Machine learning prediction of electron density and temperature from He I line ratios

Some He I line ratios, such as 667.8 nm/728.1 nm and 728.1 nm/706.5 nm, are known to be sensitive to the electron density, n_e, and temperature, T_e, respectively.¹ The dependence of He I line ratios on n_e and T_e can be calculated with a He I collisional–radiative model (CRM). From comparison between measured and calculated line intensity ratios, n_e and T_e can be estimated. This He I line ratio technique has been widely applied both in the boundary plasma of magnetically confined fusion devices^1–9 and in laboratory linear plasma devices.^10–16 While He I CRM codes have been improved by taking into account high energy electron components,¹¹ radiation trapping,^11–17 heavy particle collisions,⁶ etc., agreement of n_e and T_e from this technique with other diagnostics, including Langmuir probes and laser Thomson scattering (LTS), is not always satisfactory. In particular, radial profiles of n_e and T_e measured using probes and LTS in linear plasma devices are not well reproduced.^13,16

Instead of using a He I CRM for the evaluation of n_e and T_e, we have pursued a different approach: machine learning (ML). As described in Fig. 1, the procedure is simple. Training data for ML consist of measured He I line ratios (input) and n_e and T_e (desired output) from other diagnostic method(s), which are a Langmuir probe in this study. Thus, this is categorized as supervised learning. Support vector machine (SVM) regression analysis is, then, performed to develop a predictive model for n_e and T_e, respectively. SVM is one of ML algorithms and is relatively simple to set up compared to neural network (NN). Since SVM requires fewer training data than for NN, SVM is thought to be useful, especially, when the number of data is limited.

ML was utilized to predict n_e and T_e in the plasma column center of the Magnum-PSI linear plasma facility from main machine settings, including the magnetic field strength, gas flow rate, and source current.¹⁸ In that study, the artificial NN (ANN) was used for data fitting. In the combustion research field, the temperature and density are important for characterizing flames, and ML has also been applied to retrieve those parameters. For instance, an inverse radiation model based on the multi-layer perceptron NN method was developed, and the spatial distributions of the temperature and the species concentrations were derived from infrared spectral emission for a gas mixture of CO₂, H₂O, and CO.¹⁹ 

In confinement fusion devices, ML has also been applied for various purposes, and a couple of studies are briefly introduced below. For fast analysis of collective Thomson scattering (CTS) spectra, ANNs were utilized to provide a mapping between synthetic (simulated) CTS spectra and the bulk ion temperature in the W7-X stellarator.²⁰ In comparison with function parameterization, ANNs obtained a higher accuracy and noise robustness. Deep learning has been applied to plasma radiation data measured using a bolometer system with multiple channels on a poloidal cross section in the JET tokamak.²¹ Convolutional (C) and recurrent (R) NNs were used to reconstruct the plasma radiation profile and to perform disruption prediction, respectively. It was reported that, while results obtained with CNN were satisfactory, the performance of RNN was lower than that of other methods used at JET. To improve disruption prediction with RNN, it was expected to involve more plasma parameters. CNN was also tested to predict the scrape-off layer (SOL) heat flux width from data obtained with thermocouples embedded in graphite plasma-facing components (PFCs) of the NSTX-U tokamak.²² In that study, NSTX-U discharges were simulated to produce synthetic thermocouple data based on real PFC designs. Then, the heat flux profile was successfully reconstructed from the thermocouple data using the trained CNN with over 8000 simulated NSTX-U discharges.

In this paper, He I line ratios coupled with ML, more specifically SVM regression analysis, are examined for n_e and T_e predictions. To our knowledge, the coupling of He I line ratios with ML has never been tested. Experimental data were collected in the PISCES-A linear plasma device,²³ where pure He plasmas were produced at several different conditions, as listed in Table I. To vary n_e and T_e, the neutral He gas pressure, P_He, was scanned from 1.5 mTorr to 4.0 mTorr. At each plasma condition, a range of n_e and T_e is given in the table, since radial profiles are taken. In short, n_e was changed by more than an order of magnitude from 0.28 × 10¹⁸ m⁻³ to 3.8 × 10¹⁸ m⁻³, while T_e ranged from 3.2 eV to 7.5 eV. In general, it is easier to vary n_e in linear plasma devices with no auxiliary heating. The first six conditions from A to F in Table I are assigned to training data. The total number of training data is 342. Condition G, not included in the training data, is used to validate ML-predictive models for n_e and T_e in terms of both absolute values and radial profiles. In addition to ML, a scaling law with a power function for each line ratio is also tested for comparison.

In the following, we will first describe measurements of He I line ratios (Sec. II A) as well as n_e and T_e (Sec. II B). Then, the results of ML and a scaling law will be presented in Secs. III and IV, respectively. In Sec. V, the conclusion of this work will be given.

Plasma emission spectra were measured with a hyperspectral imaging (HSI) camera (Specim IQ), which was recently characterized for the application to steady-state plasma observations.²⁴ Since the Specim IQ camera contains 204 spectral bands for each pixel, two-dimensional images of multiple emission lines are simultaneously observed.

Figure 2 shows a photo of the plasma–target interaction region during a He plasma discharge of PISCES-A, taken with the built-in red–green–blue (RGB) camera of the Specim IQ. In this study, vertical (v) profiles of emission spectra at an axial location (z ∼ 50 mm) were analyzed, as indicated with the dashed square in Fig. 2, since a reciprocating Langmuir probe system is also located at z ∼ 50 mm. Note that spectra at z ∼ 50 ± 5 mm were averaged out to reduce the scattering of data. The spatial resolution was ∼0.83 mm/pixel in this experimental setup. The intensity calibration of the Specim IQ camera including the vacuum window was performed with an integrating sphere standard light source (Optronic Laboratories OL 455-12).

An example of He plasma emission spectra is presented in Fig. 3, which was observed in Condition G at z ∼ 50 mm and v ∼ 0 mm (the center of the plasma column). Eight He I line peaks are observed in this wavelength range. The spectral resolution in FWHM (full width at half maximum) is 7 nm, which is not high enough to resolve two He I lines at 501.6 nm and 504.8 nm. Thus, the two lines will be treated as one line for further analyses. Note that the 501.6 nm line is generally stronger than the 504.8 nm line. The strongest line at 587.6 nm often saturated in this experiment and thus will not be used.

The following six line ratios made of the seven lines will be considered: r₀ = 667.8 nm/728.1 nm, r₁ = 728.1 nm/706.5 nm, r₂ = (501.6 nm + 504.8 nm)/728.1 nm, r₃ = 492.2 nm/728.1 nm, r₄ = 492.2 nm/471.3 nm, and r₅ = 492.2 nm/447.1 nm. Note that the line-integrated intensity is used in this study without converting it to the local emissivity by means of the Abel inversion, since the Abel inversion can cause an additional scattering of radial profiles by differentiating vertical profiles. This means that we will treat vertical profiles of the line ratios as radial profiles, namely, |v| is equal to r. The total emission intensity is calculated by taking the area under the line shape with the background emission component subtracted.²⁴ Vertical profiles of these He I line ratios in Condition G are plotted in Fig. 4. Although the vertical profiles are not symmetric with respect to v = 0 mm, all the data points will be used for ML without any further adjustments.

The reciprocating Langmuir probe system is horizontally (the h direction in Fig. 2) plunged into the plasma column and provides radial profiles of n_e and T_e at z ∼ 50 mm. The location is indicated with the black circle in Fig. 2. As an example, measured radial profiles of n_e and T_e in Condition G are presented in Fig. 5. These are constructed from five probe plunges to increase the number of data points. It is seen that radial profiles of n_e and T_e are generally flat at r ≤ 15 mm in PISCES-A, which is larger than the radius (∼11 mm) of the plasma-exposed area of a target. The large error bars (up to around ±50%) of n_e are caused by a large scattering of the ion saturation current, while the typical error of T_e is around ±0.5 eV.

Measured radial profiles of both n_e and T_e are fitted with the following function:

where m_i (i = 0, 1, 2, 3) are the fitting parameters. In Fig. 5, fitted radial profiles, as well as their ±20%, are shown with the solid and dashed curves, respectively. Most of the data points fall into the ±20% boundaries. The coefficient of determination, R², for each fit is 0.89 for n_e and 0.82 for T_e. It should be noted that fitted values of probe-measured n_e and T_e will be used for desired outputs to develop predictive models and for comparison with predicted values in the following sections.

Now, we are ready for ML predictions of n_e and T_e. To develop predictive models for n_e and T_e from the training data (Conditions A–F), SVM regression analysis was performed using a routine (IDLmlSupportVectorMachineRegression), with a radial kernel (IDLmlSVMRadialKernel), available in IDL (Interactive Data Language). It should be reminded that the total number of training data is 342 with 0.28 × 10¹⁸ ≤ n_e (m⁻³) ≤ 3.8 × 10¹⁸ and 3.2 ≤ T_e (eV) ≤ 7.5. In this study, predictive models were created with two (r₀ and r₁), four (r₀, r₁, r₂, and r₃), and six (r₀, r₁, r₂, r₃, r₄, and r₅) He I line ratios. A set of (r₀ and r₁) is most commonly used in He I CRM calculations. In Ref. 25, r₂ and r₃ were introduced to consider radiation trapping, since the upper states of 501.6 nm and 492.2 nm are 3¹P and 4¹P, respectively. Additional r₄ and r₅ are singlet-to-triplet line ratios.¹¹ In Fig. 6, probe-measured n_e and T_e are compared with ML-predicted ones for Conditions A–F. As an index of goodness of fit, χ² was calculated by,

In Eq. (2), “probe” and “ML” represent probe-measured and ML-predicted n_e or T_e, respectively. While the six He I line ratios yield the best fit (the lowest χ²) for both n_e and T_e, agreement is satisfactory even with the two line ratios. The asymmetric vertical profiles of the He I line ratios [see Fig. 4] are thought to cause the deviation at n_e ∼ 2 × 10¹⁸ m⁻³, as clearly seen in Fig. 6(a). This deviation is mitigated by including more He I line ratios for ML, as demonstrated in Figs. 6(c) and 6(e).

Note that the uncertainty of probe-measured n_e and T_e data as well as of their fits [see Fig. 5] can also lead to the scattering of ML-predicted values. To assess the effect of the probe data scattering, we artificially introduce random noise to the fitted values of probe-measured n_e and T_e. The noise is assumed to be a normal (Gaussian) distribution with σ = 20% of the fitted n_e and T_e values. For the six He I line ratios, the resulting χ² values for n_e and T_e increase to 2.0 and 6.6, respectively. This indicates that T_e is more susceptible to the probe data scattering.

The developed predictive models were, then, applied to the evaluation data (Condition G). ML-predicted radial profiles of (a) n_e and (b) T_e are compared with probe-measured (fitted) ones in Fig. 7. Note that, since the line ratios at the positive and negative v locations are separately analyzed [see Fig. 4], there are two data points at each r location. This is also applied to Fig. 9. It is found that the radial profiles of both n_e and T_e are well reproduced, while n_e from the two line ratios shows a larger deviation. It should be noted that a He I CRM code with radiation trapping¹⁶ was not able to well reproduce the probe-measured radial profiles of n_e and, in particular, T_e. With an optical escape factor radius of 20 mm, the radial profile of n_e, predicted with the two line ratios, was nearly consistent with the probe-measured profile, while n_e was underestimated by a factor of ∼2 with the four and six line ratios. Regardless of the line ratios used, CRM-predicted T_e at the center was overestimated by a factor of ∼3 and increased to ∼30 eV toward the edge of the plasma column. Since He I CRM is not the main subject of this work, we do not discuss or investigate the reasons for the poor agreement between He I CRM and probe measurements.

In addition, we will examine a scaling law for the prediction of n_e and T_e from He I line ratios. Contrary to ML, a fit function needs to be pre-defined, and a power function is assumed for each line ratio as follows:

where α and c0, c1, etc., are fitting parameters. By first taking the log of both sides of Eq. (3), a multiple linear regression fit was performed to determine the fitting parameters. As for ML, two (r₀ and r₁), four (r₀, r₁, r₂, and r₃), and six (r₀, r₁, r₂, r₃, r₄, and r₅) He I line ratios were tested with the same training data (Conditions A–F). Figure 8 shows the comparison of n_e and T_e between the scaling law and the probe data. As indicated by the larger χ², the prediction by the scaling law is found to be less satisfactory than ML. Note that, as for ML, agreement becomes better with more line ratios.

The derived scaling laws were examined with the evaluation data (Condition G). As demonstrated in Fig. 9, the scaling laws also reproduce the radial profiles of the probe-measured n_e and T_e in this example.

Since the dependence of the line ratios on n_e and T_e is obtained from the scaling laws, we briefly discuss it for the most commonly used set of r₀ = 667.8 nm/728.1 nm (n_e-sensitive) and r₁ = 728.1 nm/706.5 nm (T_e-sensitive) in He I CRM. In the parameter ranges studied here, He I CRM predicts that r₀ and r₁ increase with increasing n_e and T_e, respectively.^13,14 Thus, the obtained n_e dependence of r₀ (c₀ = 2.68 ± 0.06) is consistent with He I CRM, as presented in Fig. 8(a). However, as shown in Fig. 8(b), the T_e dependence of r₁ has a negative value of c₁ = −1.73 ± 0.06, which contradicts He I CRM calculations. As a result, the scaling law reproduces the decrease in T_e toward the edge of the plasma column [Fig. 9(b)], in contrast to He I CRM, as mentioned above. The negative value of c₁ is actually expected from the vertical profile of r₁ in Fig. 4(a), where r₁ increases toward the edge, and the radial profile of T_e in Fig. 5(b), where T_e decreases toward the edge.

Machine learning prediction of n_e and T_e from He I line ratios has been examined in the ranges of 0.28 × 10¹⁸ ≤ n_e (m⁻³) ≤ 3.8 × 10¹⁸ and 3.2 ≤ T_e (eV) ≤ 7.5. Support vector machine regression analysis was performed with measured He I line ratios (input data) and probe-measured n_e and T_e (desired output data). It was demonstrated that the developed predictive models well reproduced both absolute values and radial profiles of n_e and T_e. In addition to ML, the scaling laws were also developed with a power function for each He I line ratio. While χ² values were larger with the scaling laws compared to ML, agreement of n_e and T_e with probe measurements was also satisfactory in the parameter ranges studied here. It should be noted that He I CRM could not correctly predict n_e and T_e with the same data set of He I line ratios.

In this study, two, four, and six He I line ratios were employed from the seven available lines (501.6 nm and 504.8 nm were treated as one line) in our measurements. The lowest χ² values were obtained with the six line ratios. Since a combination/selection of line ratios was not optimized here, it may be possible to obtain better predictions with an optimized combination/selection of line ratios. While the present experiments were conducted in pure He plasmas, the number of available He I lines may be reduced due to spectral overlapping with other species emission lines in mixed plasmas. Thus, a combination of line ratios needs to be carefully selected in each application.

This alternative technique does not require the intensity calibration of a spectroscopic system, unless the wavelength dependence of, e.g., a vacuum window transmission changes. However, the intensity calibration is necessary when predictive models are applied to other devices and are developed with data from multiple devices. Since the achievable ranges of n_e and T_e are typically limited in a single linear plasma device, collaborations involving multiple devices will be required to develop ML-predictive models in wider ranges of n_e and T_e. For instance, it is interesting to see if ML can develop a predictive model covering from ionizing (like in this study) to recombining plasmas.

This work was supported by the U.S. DOE under Grant No. DE-FG02-07ER54912. This work was also supported, in part, by a Grant-in-Aid for Scientific Research (Grant No. 19H01874), and the Fund for the Promotion of Joint International Research (Grant No. 17KK0132), from the Japan Society for the Promotion of Science (JSPS).

The data that support the findings of this study are available from the corresponding author upon reasonable request.