# Study of line spectra emitted by hydrogen isotopes in tokamaks through Deep-Learning algorithms

N. Saura, M. Koubiti, Sadruddin Benkadda

► To cite this version:

N. Saura, M. Koubiti, Sadruddin Benkadda. Study of line spectra emitted by hydrogen isotopes in tokamaks through Deep-Learning algorithms. Nuclear Materials and Energy, 2025, 43, pp.101935. 10.1016/j.nme.2025.101935. hal-05196897

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# Study of line spectra emitted by hydrogen isotopes in tokamaks through Deep-Learning algorithms☆

N. Saura, M. Koubiti*, S. Benkadda

Aix-Marseille Université, CNRS, PIIM UMR 7345, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France

## ABSTRACT

Artificial Intelligence (AI) is increasingly used in various plasma physics topics, including applications in spectroscopy and diagnostics in magnetically confined fusion plasmas. The paper focuses on the application of the convolutional neural network (CNN) algorithm to emission spectroscopy from the divertor regions of magnetic fusion devices. Specifically, we use a CNN to determine hydrogen isotopic ratios from the theoretical emission spectra of the Balmer-α line in hydrogen-deuterium (HD) plasmas. The motivation for coupling AI with spectroscopy is to develop novel frameworks that can outperform existing classical methods based on spectral line fitting, in terms of accuracy, speed, or adaptability. In a previous work, we have used a fully connected neural network algorithm for theoretical $H\alpha/D\alpha$ line spectra emitted by HD plasmas which have been generated for conditions relevant to divertor plasmas in tokamaks (magnetic field, neutral temperatures and fractions and hydrogen concentration). The generated spectra were preprocessed to extract few spectroscopic features which were then used as input data by the neural network. In the present work, we apply for the first time a CNN model to raw synthetic $H\alpha/D\alpha$ line spectra theoretically emitted by HD plasmas to predict the corresponding isotopic ratios. In this context, we show that the trained CNN predicts hydrogen isotopic ratios with deviations of up to 5% from the true values. Additionally, our model can generalize its predictions to spectra corresponding to any observation angle relative to the magnetic field, despite being trained solely on spectra from parallel observations. The prediction accuracy in these cases is comparable to the training accuracy.

## 1. Introduction

Over the past few decades, the use of machine learning (ML) and deep learning (DL) algorithms has rapidly expanded into various domains of plasma sciences. These algorithms have been applied to a wide range of plasmas, including high energy density pulsed-power-driven aluminum plasmas [1], laser-produced plasmas [2] and many others [3,4]. In the domain related to magnetic fusion research, machine learning was used to predict the plasma parameters using the intensities of neutral helium ($He~i$) spectral lines [5] emitted in the linear plasma devices PISCES-B and NAGDIS-II [6-8]. In other examples, ML models were combined with 2D beam emission spectroscopy (BES) for real-time inference of plasma dynamics in the DIII-D tokamak [9]. In another application related to DIII-D [10], a neural network was used to predict the electron temperature from extreme ultraviolet/vacuum ultraviolet emission spectroscopy and identify plasma detachment. Deep learning has also been used as a preprocessing tool to denoise synthetic atomic spectra thereby increasing the accuracy of known element identification methods [11]. In addition, deep neural networks were applied recently [12] to estimate physical parameters from complex spectra of the Balmer of the Balmer-β lines of hydrogen and deuterium ($H\beta/D\beta$)

emitted in the WEST tokamak.

In a previous work [13-15], we applied a neural network model to the Balmer-α line emitted by hydrogen isotopes in tokamaks to determine and predict the hydrogen isotopic ratio of the plasma. In that work, we extracted spectroscopic features from the $H\alpha/D\alpha$ line spectra, which were then used as inputs for the neural network algorithm to determine the isotopic ratios. In the current paper, we apply a one-dimensional CNN to predict the same ratios from synthetically generated $H\alpha/D\alpha$ line spectra emitted by HD plasmas. This synthetic data generation enables us to explore wide domains of parameters, covering most anticipated scenarios in tokamak divertors. The parameters of interest are the neutral temperatures and fractions, magnetic field strength, and hydrogen isotopic ratio. The use of experimental spectra will be considered in future work, which is currently ongoing, along with similar applications to $D\alpha/T\alpha$ line spectra emitted by DT plasmas. Our work on experimental spectra will aim to complete or even replace the standard method based on spectral fitting confronted to limitations especially for DT or even HDT mixtures because of the short separation of the $D\alpha$ and $T\alpha$ lines (~0.6 Å).

The remaining of the paper is organized as follows. In section 2 we describe the computation of Balmer-α line profiles for a large domain of

<footnote>* This article is part of a special issue entitled: 'PSI-26' published in Nuclear Materials and Energy.</footnote>

<footnote>* Corresponding author.
E-mail address: mohammed.koubiti@univ-amu.fr (M. Koubiti).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Profiles of the Balmer-α line computed for HD (left) and DT (right) mixtures for typical values of the neutral population temperatures (3 eV and 30 eV) and a magnetic field of 2 T. The angle of observation corresponds to θ = 0° or θ = 180°. The vertical lines indicate the positions of the non-perturbed wavelengths of the Balmer-α line for each hydrogen isotope.</figcaption></figure>

parameters. This theoretical database is used to generate the different data collections featured in this paper. Section 3 is dedicated to describing the CNN model and the selected architecture: Visual Geometry Group (VGG). In section 4, we discuss the accuracy of the deuterium concentration predicted by the CNN, both for the validation set and for data generated for arbitrary angles of observation with respect to the magnetic field. Eventually, a conclusion is drawn in section 5.

## 2. Generation of the training and test data: Balmer-α line profiles

We describe briefly here the computation of the Balmer-α line profiles which will be used as a training dataset by the neural network model. We are interested here by the Balmer-α line emitted by hydrogen isotopes in tokamak divertors. Therefore, it is necessary to explain quickly the origin of hydrogen neutrals in such peripheral regions of the magnetically confined plasmas in tokamaks. In a few words, for hydrogen plasmas, hydrogen neutrals can be released through the interactions of the plasma and its facing components thanks to chemical and physical sputtering as well as the consequence of the impact of H⁺ ions hitting these components which are neutralized and reflected as H neutrals. The chemical desorption of H₂ molecules is followed by different dissociation and thermalization processes leading to a cold population of H neutrals which can be described with a Maxwellian velocity distribution function with a temperature in the range 0.5 to 3 eV approximatively. In addition to this cold population, another one is composed of H neutrals released from the target material following H⁺ ion impact (reflection) and charge exchange (CX) processes between H atoms and H⁺ ions. This H neutral population termed warm population is also usually described by a Maxwellian velocity distribution function having a typical temperature in the range 10–100 eV. The same thing applies if the tokamak is operated with deuterium instead of hydrogen. We will have two deuterium neutral populations: a cold population and a warm population. However, as tokamaks are often operated with a mixture of hydrogen isotopes (e.g., hydrogen and deuterium), the previously described neutral populations hold for each isotope. The fraction of cold neutrals with respect to the total number of neutrals is the same for both hydrogen and deuterium and of course the fractions of warm atoms are also identical for both. This is also true for plasmas operated with a mixture of deuterium and tritium. In addition, because of the mass differences between the hydrogen isotopes, the wavelengths of the Balmer-α line they emit are not the same. Indeed, the wavelengths of the Hα, Dα and Tα are respectively 6562.85 Å, 6561.01 Å and 6560.42 Å. The strong magnetic fields greater than 1 T applied in tokamaks affect the atomic energy levels and consequently split the radiative transitions. Therefore, the profiles of the Balmer-α line are subject to the Zeeman

effect which splits the line and subject to Doppler effect, which is the dominant broadening mechanism, the Stark broadening being negligible for electron densities lower than 10¹⁴ cm⁻³. Readers interested by more details concerning the spectroscopic analysis of the Balmer-α line emitted in tokamak divertors and its modeling can refer to the following papers [13,16,17,18]. However, the electron density in the divertor may reach values of the order of 10¹⁵ cm⁻³ like in C-Mod or even higher (~3x10¹⁵ cm⁻³) like in JET. At such densities, Stark broadening competes with Doppler broadening and even largely exceeds it for the higher members of the Balmer series. As in this paper, Stark broadening is not considered, one needs to be cautious when extending the results obtained in the present work to measurements from such high densities. The presence of the magnetic field is also responsible for the polarization of the emitted photons and hence the line profiles depend also on the angle of observation with respect to the magnetic field line. To illustrate the differences between the Balmer-α line profiles emitted by hydrogen-deuterium (HD) and deuterium-tritium (DT) plasmas, we show on Fig. 1, the profiles calculated assuming a parallel observation (θ = 0° or θ = 180°), a magnetic field value of 2 T for mixtures of 90 % of deuterium and 10 % of hydrogen (Fig. 1a) and 90 % of deuterium plus 10 % of tritium (Fig. 1b). Each isotope of these mixtures is composed of 80 % of slow atoms (cold population having a temperature of 3 eV) and 20 % of fast atoms (warm population with a temperature of 30 eV). As the Tα line is separated by only about 0.6 Å from the Dα line, its σ components (left shoulder of the figure on the right) are not resolved as is the case for those of the Hα line (right shoulder of the left figure). One can also notice that the σ components of the Dα line (highest central peaks) are symmetric for HD and asymmetric for the case of DT mixture. This is due to the overlapping of the σ components of Dα and Tα lines.

For an arbitrary observation angle θ, the line intensity can be written as:

$$I_{\theta} = \sin^2(\theta) \times I_{\perp} + \cos^2(\theta) \times I_{\parallel}$$

where $I_{\perp}$ and $I_{\parallel}$ represent the perpendicular (θ = 90°) and parallel (θ = 0° ou θ = 180°) line profiles respectively.

In this paper, we will focus on HD mixtures. More precisely, we aim to develop a machine learning algorithm to predict deuterium concentrations from synthetic spectra generated by considering the following ranges of the problem's physical parameters with various angles of observation with respect to the magnetic field line. We have varied the strength of the magnetic field B in the range 1–4 T, the temperatures of the cold and warm populations in the ranges 0.5–3 eV and 10–30 eV respectively and allowing the cold population to represent between 40 to 100 % of the total neutrals. For these parameters, a uniform distribution was used to select their values in the ranges mentioned above.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Spread plot of the deuterium concentrations $f_D$ in a HD mixture as predicted by the CNN model versus the expected values (those used to generate the spectra) for validation the dataset. Each prediction point is colored by the corresponding warm neutral temperature to emphasize its impact on the CNN prediction accuracy.</figcaption></figure>

The concentration $f_D$ of deuterium varied in the range 1–99 % by considering twenty values. Here the concentration of deuterium atoms is defined as the percentage of deuterium atoms with respect to all hydrogen isotope (here hydrogen and deuterium) atoms i.e., $f_D = \frac{n_D}{n_H+n_D} \times 100$, where $n_H$ and $n_D$ represent respectively the number densities of hydrogen and deuterium atoms respectively. In this paper, we indifferently use the terms of hydrogen isotopic ratio or deuterium concentration for HD mixtures. Note that concerning Doppler effect, only Doppler broadening was considered and not the Doppler shift and a Gaussian noise of about 1 % was used to generate the profiles.

### 3. Description of the neural network

A neural network (NN) is a parametric machine learning algorithm that uses layers of neurons to combine linear and non-linear operations, approximating the hidden relationship between input features and the optimization targets. Each layer of neurons is connected by tunable weights. A convolutional neural network (CNN) is known to perform better feature extraction than traditional NNs due to its architecture, which includes convolutional layers that apply filters to input data, capturing spatial hierarchies and patterns. Each filter in a convolutional layer scans the input data, generating feature maps that highlight specific aspects of the input. This allows CNNs to efficiently recognize complex patterns in images [19] or spectral lines [11]. In the context of our work, among all machine-learning/deep-learning algorithms CNN appears to be the most appropriate algorithm to process Balmer $\alpha$ line spectral features to predict the hydrogen isotopic ratio accurately. We have chosen to implement the VGG (Visual Geometry Group) architecture [20]. This choice is motivated by its controlled number of hyperparameters to tune. While it is not the most efficient architecture compared to more recent models, such as different versions of Inceptions, Xception net and others [21], VGG has been proven effective in various tasks (see [21] and references therein, as well as [22] for example), including the present one. Moreover, it is a well-known and robust architecture, often used as baseline reference for testing new models. VGG is designed to extract abstract patterns from images or signals to classify or predict out a property. To do so, the network is composed of three types of layers:

- Convolutional layers: these layers extract local pattern from their input. Their convolutional kernels consist of trainable weights, adjusted during training to capture the most relevant patterns. This feature extraction process is hierarchical: deeper convolution layers combine previously extracted patterns to generate more abstract representations.

- Pooling layers or strided convolution layers: these layers reduce the spatial dimension of the data flowing through the network. The core idea behind using such layers is to trade spatial resolution for more complex and general feature representations.

- Dense layers: these layers integrate the different local features extracted from convolution and pooling layers to output a scalar quantity, which is in our case the hydrogen isotopic ratio.

In our VGG implementation, spatial compression is done through three down sampling stages, each comprising three 1-D convolution layers using kernel of size (3x3). This configuration results from extensive testing. To introduce nonlinearity in the network, we used the Swish activation for convolution layers and the SeLU activation for dense ones. Both choices are motivated by the fact that we do not rely on Batch Normalization, partly because it significantly increases the number of parameters. Instead, these activation functions ensure better information flow within the network, mitigating the dying ReLU and vanishing gradient issues (for a broader discussion on the activation function properties see [23] chapter 3 and references within). The last layer is activated using the Swish function instead of SeLU to prevent the internal normalization effect that SeLU could introduce. The two activation functions are defined as [24,25]:

$$ \phi_{\text{Swish}}(x) = \frac{x}{1 + e^{-\beta x}} $$

and

$$ \phi_{\text{SeLU}}(x) = \begin{cases} \lambda \times x & \text{if } x \ge 0 \\ \lambda\alpha(e^x - 1) & \text{if } x < 0 \end{cases} $$

With $\beta = 1$, $\alpha \approx 1.6733$ and $\lambda \approx 1.0507$ using the default parameters of Keras

To train the model, we constructed both training and test sets by
<figure><img src="image_5.png" /><figcaption>Fig. 3. Spread plot of the deuterium concentration $f_D$ in a HD mixture as predicted by the 1D-CNN model applied to line profiles corresponding to arbitrary angles of observation relative to the magnetic field line. Each prediction point is colored by the corresponding angle (rad.) to emphasize its impact on the 1D-CNN prediction accuracy.</figcaption></figure>

sampling discrete combinations of the physical parameters within their respective ranges detailed in the previous section. For each parameter combination, the corresponding $H\alpha/D\alpha$ line profile is synthetically calculated. Random noise is applied on each profile. The resulting noisy 1D signals (each containing 500 datapoints) serve as the model's input, while the corresponding expected deuterium concentration $f_D$ is used as the target output. After generating all possible discrete combinations of the physical parameters, we randomly selected 80 % of the dataset for training, while the remaining 20 % was reserved for validation and monitor overfitting. In total, the network is trained on 360,000 $H\alpha/D\alpha$ line profile $-f_D$ pairs and tested on 90,000 unseen pairs. All profiles were computed assuming parallel observation with respect to magnetic field line.

The loss function used to monitor the CNN performance during training is a weighted mean of the absolute and the squared errors. In addition, we penalized the loss function by computing the maximum squared errors to encourage multi-scale learning. To prevent overfitting, a common approach is to regularize the trainable parameters of the model. We applied L1 regularization with a fixed strength of $5 \times 10^{-5}$ to force a reasonable number of weights to zero. This reduces the model's degrees of freedom, leading to a more generalizable mapping with a lower risk of mimicking the training set.

Finally, to optimize the trainable weights, we used the NAdam stochastic gradient descent algorithm, starting with an initial learning rate of $10^{-3}$ for the first five epochs. The learning rate was then reduced to $8 \times 10^{-4}$ to refine the learning process for two more epochs. We set the batch size to 512; meaning the model's weights are adjusted 360,000//512 = 704 times per epoch. At the end of training, the mean squared errors decreased from 0.0481 to $4.7857 \times 10^{-5}$ on the training set and from 0.0106 to $3.9103 \times 10^{-5}$ on the test set. This off-line optimization lasted for a couple of minutes. The on-line inference is instantaneous.

## 4. Results of the application of a 1D-CNN model and discussion

Once the model is trained, we generate spectra by randomly sampling the values of each of the physical parameters where we have used a uniform distribution as mentioned previously. This validation set includes the training set and unseen intermediate combinations and con-

tains 10,000 unseen profiles-f$_D$ pairs. The CNN predicts an estimation of the target quantity from each profile. These results are depicted in the spread plot Fig. 2. The solid line represents the target solution while the dots represent the actual predicted values. The more the dot cloud fits the line, the better the learning. In this figure, each dot's color is defined by the corresponding warm neutrals temperature to emphasize its impact on the prediction accuracy. The largest absolute error difference from the target values is about 5 %, occurring mainly in extreme cases where the deuterium concentration f$_D$ is below 10 % or exceeds 85 %, along with high values of B around 4T. Thanks to the point coloring, we can conjecture that one of the most influential parameters regarding the accuracy of the CNN is the warm neutrals temperature (T$_2$). In the next section, we will carry a global sensitivity analysis regarding the physical parameters and the performance of the CNN both at first and higher order to eventually physically explain the model uncertainties.

The predicted deuterium concentration values shown in Fig. 2 concern spectra corresponding to parallel observations with respect to the magnetic field line. To deepen our assessment on both the CNN learning and generalization capacities, we applied it to predict D concentrations in hydrogen isotope mixtures for arbitrary angles of observation $\theta$. We use the same ranges of physical parameters to generate $H\alpha/D\alpha$ Balmer line profiles corresponding to random angles $\theta$. The same 1D-CNN model is used to predict the D concentrations without retraining.

Fig. 3 shows the spread plot comparing target and predicted D concentrations for this scenario. The interpretation of the figure elements is like the previous one, except that the color of the dots now represents the angle value. Firstly, this figure demonstrates that the CNN can generalize the mapping between the spectra and the deuterium concentrations in the hydrogen isotope mixtures. This indicates that the patterns extracted by the convolutional layers, along with their non-local combinations formed by the dense layers, are sufficiently general to support this application. Contrary to previous results, it appears that changing the angle of observation reduces the error for extreme values of deuterium concentration. This may be caused by the effect of the angle on the impact of other parameters, in particular the magnetic field which significantly affects the spectral profiles.
<table><caption>**Table 1**
First order Sobol indices for. $f_D = 5\%$</caption><thead><tr><th>Parameters</th><th>S<sub>i</sub></th><th>S<sub>T</sub></th></tr></thead><tbody><tr><td>T<sub>1</sub></td><td>0.0028</td><td>0.1149</td></tr><tr><td>T<sub>2</sub></td><td><strong>0.6857</strong></td><td><strong>0.8489</strong></td></tr><tr><td>PT</td><td>0.0220</td><td>0.1459</td></tr><tr><td>B</td><td><strong>0.1120</strong></td><td><strong>0.3148</strong></td></tr></tbody></table>

<table><caption>**Table 2**
First order Sobol indices for. $f_D = 95\%$</caption><thead><tr><th>Parameters</th><th>S<sub>i</sub></th><th>S<sub>T</sub></th></tr></thead><tbody><tr><td>T<sub>1</sub></td><td>0.0141</td><td>0.0976</td></tr><tr><td>T<sub>2</sub></td><td><strong>0.5337</strong></td><td><strong>0.7690</strong></td></tr><tr><td>PT</td><td>0.0479</td><td>0.1738</td></tr><tr><td>B</td><td><strong>0.1775</strong></td><td><strong>0.4108</strong></td></tr></tbody></table>

## 5. Sensitivity analysis of the CNN with respect to physical parameters

In this section, we focus on the global sensitivity of the CNN on the validation set of the parallel case. Given the “black box” nature of CNNs, it is not possible to directly analyze the source of prediction errors. However, statistical tools such as Sobol indices [26,27] can provide insight into the trained network’s sensitivity to the aforementioned physical parameters. The Sobol method relies on ANOVA decomposition, which quantifies the contribution of each parameter and their interactions to the model’s output. Using this decomposition, first order Sobol indices $S_i$ can be computed for a given parameter $\lambda_i$, as well as higher-order interaction indices $S_{ij}$ for pairs of parameters. The former indicates the proportion of the error variance attributable to individual parameters, while the latter measures the interactions between pairs of parameters contributing to part of the error variance. A third type of index, called the total Sobol index $S_T$, captures the total contribution of each parameter, including both first and higher-order coefficients. We

focused on two critical values of $f_D$ for the trained neural network (refer to Fig. 2), namely $f_D = 5\%$ and $f_D = 95\%$. For each concentration value, we scanned the parameter space used to train the model and draw 20,480 combinations in total. In Tables 1 and 2, we present the first order and total Sobol indices for both values of $f_D$.

By far, the most influential parameters are the temperature of the hot neutral population ($T_2$) and the magnetic field strength ($B$). Computations reveal that the interactions between these two have the biggest index for the two $f_D$ considered. The total Sobol indices show that the percentage of cold neutrals is also an impactful parameter, however its contribution is less significant than the two first physical parameters. Finally, the cold neutral temperature appears to be the least correlated with the model errors.

Based on this analysis, and by its physical implications, we can conjecture possible explanation related to the CNN inaccuracy for both $f_D$ values. An increase in $T_2$ impacts the full width at half maximum (FWMH) of the hydrogen isotope lines, primarily due to the Doppler broadening. This effect may hinder the CNN’s ability to make accurate predictions, as the contribution from the least abundant species to the Balmer-$\alpha$ line will be less distinguishable. Similarly, higher values of $B$ alter the shape of the spectral line, which could also induce uncertainty in the prediction.

Another interesting test to assess the quality of the prediction is to compare two synthetic Balmer lines generated by using the expected and the predicted $f_D$. To this aim, we first generated the target line and compute $f_D^{\text{predicted}}$ from it. Then, we generate a new line using the same physical parameters but substituting $f_D^{\text{expected}}$ to $f_D^{\text{expected}}$. Here again, we randomly sampled two cases out of the most critical intervals, both for high valued of $B$ and $T_2$. We also considered two different angles, namely $\theta = 0$ and $\theta = \pi/3$. Comparisons are depicted Fig. 4. This figure shows that the impact of model’s errors on the expected shape of the line profiles is noticeable.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Comparison of target and reconstructed lines using respectively $f_D^{\text{expected}}$ and $f_D^{\text{predicted}}$ respectively. For each panel, we compare both lines for different angles and indicate the most important physical parameters used for these plots. The other parameters are the same for each pair of lines sharing the same color.</figcaption></figure>
# 6. Conclusion

To conclude, we have presented a proof of concept of a potential CNN-based diagnostic tool to assess with good confidence the hydrogen isotopic ratios (or deuterium concentrations in HD mixtures) from Hα/Dα Balmer line profiles. We have shown that our method can predict these ratios for a wide range of physical parameters as well as angles of observation with no model retraining. Global sensitivity analysis has been carried through the Sobol method, highlighting that both the warm neutrals temperature and magnetic field strength impact the CNN accuracy for low or high deuterium concentrations in HD mixtures. From this analysis, one might want to consider other CNN architectures, more complex such as Inception-net. Interestingly, the VGG architecture can be further adapted to incorporate other architecture specificities: such as 1x1 convolutions, Inception or Capsule blocks and many others.

In further work, we will apply a similar framework to predict tritium concentration for DT mixtures from Dα/Tα line profiles, in preparation of operations of tritium based fusion reactors such as the ITER tokamak.

Another extension of this work will be to consider experimental datasets. Such an application would require more attention on the pre-processing of such measurements. A well-documented and general approach could be the use of a non-linear convolutional autoencoder to project the potentially different and noisy experimental data into an abstract manyfold [28].

## CRediT authorship contribution statement

N. Saura: Writing – original draft, Visualization, Validation, Software, Methodology. M. Koubiti: Writing – original draft, Validation, Supervision, Methodology. S. Benkadda: Writing – review & editing, Validation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

# Acknowledgments

This work received funding from the Eurofusion consortium through the WPTE programme.

# Data availability

Data will be made available on request.

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