or with zeros (pink dashed line), i.e. the zero-padding method for (a) $B_{\rm n}$ (left panel) and (b) $B_{\rm t}$ (right panel) at 2.1 s of KSTAR shot #20341. Probes #14 and 30 for $B_{\rm n}$ and probes #4 and 8 for $B_{\rm t}$ are treated as the missing ones. The bottom panels compare histograms of MSSIM and PSNR using the imputation method (green) and the zero-padding method (pink) for all the equilibria obtained from KSTAR shot #20341.

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It is clear that the nn-EFIT with the imputation method (green line) is not only much better than that with the zero-padding method (pink dashed line) but it also reconstructs the equilibrium close to the reference (black). In fact, the zero-padding method is too far off from the reference (black line) to be relied on for plasma controls.

Motivated by such a successful result of the nn-EFIT with the imputation method on the two missing values, we have increased the number of missing values as shown in figures 16 and 17 for the same KSTAR discharge, i.e. KSTAR shot #20341. Let us first discuss figure 16 which shows (a) the eight (without probe #6) and (b) nine (all) missing values of $B_{\rm t}$. The color codes are the same as in figure 15, i.e. the reference is black, and the nn-EFIT with the imputation method is green or with the zero-padding method is pink. It is evident that the nn-EFIT with the imputation method performs well at least up to nine missing values. Such a result is, in fact, expected since the imputation method has inferred the missing values well as shown in figure 14(b) in addition to the fact that a well-trained neural network typically has a reasonable degree of resistance on noise. Again, the nn-EFIT with the zero-padding method is not reliable.

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Figures 17(a) and (b) are the results with the eight (without probe #37) and nine (all) missing values of $B_{\rm n}$, respectively. The color codes are the same as in figure 15. We find that the nn-EFIT with the eight missing values reconstructs the equilibrium similar to the reference one, while the reconstruction quality becomes notably worse for the nine missing values. This is caused mostly due to poor inference of probe #37 by the imputation method (see figure 14(a)). Nevertheless, the result is still better than that with the zero-padding method. Figure 18 shows the reconstruction results with the same color codes as in figure 15 when we have (a) 4  +  4 and (b) 9  +  9 combinations of $B_{\rm n}$ and $B_{\rm t}$ missing values simultaneously.

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All these results suggest that the nn-EFIT with the imputation method reconstructs equilibria reasonably well except when the imputation infers the true value poorly, e.g. $B_{\rm n}$ probe #37 in figure 14(a) and table 2. In fact, the suggested imputation method [50] infers the missing values based on the neighboring intact values (using Gaussian processes) while satisfying Maxwell's equations (using Bayesian probability theory). Consequently, such a method becomes less accurate if (1) the neighboring channels are also missing AND (2) the true values change fast from the neighboring values. In fact, $B_{\rm n}$ probe #37 happens to satisfy these two conditions, i.e. probe #35 is also missing, and the true values of probe #35, #37 and #38 are changing fast as one can discern from figure 14(a).

To adjust the signal drifts, we deem a priori that the signals drift linearly in time [63–65]. Of course, non-linear drift may well exist in the signals. However, we need to come up with a very simple and fast solution to adjust the drifts in real time with the limited amount of information. One can consider such linearization in time as taking up to the first order of Taylor-expanded drifting signals. Therefore, we take the drifting components of the signals ($y_i^m$) from various types (the magnetic pick-up coils or the flux loops) of magnetic diagnostics to follow:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:lineardrift} y_i^m = a_i^m t + b_i^m, \nonumber \end{align} \tag{ A.1 }$$

where t is the time. $a_i^m$ and $b_i^m$ are the slope and the offset, respectively, of a drift signal for the ith magnetic sensor of type m (magnetic pick-up coils or flux loops). Then, our goal simply becomes finding $a_i^m$ and $b_i^m$ for all i and m of interest before a plasma starts or the blip time (t  =  0) so that $y_i^m$ can be subtracted from the measured magnetic signals in real time, i.e. preprocessing the magnetic signals for the neural networks.

We use two different time intervals during the initial magnetization stage, i.e. before the blip time, for every plasma discharge to find $a_i^m$ and $b_i^m$, sequentially. Figure A1 shows an example of temporal evolutions of currents in the poloidal field (PF) coils, $B_{\rm n}$ and $B_{\rm t}$ and poloidal magnetic flux up to the blip time (t  =  0) of a typical KSTAR discharge.

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During the time interval d1 in figure A1, all the magnetic signals must be constant in time because there are no changes in currents of any of the PF coils; in addition, there are no plasmas yet that can change the magnetic signals. Therefore, any temporal changes in a magnetic signal during d1 can be regarded as due to a non-zero $a_i^m$. With the knowledge of $a_i^m$ from the d1 time interval, we obtain the value of $b_i^m$ using the fact that all the magnetic signals must be zeros during the time interval d2 because there are no sources of magnetic fields, i.e. all the currents in the PF coils are zeros.

Summarizing our procedure, (1) we first obtain the slopes $a_i^m$ based on the fact that all the magnetic signals must be constant in time during d1 time interval, and then (2) find the offsets $b_i^m$ based on the fact that all the magnetic signals, after the linear drifts in time are removed based on the knowledge of $a_i^m$, must be zeros during d2 time interval.

Bayesian probability theory [51] has a general form of

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \newcommand{\rp}{\right)} \newcommand{\lp}{\left(} \displaystyle p\lp\mathcal{W}|\mathcal{D}\rp=\frac{p\lp\mathcal{D}|\mathcal{W}\rp p\lp\mathcal{W}\rp}{p\lp\mathcal{D}\rp}, \label{eq:Bayes} \nonumber \end{align} \tag{ A.2 }$$

where $\mathcal{W}$ is a (set of) parameter(s) we wish to infer, i.e. $a_i^m$ and $b_i^m$ for our case, and $\mathcal{D}$ is the measured data, i.e. measured magnetic signals during the time intervals of d1 and d2 in figure A1. The posterior $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} p\lp\mathcal{W}|\mathcal{D}\rp$ provides us the probability of having a certain value for $\mathcal{W}$ given the measured data $\mathcal{D}$ which is proportional to a product of likelihood $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} p\lp\mathcal{D}|\mathcal{W}\rp$ and prior $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} p\lp\mathcal{W}\rp$. Then, we use the maximum a posterior (MAP) to select the value of $\mathcal{W}$. The evidence $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} p\lp\mathcal{D}\rp$ (or marginalized likelihood) is typically used for a model selection and is irrelevant here as we are only interested in estimating the parameter $\mathcal{W}$, i.e. $a_i^m$ and $b_i^m$.

We estimate values of the slope $a_i^m$ and the offset $b_i^m$ based on equation (A.2) in two steps as described above:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:baye-slope} {{\rm Step~(1)}}\:: \: p(a_i^{m}|\mathcal{\vec {D}}_{i, d1}^m) \propto p(\vec {D}_{i, d1}^m|a_i^{m})\,p(a_i^{m}), \nonumber \end{align} \tag{ A.3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:baye-off} {{\rm Step~(2)}}\:: \: p(b_i^{m}|\mathcal{\vec {D}}_{i, d2}^m, a_i^{m*}) \propto p(\vec {D}_{i, d2}^m|b_i^{m}, a_i^{m*})\,p(b_i^{m}), \nonumber \end{align} \tag{ A.4 }$$

where $\mathcal{\vec {D}}_{i, d1}^m$ ($\mathcal{\vec {D}}_{i, d2}^m$) are the time series data from the ith magnetic sensor of type m (magnetic pick-up coils or flux loops) during the time intervals of d1 (d2) as shown in figure A1. $a_i^{m*}$ is the MAP, i.e. the value of $a_i^m$ maximizing the posterior $p(a_i^{m}|\mathcal{\vec {D}}_{i, d1}^m)$. Since we have no prior knowledge of $a_i^m$ and $b_i^m$, we take priors, $p(a_i^{m})$ and $p(b_i^{m})$, to be uniform allowing all the real numbers. Note that a correct $p(a_i^{m})$ would be equal to $\newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lsb}{\left[} \newcommand{\rsb}{\right]} 1/\lsb \pi\lp1+\lp a_i^{m} \rp^2 \rp \rsb$ [66], but we sacrifice rigor to obtain a fast solution. Furthermore, the posterior for $b_i^m$ should, rigorously speaking, be obtained by marginalizing over all possible $a_i^m$, i.e. $p(b_i^{m}|\mathcal{\vec {D}}_{i, d2}^m)=\int p(b_i^{m}|\mathcal{\vec {D}}_{i, d2}^m, a_i^m)\,p(a_i^{m}|\mathcal{\vec {D}}_{i, d1}^m) da_i^m$. Again, as we are interested in real-time application, such a step is simplified just to use $a_i^{m*}$.

With equation (A.1), we model likelihoods, $p(\vec {D}_{i, d1}^m|a_i^{m})$ and $p(\vec {D}_{i, d2}^m|b_i^{m}, a_i^{m*})$, as Gaussian:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle & p(\vec {D}_{i, d1}^{m}|a_i^m) = \frac{1}{\sqrt{(2\pi)^L} |\sigma_{i, d1}^{m}|} \nonumber \\ & \times \exp \left(- \frac{\sum\limits_{t_l \in d1}^L \left[a_i^m(t_{l} - t_0) - \left(D_{i, d1}^m(t_l) - \left< D_{i, d1}^m(t_0)\right> \right)\right]^2}{2(\sigma_{i, d1}^m)^2}) \right), \label{eq:like-slope} \nonumber \end{align} \tag{ A.5 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle p(\vec {D}_{i, d2}^m | b_i^m, & a_i^{m*}) = \frac{1}{\sqrt{(2\pi)^K} |\sigma_{i, d2}^m|} \nonumber \\ & \times \exp \left(- \frac{\sum\limits_{t_k\in d2}^K \left[b_i^m - \left(D_{i, d2}^m(t_k) -a_i^{m*} t_k\right)\right]^2}{2(\sigma_{i, d2}^m)^2} \right), \label{eq:like-offset} \nonumber \end{align} \tag{ A.6 }$$

which simply state that noises in the measured signals follow Gaussian distributions. Here, $\sigma_{i,d1}^m$ and $\sigma_{i,d2}^m$ are the experimentally obtained noise levels for the ith magnetic sensor of type m (magnetic pick-up coils and flux loops) during the time intervals of d1 and d2 in figure A1, respectively. t_l and t_k define the actual time intervals of d1 and d2, i.e. $t_l \in [-6, -1]$ s and $t_k \in [-14, -13]$ s with L and K being the numbers of the data points in each time interval, respectively. t₀ can be any value within the d1 time interval, and we set t₀  =  −2 s in this work. $\left< D_{i, d1}^m(t_0)\right>$, removing the offset effect to obtain only the slope, is the time-averaged value of $D_{i, d1}^m(t)$ for $t\in [t_0-0.5, t_0+0.5]$ s. We use the time-averaged value to minimize the effect of the noise in $D_{i, d1}^m(t)$ at t  =  t₀.

With our choice of uniform distributions for priors in equations (A.3) and (A.4), MAPs for $a_i^m$ and $b_i^m$, which we denote as $a_i^{m*}$ and $b_i^{m*}$, coincide with the maximum likelihoods which can be analytically obtained by maximizing equations (A.5) and (A.6) with respect to $a_i^m$ and $b_i^m$, respectively:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:direct-slope} a_i^{m*} = \frac{\sum\limits_{t_l \in d1}^L \left[\left(D_{i, d1}^m(t_l) - \left< D_{i, d1}^m(t_0)\right>\right)\left(t_l - t_0 \right) \right]}{\sum\limits_{t_l \in d1}^L \left[t_l - t_0 \right]^2}, \nonumber \end{align} \tag{ A.7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:direct-off} b_i^{m*} = \frac{1}{K} \sum\limits_{t_k \in d2}^K \left[D_{i, d2}^m(t_k) - a_i^{m*} t_k \right]. \nonumber \end{align} \tag{ A.8 }$$

Now, we have attained simple algebraic equations based on Bayesian probability theory which can provide us values of the slope $a_i^m$ and the offset $b_i^m$ before the blip time, i.e. before t  =  0.

Since the required information ($a_i^m$ and $b_i^m$) to adjust drifts in the magnetic signals is obtained before every discharge starts, we can preprocess the magnetic signals in real time. This is how we have adjusted the drift signals shown in figure 2.

PSNR is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:psnr} {\rm PSNR}=10\times\log_{10}\left[\frac{\max\left(y^{\rm Target}\right)^2}{\frac{1}{M}\sum\nolimits_{i=1}^M\left(y_i^{\rm Target} - y_i^\star \right)^2}\right], \nonumber \end{align} \tag{ B.1 }$$

where y _i is the value of either $\psi$ or $\Delta^*\psi$ at the ith position of the spatial grid (analogous to a pixel value of an image), and M for the total number of grid points, i.e. either $286 (=22\times 13)$ or $4225 (=65\times 65)$ depending on our choice for reconstructing equilibrium. $\max(\cdot)$ operator selects the maximum value of an argument, and $y^{\rm Target}$ is an array containing 'pixel' values of a reference EFIT 'image', that is a reconstructed magnetic equilibrium. $y^\star$ is also an array, and depending on whether we wish to compare the off-line EFIT result with either the rt-EFIT result or the nn-EFIT result, we select the corresponding values.

MSSIM is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lab}{\left\langle} \displaystyle \label{eq:mssim} {\rm MSSIM}=\frac{\left(2\mu_{y^{\rm Target}} \mu_{y^\star} + C_1 \right)\left(2\sigma_{y^{\rm Target} y^\star}+C_2 \right)}{\left(\mu^2_{y^{\rm Target}} + \mu^2_{y^\star}+ C_1 \right)\left(\sigma^2_{y^{\rm Target}} + \sigma^2_{y^\star}+ C_2 \right)}, \nonumber \end{align} \tag{ B.2 }$$

where $\mu_{y^{\rm Target}}$ and $\mu_{y^\star}$ are the mean values of $y^{\rm Target}$ and $y^\star$, respectively. Here, $y^{\rm Target}$ and $y^\star$ mean the same as in section B.1. $\sigma^2_{y^{\rm Target}}$ and $\sigma^2_{y^\star}$ are the variances of $y^{\rm Target}$ and $y^\star$, respectively, while $\sigma_{y^{\rm Target} y^\star}$ is the covariance between $y^{\rm Target}$ and $y^\star$. C₁ and C₂ are used to prevent a possible numerical instability, i.e. the denominator being zero, and are set to be small numbers. Following [62], we have $C_1=10^{-4}$ and $C_2=9\times 10^{-4}$.