SUPPNet: Neural network for stellar spectrum normalisation

A machine learning model can be as good as data used for its training but not any better. Because of that, much attention was paid to preparing the training data properly. We applied the active learning technique so we prepared two datasets: the first, composed of synthetic spectra only, and the second based on automatically normalised and manually corrected observational spectra. In this work, we denote supplementation of the training process with spectra and pseudo-continua from manual normalisation as active machine learning.

We started by preparing an extensive set of synthetic continuum-normalised spectra. We used SYNTHE/ATLAS (Kurucz, 1970) codes to compute 10000 spectra with randomly selected atmospheric parameters (effective temperature, $T_{\textrm{eff}}$, ranging from 3000 K to 30000 K and logarithm of surface gravity, $\log g$, from 1.0 to 5.5). We also used BSTAR and OSTAR grids (Lanz & Hubeny, 2003, 2007), which together span effective temperature range from 15000 K to 55000 K, and logarithm of surface gravity from 1.75 to 4.75 (not in the whole range of temperatures). The main difference between these two sources of spectra is a different treatment of atomic levels populations. The ATLAS code solves the classical stellar atmosphere problem, and SYNTHE computes the synthetic spectrum assuming local thermodynamic equilibrium (LTE), which means that level populations are calculated using the Boltzmann distribution and the Saha ionisation equation. In the case of BSTAR and OSTAR grids, they were calculated using SYNSPEC/TLUSTY codes that explicitly solve rate equations for a chosen set of levels. This is especially important for hotter stars, where the radiative processes dominate the collisional transitions and non-LTE effects become prominent. Including non-LTE effects leads to changes of line depths and, what is more important, frequently results in the appearance of emission features.

Some families of analytical functions (sinusoidal, smoothed sawtooth, Akima spline with 5 knots) and also some continua coming from manual normalisation were used as artificial pseudo-continua shapes. To build the first training dataset the spectrum from the set of prepared synthetic spectra was sampled, part of it was randomly chosen. Next, it was convolved with a random broadening kernel and finally multiplied by an artificial continuum. This procedure repeated about 150000 times gave a large set composed of diverse training samples. The broadening kernel included rotation, macroturbulence and instrumental broadening. Rotation and macroturbulence velocities were chosen randomly in physically reasonable ranges. A projected rotation velocity ($v\sin i$) range depends on effective temperature and is drawn from a uniform distribution with the lower boundary equal to zero, and the upper one successively equal to 50, 100, 200, 300, 400 km s^-1 in the ranges (¡ 5000), (5000, 6000), (6000, 7500), (7500, 10000) and (¿10000) K, respectively. This distribution is based on the discussion presented by Royer (2009). The macroturbulence velocity, $\zeta_{v}$, was drawn from the same uniform distribution in the range from 0 to 30 km s^-1, regardless of effective temperature. The chosen instrumental resolution (from 40000 to 120000) covers a range of the most available high-resolution spectrographs.

The second stage of learning used the training set extended by the active dataset, which is composed of automatically normalised and manually corrected observational spectra (application and description of active learning in astronomical context can be found in the work by Škoda et al., 2020). First, the model trained on synthetic data was used to normalise spectra from UVES Paranal Observatory Project (UVES POP³³3https://www.eso.org/sci/observing/tools/uvespop.html; Bagnulo et al., 2003) (IC 2391, NGC 6475, and the brightest stars of southern sky, resolution equal 80000), and the set of FEROS⁴⁴4https://www.eso.org/sci/facilities/lasilla/instruments/feros.html spectra (153 spectra with $\textrm{SNR}>500$ without object duplicates, resolution equal 48000). Then normalisation was manually checked and carefully corrected for each automatically processed spectrum. That resulted in a set of around 250 normalised spectra and pseudo-continua fits, making up the active dataset.

Initial tests of deep learning methods in stellar spectrum processing focused on two distinct tasks: segmentation of spectrum into pseudo-continuum and non-pseudo-continuum parts, and pseudo-continuum prediction. The segmentation can be considered as a subclass of a classification problem. It aims to predict the class that a segment of input data belongs to. The segment can be as small as one pixel in the case of image segmentation or flux measurement at a given wavelength (sample) in the case of one-dimensional spectral data segmentation. In our case we classify spectrum sample-wise into two classes: ”continuum”, and ”non-continuum” (see the bottom panel of Fig. 1 for an example).

It can be given by a function $f:\leavevmode\nobreak\ \mathcal{R}^{n}\rightarrow\{0,1\}^{n}$, where $n\in\mathcal{N}$, is the number of samples in the input and the output, $1$ corresponds to the pseudo-continuum class, and $0$ to the non-pseudo-continuum class. It is important to note that the result returned by the last layer of a neural network is composed of real numbers in the range from 0 to 1. Thresholding must be applied to obtain discrete classes, . The common choice used also in this work is $0.5$. Spectrum segmentation was considered here as an auxiliary task and as a potential regularisation technique (Kukačka et al., 2017).

A functional form of continuum prediction, which is a multidimensional regression, differs slightly from the segmentation and is given by $f:\leavevmode\nobreak\ \mathcal{R}^{n}\rightarrow\mathcal{R}^{n}$, with the same notation as above. The co-domain is here over the real numbers, instead of a discrete set of values that represent different classes. An exemplary result of such a prediction for both, segmentation and a pseudo-continuum prediction task is given in Fig. 1.

A neural network learning needs an optimisation algorithm that adjusts well free parameters of the model to the approximate trained function. The quality of the obtained mapping can be evaluated using various loss functions and metrics. A loss function denotes a differentiable function whose value is optimised during the training process by the algorithm that iteratively adjusts free parameters of a neural network under the training. A metric is a function that is used to evaluate the performance of the model but is not directly used for optimisation during the training. Metric functions can be non-differentiable. The adopted naming convention comes from the TensorFlow machine learning library (Abadi et al., 2015).

The cross-entropy loss function is often used for segmentation:

where $N$ is a number of samples, $y_{i}$ is a vector of target classes for i-th sample, and $\hat{y}_{i}$ is a vector of the model’s output for the same sample. $y_{i,j}$ equals $1$ only if i-th sample belongs to j-th class, so $y_{i}\in\{0,1\}^{M}$, where M is a number of classes. The cross-entropy equals zero if a model perfectly assigns classes to samples. In this work, the binary-cross-entropy loss was applied as there are only two distinct classes. The accuracy, which is given by the number of correct predictions divided by the number of all predictions was used as a metric. Here, in the case of the binary segmentation, the $accuracy=(\textrm{TP}+\textrm{TN})/N$, where TP denotes the number of true positive predictions, TN corresponds to true negative, and $N$ gives the number of samples.

The mean squared error (MSE) was used as a loss function for the pseudo-continuum prediction. It is given by:

where $N$ denotes the number of samples, $y$ is a vector of target values, and $\hat{y}$ is an estimated vector. MSE is a standard loss function for a regression problem. The mean absolute error (MAE) was used as a metric for this task. The functional form of MAE is given by:

All performed experiments use the Nesterov-accelerated Adaptive Moment Estimation optimiser (Nadam; Dozat, 2016), which iteratively adjusts free parameters of a neural network during a training process. It is based on two main concepts. The first is Nesterov’s momentum which is beneficial in dimensions of an objective function with small curvature that consistently points one direction (Nesterov, 1983). The second concept is the adaptive learning rate that is able to compute individual learning rates based on the first and the second moments of gradients (Adam; Kingma & Ba, 2017), and works well with sparse gradients and non-stationary objectives.

Initial exploration of neural network architectures was focused on tasks of pseudo-continuum prediction and spectrum segmentation. As these problems are inherently one-dimensional (inputs and targets are sequential data), we have implemented and tested one-dimensional versions of several neural network architectures that were successfully used in the field of image segmentation. We expected that advances made in the two-dimensional domain, with necessary adaptations and some minor changes, could be adopted for the processing of one-dimensional signals that stellar spectra are. The purpose of these tests was to check this hypothesis. To some extent, the method to find the best neural network follows the work of Radosavovic et al. (2020).

The selection of potential architectures was based in particular on the paper of Hoeser & Kuenzer (2020), which gives a detailed overview of both, the historical development and current state-of-the-art solutions. The architectures selected for the experiments were: Fully Convolutional Network (FCN, Long et al., 2015), Deconvolution Network (DeconvNet, Noh et al., 2015), U-Net (Ronneberger et al., 2015), UNet++, (Zhou et al., 2018), Feature Pyramid Network (FPN, Lin et al., 2017; Kirillov et al., 2019), and Pyramid Scene Parsing Network (PSPNet, Zhao et al., 2017). Additionally, a new architecture – UPPNet, which is an extension of the U-Net architecture was proposed.

The same training and evaluation scheme was used for all tests to make the comparison informative. Each tested neural network is composed of a body and a prediction head. The body is the part sampled from the adopted parameters space, considering the type of architecture and its free hyper-parameters (e.g. a number of layers, etc.). As an input, it takes a one-dimensional spectrum (vector of the length equals 8192) and outputs several feature maps of the same length as the input spectrum (e.g. in the case of a body that return 8 feature maps, a matrix of size (8192,8) is an output). In the context of a one-dimensional spectrum, feature maps can represent different shapes of spectral lines. Schematic diagrams of all tested bodies can be found in Figs. 12-18 in Appendix A. As the main building block, all networks use residual bottleneck blocks with group convolution (Xie et al., 2016), referred here as residual blocks (RB). Hyper-parameters of RBs are: a number of feature maps, a group width and a bottleneck ratio (in all experiments the bottleneck ratio was fixed to one, see Radosavovic et al. (2020)). The head takes feature maps from the body part as an input and returns a final prediction. All heads share the same architecture, i.e. have three point-wise convolutional layers (LeCun et al., 1989) with the number of features equal 64, 32 followed by 1. In the first two layers the ReLU ($\textrm{ReLU}(x)=\max(x,0)$) activation function was used, while in the third one, it was the softmax function for segmentation and ReLU for pseudo-continuum prediction. The exception is the Fully Convolutional Network, whose architecture does not allow for this consistent approach. In this case, we skipped some of the tests (details are given in the architecture description below).

One hundred neural architecture realisations with a number of free parameters ranging from 200000 to 300000 were sampled from each architecture and trained in a low-data regime ($\sim$5000 training samples from the synthetic training dataset, 30 epochs training, where epoch means single complete pass through the training data) in a pseudo-continuum prediction task. Networks’ validation was performed on the validation dataset separated at the beginning from the training set. The obtained distribution of models that can be found in Fig. 2, which shows the probability that the loss value on a validation set of a neural network sampled from a given architecture will be lower than a corresponding abscissa value. For example, approximately 60 percent of neural networks of the UPPNet (full) architecture have a loss function value lower than $4\times 10^{-3}$. This plot shows that neural networks of different architectures differ in the concentration of high-quality models in their parameters space and in the mean squared error of the best neural network. The UPPNet (full) can be thus considered the most promising architecture as it gives many relatively good models and also has the best model among all the trained networks.

In the second step of exploration, the best neural network from each architecture was picked and trained in segmentation/pseudo-continuum prediction, and also in both tasks simultaneously. In the case of the simultaneous training neural networks were equipped with two independent prediction heads. This training covered the entire synthetic dataset and lasted 150 epochs, where epoch means a single complete pass through the training data. The metrics were calculated on the same validation dataset as in the first step of exploration. The first 100 epochs used a learning rate equal $10^{-4}$ and a learning rate equal $10^{-5}$ was used for the remaining steps.

Brief summary of the results can be found in Table 1. Mean absolute error or accuracy is shown for each architecture and task. The best value in each column is given in bold. UPPNet (full), trained on both tasks simultaneously is leading in pseudo-continuum prediction with MAE equal 0.0110, while in segmentation the best is UPPNet (sparse) trained only in the segmentation task, with accuracy equal 0.9166. The best, when trained in pseudo-continuum prediction only, is the U-Net network, with MAE equal to 0.0112. Training in both tasks simultaneously results in a systematic decrease in the segmentation quality but has little effect on the pseudo-continuum estimation. An overview of original models used in the image semantic segmentation, with a short note about their one-dimensional versions with the results and the conclusions is provided below.

Insights from the previously mentioned architectures have encouraged us to experiment with a different placement of skipped connections together with the use of Pyramid Pooling Modules. This has led to U-Net with Pyramid Pooling Modules (UPPNet). First, we hypothesised that not all skipped connections are needed, but at the same time we suspected that PPMs included at some depths may result in better predictions. The proposed architecture is derived from U-Net by randomly dropping skipped connections or by replacing them with PPM. This version of UPPNet is denoted as sparse. U-Net, DeconvNet, and PSPNet are special cases of this architecture. Next, we decided to experiment with a version of UPPNet which is derived from the U-Net by replacing all skipped connections with PPMs, with an additional PPM module at the bottom of the network. This version of UPPNet is denoted as full. The schematic diagram of UPPNet (full) can be found in Fig. 4.

Figure 2 partially justifies the idea behind sparse UPPNet as there are relatively many networks giving loss values lower that $4\times 10^{-3}$, but Table 1 shows that in full training regime this additional degree of freedom in skipped connections arrangement not necessarily leads to results better than obtained with U-Net and PSPNet (separate training: $\textrm{MAE}=0.0122$, $\textrm{accuracy}=0.9166$, C&S training: $\textrm{MAE}=0.0126$, $\textrm{accuracy}=0.8857$).

The regular UPPNet (full) architecture generally gives better results. Figure 2 shows that there are relatively many models with low loss value and that the best single network is of this kind. In the training on the full synthetic dataset (see Table 1) the advantage of this network decreases and the quality of pseudo-continuum prediction is comparable to those of U-Net and PSPNet models (separate training: $\textrm{MAE}=0.0116$, C&S training: $\textrm{MAE}=0.0110$). Nonetheless, the best result in pseudo-continuum normalisation belongs to UPPNet (full) architecture when trained in both tasks. Additionally, visual inspection of the results of U-Net and UPPNet showed that the latter gives slightly smaller residuals on hydrogen H$\alpha$ and H$\beta$ spectral lines which are important for atmospheric parameters’ estimation. The quality of segmentation is moderate in the case of training only in this task ($\textrm{accuracy}=0.9065$), but is the best among models trained in both tasks simultaneously ($\textrm{accuracy}=0.9033$). Because of these findings, the final model for pseudo-continuum prediction uses UPPNet (full) architecture as its main building block and is trained in both tasks simultaneously.

SUPPNet is composed of two UPPNet (full) blocks and four prediction heads (Seg 1, Seg 2, Cont 1 and Cont 2; see Fig. 5). The UPPNet module was chosen as the main building block because of its high quality reviewed in Section 2.5. The heads compute final predictions from the high-resolution feature maps created by UPPNet blocks and have simple architecture, described in Section 2.4. The first block is responsible for the coarse prediction of a continuum/non-continuum segmentation mask (Seg 1) and pseudo-continuum (Cont 1). The mentioned outputs are used only while training, for deep supervision (intermediate supervision) (Newell et al., 2016; Zhou et al., 2018), which is beneficial in two ways. First, it helps to deal with the vanishing gradient problem. Second, the deep supervision regularises the neural network model. As an input, the second block takes spectrum, intermediate predictions (Seg 1 and Cont 1) and high-resolution feature maps from the first UPPNet block concatenated together. It outputs improved features that are used by the Seg 2 and Cont 2 heads for final predictions.

Both UPPNet blocks shared hyper-parameters (e.g. a depth, a number of residual blocks in each residual stage, etc.) but not their weights. The choice of hyper-parameters was based on a sampling of 400 UPPNet models that were trained in a low-data regime in a pseudo-continuum prediction task, following the procedure from Section 2.4. Subsequently, the best UPPNet was selected. The depth of this model equals 8 (its narrowing path has 8 pooling layers in total), with the number of residual blocks in residual stages successively equal 1, 1, 1, 2, 2, 5, 6, 7, 10, 7, 6, 5, 2, 2, 1, 1, 1 with the number of filters at each stage equal respectively 12, 16, 16, 20, 24, 32, 44, 44, 44, 44, 44, 32, 24, 20, 16, 16, 12. The narrowing and widening parts are symmetrical with respect to the bottom residual stage. PPM modules in SUPPNet have four filters and use residual stages that are composed of a single residual block. Their group width equals 64 so residual stages effectively use non-grouped convolutions. The total number of free parameters of the proposed SUPPNet approximately equals one million.

First, the SUPPNet model was trained on a synthetic dataset only. The training data was fed into the model using a data generator that augmented training examples by randomly cropping the spectra with ratio ranging from 0.7 to 2.5, reflecting the spectrum over the y-axis with 0.5 probability, and adding Gaussian noise to obtain a signal-to-noise ratio from 30 to 500. The ratio equals one means that an input sampling of 0.03 Å was kept. The ratio 0.7 corresponds to a sampling 0.021 Å and a spectral range $170$ Å ($170\approx 0.021\times 8192$), and 2.5 to a sampling of 0.075 Å  that corresponds to an $610$ Åinput window width. In this setting, the average sampling of the training data was equal $0.05$ Åand this is the value used later for prediction.

The Nadam optimiser was used for training as in all other experiments described in this work. A learning rate (lr) equals $10^{-4}$ for the first 100 epochs and was decreased to $10^{-5}$ for the remaining 50 epochs. After these 150 epochs, a learning curve flattened. We did not experiment with different parameters of the Nadam optimiser and left them equal to their default values ($\beta_{1}=0.9$, $\beta_{2}=0.999$). A batch size equals 128. We denote this version SUPPNet (synth, S).

For active learning, the SUPPNet (synth) model was loaded and the learning continued for the next 100 epochs (for first 50 epochs $lr=10^{-4}$, and later $lr=10^{-5}$), using the same augmentations but with 10 percent of the synthetic training, examples replaced with spectra and pseudo-continua acquired from manual normalisation. This resulted in SUPPNet (active, A) network.

All neural network outputs were used in the training process. As in previous experiments, a mean squared error was used as a loss for pseudo-continuum regression, and mean binary cross-entropy for segmentation. The MSE loss was multiplied by a factor of 400 to give it the same order of magnitude as loss used in segmentation branches.

The proposed spectrum normalisation method consists of three stages. First, the spectrum is re-sampled with the sampling of 0.05 Å, and the sliding window of the size equal 8192 samples with the shift of 256 samples is applied. This particular shift means that each sample is normalised $32=8192/256$ times. The data prepared in this way is then re-scaled (min-max normalisation is applied). The second step is the application of the SUPP Network. As the post-processing step, the predicted pseudo-continua are scaled back. Finally, the result is calculated as a weighted average over all predictions for each sample. This is also used to obtain a qualitative measure of the model’s inherent uncertainty of both predictions (pseudo-continuum and segmentation).

The processing described above gives final pseudo-continuum and segmentation mask predictions. Because of the noisy pattern of the order of 0.001 of the relative magnitude present in the final result, we recommended an application of additional final post-processing. We used a smoothing spline available in SciPy Python module (Virtanen et al., 2020, function UnivariateSpline). A useful advantage of this approach toward smoothing is the possibility of incorporating the estimated pseudo-continuum errors in the smoothing process. A smoothing spline fit contains a knots arrangement that takes into account the uncertainty of the predicted pseudo-continuum. For example, it places fewer knots in the ranges of greater uncertainty. Example SUPPNet result can be found in Fig. 1.

To measure normalisation quality and minimise uncertainty introduced by the manual normalisation, the first test used only synthetic spectra. In the beginning, six chosen stars were modelled using ATLAS/SYNTHE codes. Parameters for synthetic spectra were taken from Table 2 and related articles. Missing parameters were manually estimated to be equal $T_{\textrm{eff}}=30000$ K, $\log g=3.15$ in the case of HD 155806 (O7.5 IIIe) star, and $\log g=3.90$ for HD 90882 (B9.5 V). Then, the synthetic normalised spectrum for each star was multiplied by about 200 different pseudo-continua derived from the manual normalisation. That gave about 1200 spectra in total, which were later normalised using the two versions of SUPPNet (synth and active), and used to calculate normalisation metrics.

The summary statistics of this experiment can be found in the top two rows of Table 3, in Fig. 6 and Fig. 19 in Appendix B. Medians of residuals, which measure a normalisation bias, are between $-0.0016$ (SUPPNet active, HD 25069, K0 III) and 0.0017 (SUPPNet synth, HD 27411, A3m). Residuals in the case of SUPPNet trained using active learning are systematically smaller than when trained using only synthetic dataset. This means that the latter places pseudo-continuum systematically lower. The dispersion of residuals measured as a difference between 84.13 and 15.87 percentiles are between 0.0022 (SUPPNet synth, HD 155806, O7.5 IIIe) and 0.0095 (SUPPNet synth, HD 27411, A3m). The residuals are often slightly smaller when active learning is applied. For SUPPNet trained using the active learning the dispersion of residuals is slightly but significantly smaller in the case of HD 27411 (A3m), HD 37495 (G4 V) and HD 25069 (K0 III), is significantly larger for HD 90882 (B9.5 V). There are no statistically significant differences between dispersions of residuals for HD 155806 (O7.5 V) and HD 37495 (F4 V). The detailed inspection of Fig. 6 shows that systematic errors vary significantly across both wavelength and spectra parameters. They are especially prominent for A3m (HD 27411) and F4V (HD 37495) stars, in wavelength shorter than 4500 Å, where they reach 0.03. Figure 7 is a close-up of the 3900–4500 Å wavelength range of the A3 V synthetic spectra with the mean normalisation result and residuals. It can be seen, that a significant normalisation bias arises in a range where the whole spectrum is below the continuum level. Average biases in other wavelength ranges of synthetic spectra’s residuals are generally below 0.01.

We consider this statistics to be close to the realistic normalisation uncertainty in wavelength ranges with spectral features well represented in synthetic models. Nonetheless, normalisation of the synthetic spectra does not allow us to draw conclusions about the normalisation quality of the observed spectra, as they contain additional features (e.g. complex emission spectral lines).

Approximately a hundred spectra from the UVES POP library were used to assess the quality of SUPPNet on observed stellar spectra. In the beginning, the majority of spectra from spectral type O to G were manually normalised. Then spectra were normalised using both versions of SUPPNet, and residuals’ statistics were calculated. The results are summarised in Fig. 8.

In the case of the model trained using only synthetic data some biases are prominent around 6750 Åand for wavelengths shorter than 4500 Å. Strong telluric lines at approximately 6870 Å, which were absent in synthetic spectra, are the cause of the first bias. The second problem appears because SUPPNet (synth) systematically places pseudo-continuum too low in wide hydrogen absorption lines of A and F spectra types. Check Figs. 20-24 in Appendix B for separate plots for each spectral type. The median of residuals is significantly different from 0 and equals $-0.0006$, which means that the model places pseudo-continuum slightly above levels chosen by astronomers. The dispersion of residuals measured as a difference between 84.13 and 15.87 percentiles equals 0.0124. Root mean squared error equals 0.0122.

SUPPNet trained using an active learning approach shows slightly better characteristics. Dispersion of its residuals (0.0118, $\textrm{RMS}=0.0128$) is not significantly different from the SUPPNet (synth), but most of the systematic effects that can be seen in Fig. 8 are reduced (compare bias in wavelengths shorter than 4500 Å) after training that included real spectra. Nonetheless, SUPPNet (active) shares the tendency of the model trained on a synthetic dataset, to place the pseudo-continuum higher than when a spectrum is manually normalised. This tendency is measured by a median of residuals which equals $-0.0018$ and is significantly (for a 95% confidence interval) different from 0. This effect can potentially be introduced by the human, who often model pseudo-continuum to lie lower than in reality. Differences in medians of residuals for both SUPPNet versions, although statistically significant, are at most comparable and often smaller than an intrinsic error of manual normalisation described in detail in Section 4.3.

Normalisation quality and active learning importance vary significantly across spectral types. In general the later the spectral type is the larger is the dispersion of normalisation residuals. Differences between residuals’ medians of both versions of SUPPNets for O, B, and A type stars are statistically significant. However, SUPPNet synth and active are not significantly different in terms of residuals’ dispersions and remaining medians. The results are summarised in Table 4 and summary plots for each spectral type can be found in Figs. 20-24 in Appendix B. The tendency to place pseudo-continuum higher than during manual normalisation holds for most spectral types, with a notable exception of G type stars. Inspection showed that in spectral ranges with substantial narrow absorption lines blending, where the continuum is absent across wide parts of a spectrum, the model tends to place pseudo-continuum below the correct level. A simple workaround for this SUPPNet limitation is to reduce default sampling from 0.05 Å to 0.03–0.04 Å, when working with spectral types later than G5.

The second issue is the relatively high uncertainty in the H$\alpha$ Balmer line, prominent especially in the case of F and A type stars. Closer examination showed that this can be explained by erroneous manual normalisation. For this spectral range, the manual normalisation is impossible as a wavy pattern introduced by imperfections of an instrument pipeline crosses and changes this spectral feature significantly (see Fig. 9 for details).

The positive influence of active learning on SUPPNet normalisation quality is the most prominent in the case of O type stars, where strong emission features are often present. A spectral range of the HD 148937 (O6.5) star, with H$\alpha$ Balmer and He I 6678 Å lines in emission is shown in the Fig. 10 . SUPPNet (active) relatively well predicts pseudo-continuum across these features while SUPPNet, trained using synthetic data only, treats these features as a part of pseudo-continuum.

In the last step of the SUPPNet analysis, six stars were normalised automatically and manually by three of us (TR, EN and NP), and carefully compared. Their spectral types range from O7.5 V to K0 III and show most of the typical spectral features. The main characteristics of the selected stars are gathered in Table 2.

HD 155806 (HR 6397, $V=5.53$ mag) is the hottest Galactic Oe star (Fullerton et al., 2011). Its initial spectral classification O7.5 V[n]e (Walborn, 1973) was changed to O7.5 IIIe because of the strength of its metallic features (Negueruela et al., 2004). Stars of this type are rare and show a double-peaked or central emission in their Balmer lines.

HD 27411 (HR 1353, $V=6.06$ mag, A3m) is an example of a chemically peculiar (CP) star. Its atmospheric parameters and abundances were investigated in detail in the context of diffusion theory in the work of Catanzaro & Balona (2012).

HD 90882 (HR 4116, $\delta$ Sex, B9.5 V), HD 37495 (HR 1935, $\nu^{2}$ Col, F5 V), and HD 59967 (HR 2882, G3 V) are typical representatives of their spectral types. The first is a rapidly rotating B type star, and the last is a young ($\approx 0.4$ Gyr), active, slowly rotating solar-twin star.

The last star for detailed analysis is HD 25069 (HR 1232, $V=5.83$ mag). In the UVES POP catalogue, its spectral type is G9 V, while SIMBAD’s database sources give K0 III or K0/1 III. Here K0 III is used. This star is a representative example of a late G and early K spectral type.

The results for all stars can be found in the four bottom rows of Table 3. Figure 11 contains detailed pseudo-continua and residuals for the HD 27411 (A3m) star. The results show that the differences between manually normalised spectra are between $-0.0042$ and $0.0016$ in the median with typical dispersion, defined by a 15.87–84.13 percentiles band, ranging approximately from 0.0040 to 0.0330 (see NP–TR and EN–TR rows in Table 3). This is the scale of uncertainty inherent in manual normalisation. In terms of residual’s statistics, the quality of the SUPPNet normalisation method is superior in comparison to the quality of the manual normalisation.

The left panels of Fig. 11 and Figs. 25-29 in Appendix B convince why the pseudo-continuum fitting is a challenging task. In HD 27411 most of the spectrum is disturbed by a semi-periodic pattern that arises from imperfect orders merging and a blaze function removal. SUPPNet models this pseudo-continuum type relatively well for wavelengths longer than the Balmer H$\beta$ line and significantly worse in a spectral range from 4400 to 4800 Å, where the amplitude and frequency of this pattern significantly increase. Nonetheless, the error amplitude is of the order of 0.02, both for manual and SUPPNet normalisation. For wavelengths shorter than 4200 Å  the dispersion between different normalised fluxes grows considerably. In this range, it is difficult to assess the normalisation quality without referring to the synthetic spectral model, which could potentially guide manual normalisation.

The uncertainty estimated by SUPPNets has only qualitative meaning and informs the users where they can expect normalisation results to be most uncertain, but not necessarily where the model failed in predicting pseudo-continuum. As can be seen in the bottom panel of Fig. 11 the estimated uncertainty is the highest in the wide absorption lines. However, in the spectral range from 4400 to 4800 Å the proposed method did not capture the ambiguous character of the predicted pseudo-continuum.

Stellar spectra present in databases have various levels of noise, resolutions, and projected rotational velocities. We tested the consistency of normalisation with respect to those parameters. For that purpose synthetic spectra calculated for parameters of HD 27411 (A3m) with a signal-to-noise equal 30 and 500, a projected rotational velocity equal 0, 50, 100, 200 km s^-1, and resolution equal $10^{4}$ and $10^{5}$ were used.

Predicted pseudo-continuum is generally consistent for the tested signal-to-noise ratios. The differences between results increase for low-resolution spectra and in heavily blended line regions where there is no real continuum in the flux. The test with low-quality medium-resolution spectra showed that in this case, the proposed method places the pseudo-continuum significantly too high. This problem can be suppressed by increasing a sampling step or by training SUPPNet using low-resolution, noisy spectra.

For a spectrum broadened due to the projected rotational velocity, the differences between predicted pseudo-continua are generally smaller than 0.01, except for the wavelengths shorter than 4300 Å where a flux is well below expected pseudo-continuum.

The results described above are summarised in Fig. 30 in Appendix C.