© The Authors 2024

1 Introduction

In recent years, an increasing number of large spectroscopic surveys have been started as well as completed. These surveys have generated an unprecedented number of spectra. For instance, the ongoing LAMOST Galactic Spectroscopic Survey of the Galactic Anti-centre (LSS-GAC) (Luo et al. 2015) has already collected millions of low-resolution (R ~ 1800) spectra of stars. Faced with massive amounts of stellar spectroscopic data, researchers need automated analysis tools to efficiently extract useful astrophysical information about target stars. The stellar spectral template library is able to perform analytical tasks by matching observed spectra with template spectra, such as classification of stellar types and determination of stellar atmospheric parameters. These pieces of information are essential for understanding the formation, structure, and evolution of the Milky Way (Jofré et al. 2019). Several stellar parameter pipelines, such as SSPP (Lee et al. 2008), LASP (Luo et al. 2015), and LSP3 (Xiang et al. 2015), have emerged to provide convenient solutions for spectral parameter determinations. These mainstream parametric measurement pipelines rely on synthetic template libraries or empirical template libraries; however, they all have corresponding shortcomings.

The synthetic template libraries comprise stellar spectra generated by theoretical models based on stellar properties. The synthetic template libraries offer noiseless data covering a wide range of stellar parameter values. In a related work on template library construction based on LAMOST data, it has been pointed out that the influence of theoretical and instrumental factors could lead to inconsistencies between the synthetic model and observed spectra at the wavelengths of some features (Du et al. 2019). This mismatch may lead to unrealistic parameter measurements. On the theoretical side, the parameters that have been considered during spectral modelling are not comprehensive, as they lack factors such as microturbulence, rotation velocities, and convection. Assumptions regarding the physics of stellar models both in their interiors and atmospheres also impact the accuracy of any synthetic grid. These theoretical limitations have been mentioned in previous studies (Kurucz 2013; Prieto et al. 2018; Franchini et al. 2018). On the instrumental side, the modelling of instrumental factors and noise lacks perfection (Ballester et al. 2000; Martioli et al. 2012).

The empirical template libraries such as MILES (Sánchez-Blázquez et al. 2006; Falcón-Barroso et al. 2011) and ELODIE (Prugniel & Soubiran 2001; Prugniel et al. 2007) consist of high-quality, real observed spectra with stellar atmospheric parameters (Du et al. 2019; Royer et al. 2024; Prieto et al. 2004; Verro et al. 2022), having been created through the efforts of observers. However, a good empirical template library requires coverage of a wide range of parameters, and acquiring observed spectra that satisfy this requirement is a challenging task. Furthermore, the stellar labels determined using empirical template libraries are limited in accuracy. This is due to the errors present in the parameter labelling of the observed spectra, which are passed on to the spectra to be measured during the parameter measurement. Jofré et al. (2019) reviewed the latest efforts in this field (Ren et al. 2016; Jönsson et al. 2018) in order to assess the accuracy of stellar parameter measurement results and discussed the uncertainties of the measurement results from the perspectives of random uncertainties, systematic uncertainties, and biases.

Some works have been done by researchers to overcome the shortcomings of the current synthetic spectra. Works using traditional methods typically attempt to take into account a variety of complex physical effects and data properties when modelling spectra. Kovalev et al. (2019) incorporated non-local thermodynamic equilibrium (NLTE) effects into the model atmospheres when determining stellar parameters and abundances in medium-resolution spectra of FGK-type stars. Similarly, Amarsi (2016) investigated the departure from local thermodynamic equilibrium (LTE) of atomic oxygen and its impact on various oxygen lines in FGK-type stars, offering grids predicting 3D NLTE-based equivalent widths and abundance corrections. Additionally, Bialek et al. (2020) bridged the synthetic gap by augmenting the synthetic spectra with real observational features, such as by adding Gaussian noise, applying rotational and radial velocities, removing the continuum, and masking geodesic regions. However, these methods usually require a great deal of a priori knowledge and manual adjustments, and it is difficult for them to model the various effects in the spectra, which tend to overlook certain details or effects.

Compared with traditional methods, deep learning methods have stronger non-linear modelling capabilities and are able to learn features and laws automatically without many manual adjustments. Deep learning technologies, including neural networks, convolutional neural networks (CNNs), and generative adversarial networks (GANs), are widely utilised in astronomy, particularly in spectral analysis. Recent studies have introduced innovative algorithms, such as ZETA-PAYNE by Straumit et al. (2022), that use CNNs to accurately analyse spectra from hot OBAF-type stars in SDSS-V data. Additionally, Gebran et al. (2022) have demonstrated the effectiveness of CNNs in determining stellar parameters from observed and synthetic spectra, offering practical guidance for their application in astronomy. Furthermore, deep learning techniques, such as LSTM neural networks, have been employed in studies, including that of Hu et al. (2022), to analyse spectral time series of Type Ia supernovae, enabling precise reconstruction of spectral sequences from a single observed spectrum. In response to the small number of O-type stars in the LAMOST data release, Zheng et al. (2020) proposed the SGAN for generating artificial O-type spectra. O’Briain et al. (2021) proposed Cycle-StarNet, a domain-adaptive method to narrow down the discrepancy between synthetic spectra and observed spectra. Gebran (2024) established a connection between stellar parameters and spectra by training a fully connected neural network, which can be used for library construction. Nevertheless, the reduction of the mismatch between the synthetic model and observed spectra is worth exploring.

In this paper, we propose a synthetic-to-observed spectral translation method called SOST. This method is based on a GAN and uses Kurucz synthetic spectra¹ (Castelli 2005) (Kurucz model atmospheres with the radiative code SYNTHE) and LAMOST observed spectra as data sources. The well-trained SOST method can guide the calibration of synthetic spectra by providing observed spectra and correct errors that may arise during the synthetic spectrum modelling process. The SOST-corrected synthetic spectra can achieve a favourable match with the observed spectra. This addresses the past shortcomings of relying on either synthetic spectra or observed spectra data.

Additionally, we construct a stellar spectral template library using the SOST method. Our spectral library offers a set of stellar spectra with uniformly and extensively distributed parameters. The effective temperature (T_eff) ranges from 3500 to 8000 K, the surface gravity (log g) spans 0.0–5.0 dex, and the metallicity ([Fe/H]) varies from −2.0 to 0.5 dex. The wavelength coverage for the templates spans 3900–8700 Å at a resolution of approximately R = 1800. To assess the accuracy of our constructed library, this study employs a template-matching method based on 𝒳² minimisation (Jofré et al. 2010) to compare the parameters obtained using SOST with measurements from PASTEL (Soubiran et al. 2016) and the Large Sky Area Multi-Object Fibre Spectroscopic Telescope Data Release 8 (LAMOST DR8; Yan et al. 2022). The template-matching method determines the parameters by comparing the target spectrum with the template spectrum (Zwitter et al. 2008; Lee et al. 2008). Through comparison with external databases, the accuracy of this template was determined to be 121 K, 0.26 dex, and 0.13 dex for T_eff, log g, and [Fe/H], respectively. This work provides a powerful support for stellar studies and a powerful tool for a deeper understanding of stars in the Universe (Suda et al. 2008; Yoon et al. 2018).

This paper is organised as follows: the SOST and the stellar spectral template library construction method are introduced in Sect. 2. The spectral data used for the experiments as well as the designed experiments for stellar spectral parameter determinations are described in detail in Sect. 3. Finally, we summarise this work in Sect. 4.

2 Method

In this section, the SOST, the SOST training process, and the stellar spectral template library construction method are described in detail. To facilitate the method construction, certain reasonable assumptions needed to be made. Synthetic and observed spectra lie in two related but distinct distribution domains, which we call the synthetic and observed spectral domains. Synthetic spectra and observed spectra both have some underlying physical characteristics (such as effective temperature, metallicity, and surface gravity), but they also contain distinctive information. This distinctiveness could arise from instrumental factors (such as line spread functions), errors introduced during data processing (such as pseudo-continuum normalisation), or variations not present in synthetic spectra (such as atmospheric absorption lines from Earth’s atmosphere). Learning the transformation between these two domains directly is complex and difficult. Therefore, we assumed that the spectrum can be decomposed into a content code and a style code. Specifically, content codes refer to the underlying spatial structure. Content codes determine the overall trend of the spectrum, reflecting the physical characteristics inherent in the spectrum (robust spectral features). Style codes influence the specific details of the spectrum, representing the unique properties of both synthetic and observed spectra. The unique properties include the features resulting from assumptions made during synthetic spectrum modelling and the instrumental factors of observed spectra.

Based on this assumption, we designed the SOST with an auto-encoder (Kingma & Welling 2013) and discriminator model in both the synthetic and observed spectra domains. The autoencoder model can realise the separation and reorganisation of spectral content and style features. The content features extracted by the encoder have the same distribution, and the style features have different distributions. Cross-domain content features and style features can be combined by one of the generators in order to achieve cross-domain transfer of spectra. The discriminator plays the role of a teacher, evaluating the generated spectra, which in turn improves the quality of the generated domain migration spectra. The SOST allows us to construct a stellar spectral template library by transferring the desired synthetic spectra across domains to the corresponding observed spectra, enabling correction of the synthetic spectra.

Fig. 1 Architecture of SOST. The style encoder utilizes strided convolutional layers followed by a global average pooling layer and a fully connected layer. In contrast, the content encoder employs strided convolutional layers alongside residual blocks. In the decoder, a multi-layer perceptron is employed to compute a set of AdaIN parameters based on the style code. Subsequently, the content code is processed through residual blocks with AdaIN layers to achieve cross-domain spectral translation.

2.1 SOST

As shown in Fig. 1, the SOST consists of a content encoder, $E_{\text{synth}}^{\text{c}}$ and $E_{\text{obs}}^{\text{c}}$, a style encoder, $E_{\text{synth}}^{\text{c}}$ and $E_{\text{obs}}^{\text{c}}$ a generator G_synth and G_obs, and a discriminator, D_synth and D_obs, for the synthetic and observed spectra domains. The content encoder extracts the common features of the two domains, and the style encoder extracts the unique style features of the two domains. The common features of the two domains and the unique style features align with our assumption regarding ‘related but distinct domains’. The content features and the cross-domain style features are combined by the generator using the instance normalisation (AdaIN) (Huang & Belongie 2017) to transform the domains. The equation of AdaIN is as follow: $$\mathrm{AdaIN}(z,\gamma ,\beta )=\gamma \left(\frac{z-\mu (z)}{\sigma (z)}\right)+\beta ,$$(1)

where z is the activation value obtained by the content feature after going through the convolutional layer in the generator, µ and σ are the channel mean and standard deviation, γ and β are the parameters generated by the style feature through the multi-layer perceptron (MLP) layer in the generator, which can be viewed in MUNIT (Huang et al. 2018) for specific implementations.

The generation of spectra is countered by a discriminator that evaluates the quality of the generated spectra. The method is based on MUNIT (Huang et al. 2018). MUNIT constructs a multi-modal unsupervised image translation framework that is capable of transforming images across different seasons while maintaining photo content. Similarly, in our spectral calibration, robust spectral features correspond to content features in photo images, while the calibration of synthetic spectra factors correspond to style features, akin to seasons in photo images. In order to adapt MUNIT to the spectral generation work, we modified its internal network structure (convolutional as well as normalisation methods, etc.), and added cycle consistency loss so that the method is applicable to domains between the observed spectra of LAMOST and the synthetic spectra of the Kurucz synthetic spectra transfer learning. The components of the SOST are briefly described in this subsection, and the functions of each component and the training process of the method are described in detail in Sect. 2.2.

2.2 Architecture and training

In order to achieve domain transfer between spectra while ensuring that the generated spectra do not lose their own information, we designed multiple loss functions to constrain the training process. We used spectral reconstruction loss (Eq. (3)) and cross-domain feature reconstruction loss (Eqs. (4) and (5)) to train the auto-encoder for feature extraction and reduction. We used cycle consistency loss (Eq. (6)) to constrain the generated spectra and generative adversarial loss (Eq. (7)) to train the antagonism between the generated spectra and the real spectra so that the distribution of the cross-domain spectra matches the distribution of the target spectra. The guidance of the method during training was achieved through these loss functions. In the following section, we provide a detailed explanation of the components of Eq. (2). The overall loss during training is as follows: $$\begin{array}{l}ℒ={\lambda }_{\text{GAN}}\left({ℒ}_{\text{GAN}}^{\text{synth}}+{ℒ}_{\text{GAN}}^{\text{obs}}\right)\hfill \\ +{\lambda }_{\text{id}}\left({ℒ}_{\text{recon }}^{\text{synth }}+{ℒ}_{\text{recon }}^{\text{obs }}+{ℒ}_{\text{cyc }}^{\text{synth }}+{ℒ}_{\text{cyc }}^{\text{obs }}\right)\hfill \\ +{\lambda }_{\text{cont }}\left({ℒ}_{\text{recon }}^{c_{\text{synth }}}+{ℒ}_{\text{recon }}^{c_{\text{obs }}}\right)\hfill \\ +{\lambda }_{\text{style }}\left({ℒ}_{\text{recon }}^{s_{\text{synth }}}+L_{\text{recon }}^{S_{\text{obs }}}\right),\hfill \end{array}$$(2)

where the weights λ_GAN, λ_id, λ_cont, and λ_style are hyperparameters that control the importance of each part of the loss function. The individual loss functions are described in detail in the following subsections. After conducting several empirical tests, we found that setting λ_GAN = 1, λ_id = 10, λ = 1, and λ_style = 1 resulted in satisfactory performance.

Fig. 2 Auto-encoder model. The SOST has two auto-encoders. The blue and yellow colours represent the synthetic and observed spectra domains, respectively. The auto-encoder extracts the corresponding content features and style features from the input spectra and generates the corresponding spectra through feature combinations.

2.2.1 In-domain spectra reconstruction loss

The encoder-decoders should be able to map spectra to latent representations and then back to original spectra within each domain. This process enables the auto-encoder model to learn both the feature extraction and generative capabilities of spectra. As shown in Fig. 2, E^c and E^s compress the data from the high-dimensional space to the potential space, and generator G does the opposite; that is, it converts the potential space back to the high-dimensional space. The encoder and generator pairs that are inverse to each other are learned by minimising the difference between the reconstructed spectrum and the input spectrum. In the next step, the learned encoder and generator pairs can be used for feature extraction and restoration of the spectral data. Equation (3) demonstrates the detailed process of in-domain spectra reconstruction, which involves feature extraction by the encoder followed by reconstruction through the generator and computing the L1 loss with the original spectrum. The loss within the synthetic spectra domain can be written as $$\begin{array}{l}{ℒ}_{\text{recon }}^{\text{synth }}={\text{𝔼}}_{{𝒳}_{\text{synth }}~p\left({𝒳}_{\text{synut }}\right)}\hfill \\ \left[{\Vert G_{\text{synth }}\left(E_{\text{synth }}^{\text{c}}\left({\mathcal{𝒳}}_{\text{synth }}\right),E_{\text{synth }}^{\text{s}}\left({\mathcal{𝒳}}_{\text{synth }}\right)\right)-{\mathcal{𝒳}}_{\text{synth }}\Vert }_1\right],\hfill \end{array}$$(3)

where 𝒳_synth is the synthetic spectrum sampled from the distribution of synthetic spectra p(𝒳_synth) and $E_{\text{synth}}^{\text{c}}$, $E_{\text{synth}}^{\text{s}}$, and G_synth are the content encoder, style encoder, and generator within the synthetic spectra domain, respectively. The L1 loss (mean absolute error) is utilised to compute the loss function.

Fig. 3 Reconstruction of cross-domain features. In this figure, the dashed arrows point to the target where the reconstruction loss is to be performed. The combination of content features and style features belonging to the two domains in the figure achieves the cross-domain transfer of the spectrum. After the cross-domain translation, the spectrum should be able to reconstruct the corresponding content and style features when the encoder extracts the features again.

2.2.2 Cross-domain feature reconstruction loss

For the content and style features obtained from sampling during cross-domain translation, we wanted to be able to reconstruct them after generating the cross-domain transfer spectra (generated by combining features across domains) by encoding them again as shown in Fig. 3. Reconstruction of features ensures that the encoder is able to extract the content features of the shared latent space and the style features of the unique latent space, which is consistent with the hypothesis presented at the beginning of this section. Instead of using observed spectra for style feature extraction during training, we directly adopted the standard Gaussian distribution as the prior distribution. This constraint ensures that the style potential space is regularised, which helps generate diverse spectra when transferring across domains. Our goal is that synthetic spectra can be generated diversely by different features provided by different observed spectra rather than through a simple one-to-one correspondence between synthetic and observed spectra. Equations (4) and (5) show the process of the generator generating spectra in the cross-domain followed by feature extraction by the encoder and computing the L1 loss with the input features. The cross-domain feature reconstruction can be achieved according to the following equation: $$\begin{array}{l}{ℒ}_{\text{recon}}^{{\mathcal{C}}_{\text{synth}}}={\mathbb{E}}_{{\text{C}}_{\text{synth}}\sim p\left(c_{\text{synth}}\right),s_{\text{obs}}\sim q\left(s_{\text{cbs}}\right)}\hfill \\ \left[\parallel E_{\text{obs}}^{\text{c}}\left(G_{\text{obs}}\left(c_{\text{synth}},s_{\text{obs}}\right)-c_{\text{synth}}\right){\parallel }_1\right]\hfill \end{array}$$(4) $$\begin{array}{l}{ℒ}_{\text{recon}}^{{\text{s}}_{\text{obs}}}={\mathbb{E}}_{{\text{C}}_{\text{synth}}\sim p\left(c_{\text{synth}}\right),s_{\text{obs}}\sim q\left(s_{\text{obs}}\right)}\hfill \\ \left[\parallel E_{\text{obs}}^{\text{s}}\left(G_{\text{obs}}\left(c_{\text{synth}},s_{\text{obs}}\right)-s_{\text{obs}}\right){\parallel }_1\right],\hfill \end{array}$$(5)

where c_synth and s_obs are the content code in the synthetic spectra domain and the style code in the observed spectra domain, respectively. The value of p(c_synth) is given by $E_{\text{synth}}^{\text{s}}$ (𝒳_synth), and q(s_obs) is the prior 𝒩(0, 1).

Fig. 4 Graphical representation of cycle consistency loss. This figure demonstrates the process of spectra being converted from one domain to another and then reversed again to be converted back to the original domain. At this point, the spectra should be consistent with those before the cross-domain translation.

2.2.3 Cycle consistency loss

Our goal is to ensure that spectra that have undergone cross-domain translation also retain their own information, rather than simply being mapped to the target domain (see Fig. 4). To achieve this goal, it was necessary to ensure that spectra can be transformed from one domain to another and can be reverse-transformed back to the original domain again. The principle of cycle consistency (Zhu et al. 2017) helped the generative model better understand the properties shared by the spectra of the two domains and preserve these properties during the generation process. If the generated spectrum is irrelevant to the original input spectrum, it will be difficult to return to the original domain by reverse conversion. This two-way conversion constraint ensures that the generated spectra are associated with the original input spectra, which enhances the robustness and interpretability of the model. Equation (6) is the L1 loss between the cycle reconstructed synthetic spectra and the original synthetic spectra. The cycle consistency loss for synthetic spectra can be written as $$\begin{array}{l}{ℒ}_{\text{cyc}}^{\text{synth}}={\mathbb{E}}_{{𝒳}_{\text{synuh}}\sim p\left({𝒳}_{\text{synth}}\right),{𝒳}_{\text{synth}\to \text{ols}}\sim p\left({𝒳}_{\text{synth-obs}}\right)}\hfill \\ \left[\parallel G_{\text{synth}}\left(E_{\text{obs}}^{\text{c}}\left({\mathcal{𝒳}}_{\text{synth}\to \text{obs}}\right),E_{\text{synth}}^{\text{s}}\left({\mathcal{𝒳}}_{\text{synth}}\right)\right)-{\mathcal{𝒳}}_{\text{synth}}{\parallel }_1\right],\hfill \end{array}$$(6)

where 𝒳_synth→obs is the synthetic spectrum transformed across domains to the observed spectral domain, which is the main focus of this paper. The value of p(𝒳_synth→obs) is given by G_obs(c_synth, s_obs)

2.2.4 Generative adversarial loss

The method, in addition to the auto-encoder model mentioned above, includes the discriminator model, as shown in Fig. 5, and the two parts are trained alternately to form the GAN together. The goal of the GAN is for the generator to produce spectra that are sufficient to deceive the discriminator. We wanted the spectra transferred across domains to have the same data distribution as the spectra in the target domain, which requires matching the distribution of the synthetic spectra with the distribution of the observed spectra by the discriminator. In short, taking the generating adversarial process of synthetic spectra as an example, the task of the auto-encoder model is to generate synthetic spectra that are sufficiently similar to the observed spectra in order to fool the discriminator. The task of the discriminator is exactly the opposite, namely to distinguish the spectra generated by the generator from the real observed spectra, and the two models are trained alternately in order to generate the confrontation.

Taking the discriminator D_obs as an example, the loss function in Eq. (7) consists of two terms. One of the terms is related to the generator, and D_obs(𝒳_synth→obs) is the probability that the discriminator will judge whether the generated spectra are observed spectra or not. We wanted the discriminator to be able to predict the generated spectra as negative samples, that is, the output of the discriminator tends to zero. For the generator, we wanted it to generate spectra that are sufficient to deceive the discriminator, that is, the result of D_obs(𝒳_synth→obs) to tends to one. In this way, the generator and the discriminator form an antagonistic relationship. For the second term, we required that the discriminator d is able to identify real observed spectra as positive samples, and the discriminator output tends to one. The same is true for the discriminator D_obs. The discriminator D_synth was trained in the same way. The generator was trained alternately with the discriminator, forming a maximum-minimum loss training setup. In conclusion, the generative adversarial loss within the observed spectral domain is given as follows: $$\begin{array}{c}{ℒ}_{\text{GAN}}^{\text{obs}}={\mathbb{E}}_{{𝒳}_{\text{synth}\to \text{obs}}\sim p\left({𝒳}_{\text{synth}\to \text{obs}}\right)}\left[\log \left(1-D_{\text{obs}}\left({𝒳}_{\text{synth}\to \text{obs}}\right)\right)\right]\\ +{\mathbb{E}}_{{\mathcal{𝒳}}_{\text{ols}}\sim p\left({\mathcal{𝒳}}_{\text{obs}}\right)}\left[\log \left(D_{\text{obs}}\left({𝒳}_{\text{obs}}\right)\right)\right],\hfill \end{array}$$(7)

where D_obs is a discriminator in the observed spectra domain. It attempts to distinguish between the synthetic spectra after cross-domain transfer 𝒳_synth→obs and the real observed spectra 𝒳_obs.

Fig. 5 Generative adversarial framework. There are respective discriminators, Dsynth and Dobs, in the synthetic spectra domain and the observed spectra domain. They give the corresponding confidence level for the incoming spectral data. The discriminators try to discriminate the original spectra as zero (i.e. true) and the spectra transferred across the domains as one (i.e. false) during the training process.

2.3 Stellar spectral template library construction method

The SOST can control the style of translation output by providing observed spectra. In order to generate better spectral data, a matching exercise between the synthetic spectra and the observed spectra is required. The matched observed spectra will be a better guide for the transfer of the synthetic spectra across domains. Here, the matching was done by minimising the 𝒳² distance. We defined 𝒳² as follows: $${𝒳}^2={\displaystyle \sum _i^N}\frac{{\left(S^i-O^i\right)}^2}{{\sigma }_i^2}$$(8)

where sⁱ and oⁱ are the flux densities of the synthetic spectra and the observed spectra of the ith pixel, respectively. The term N is the total number of pixels used to calculate the 𝒳² distance, and σ_i is the flux density error of the observed spectra of the ith pixel.

After obtaining the matched pairs of synthetic spectra and observed spectra, we input them into the corresponding encoder (as shown in Fig. 6) to get the corresponding content features and style features. The content features provided by the synthetic spectra were combined with the style features provided by the observed spectra in a generator for the observed spectra domain. This implemented a domain transfer from synthetic spectra to observed spectra, which we also refer to as calibration of the synthetic spectra. Next, we discuss how we used the calibrated synthetic spectra to construct synthetic template libraries with atmospheric parameters that are inherited from the original synthetic spectra.

Fig. 6 Cross-domain transfer of synthetic spectra. The cross-domain transfer of the synthetic spectra is achieved by the generation of cross-domain combination of the content features extracted from the synthetic spectra and the style features extracted from the observed spectra.

3 Experiments and results

In this section, we first detail how the calibration of the synthetic spectra is achieved, including the construction of the experimental dataset and the pre-processing procedure. Then we perform in-domain reconstruction, cross-domain translation, and t-SNE visualisation of the test set spectra to validate the SOST. Indomain reconstruction refers to the reduction of the test set spectra within the original spectral domain, which helps validate the performance of the method in the same spectral domain and is also a prerequisite for cross-domain translation. We also used the t-SNE visualisation method to clearly demonstrate the distribution of the synthetic spectra and observed spectra as well as the features extracted from them before and after the cross-domain translation. The visualisation enables the changes in the spectral distribution to be observed intuitively, further demonstrating the good performance of the method in the spectral domain transfer task.

However, good visualisation results do not guarantee that the spectra generated by the SOST correctly correspond to the parameters. Thus, in conclusion, we constructed a stellar spectral template library using the SOST and then estimated the parameters for the PASTEL catalogue and the observed spectra released by LAMOST DR8 (Yan et al. 2022), respectively. Through this experiment, the usefulness and reliability of the SOST in practical parameter measurement tasks were successfully demonstrated. This series of experiments provides a solid experimental foundation for our research and an important reference and tool for further research in the field of spectral generation and parameter determinations.

3.1 Calibration of synthetic spectra

In this section, we detail the experimental setup as well as the in-domain reconstruction, trans-domain transfer, and t-SNE visualisation of the domain distribution experiments. We show how SOST can compensate for the difference in the synthetic spectra with respect to the observed spectra, also known as the synthetic gap.

3.1.1 Experimental setup

The source of the observed spectra is the eighth data release of the LAMOST low-resolution spectral survey. To ensure the quality of the spectral data as well as a wider and uniform parameter distribution, we sampled the spectra according to the stratified sampling of the officially provided atmospheric parameters. We selected 50 000 spectra of the training set, 5000 spectra of the validation set, and 5000 spectra of the test set with an S/N greater than 30 in the g band. The source of the synthetic spectra is the Kurucz synthetic template library, and the parameter coverage of these synthetic spectra is 3500 K ≤T_eff ≤8000 K, in steps of 250 K; 0.0 dex ≤log g ≤5.0 dex, in steps of 0.5 dex; and -=2.5 dex ≤[Fe/H] ≤0.5 dex, in steps of 0.25 dex. For generating the Kurucz synthetic templates used in our study, we utilised the Kurucz spectral synthesis code based on the ATLAS stellar atmosphere models provided by Sbordone et al. (2004). The initial resolution of the synthetic data is 2000. To match the resolution of the LAMOST low-resolution spectra (1800), we smoothed the synthetic spectra using fast Fourier transform convolution with a Gaussian kernel (Zhang et al. 2020).

Although the Kurucz synthetic template library covers a wide range of parameters, their parameter distributions are relatively sparse, and generating synthetic spectra is relatively time-consuming. Ting et al. (2019) have demonstrated that the PAYNE model based on residual networks exhibits strong fitting capabilities, providing a good connection between physical parameters and synthetic spectra. To ensure consistency with the parameter space of observed spectra, we trained the PAYNE model using the Kurucz synthetic spectral template library. The well-trained PAYNE model generated synthetic spectral datasets that match the parameter distribution of observed spectra. In this way, datasets of synthetic and observed spectra with consistent parameter distributions were obtained.

Next, we employed linear interpolation to interpolate the synthetic spectra and observed spectra onto the same wavelength grid, ensuring they are aligned in the same positions. The covered wavelength range extended from 3900 Å to 8700 Å, with a sampling of 1 Å per pixel. For the fluxes, we performed pseudocontinuum spectral normalisation (Zhang et al. 2020, 2021; Cai et al. 2022, 2024). This removed the dispersion effect between the spectra and highlighted the chemical features, which helps when comparing and analysing the spectral features of the stars in order to study their physical properties and chemical composition. These steps play a key role in the pre-processing of spectral data.

3.1.2 Reconstruction of synthetic spectra

As discussed in Sect. 2.2.1, SOST performs spectral reconstruction during training. We first show in this section that in-domain reconstruction is valid with cycle reconstruction, which is necessary for cross-domain translation.

The validity of the in-domain reconstruction and the cycle reconstruction is demonstrated in Fig. 7. The blue end of the spectrum contains most of the information needed to constrain stellar parameters, so only the blue end of the spectrum is generally used in parameter measurements. Although the blue end of LAMOST covers wavelengths from 3700 Å ~ 5900 Å, data below 4000 Å and above 5500 Å are excluded due to the low instrumental response at both ends, leaving this part of the spectrum of greater interest to us (Yang et al. 2022, 2023a,b). The residual values corresponding to the spectral data of the test set and the histogram of the residual distribution are shown in Fig. 7. The red line indicates the average residual corresponding to each angstrom in the spectral data, and the orange line indicates the average absolute residual. The calculation results show that the average absolute residual value is less than 1% and is basically negligible, and this proves that the SOST can extract potential features from spectra and reconstruct spectra based on these features.

The phenomenon of having more residual values (exceeding one standard deviation) at certain positions of emission lines and absorption lines, such as the Balmer Hbeta line, G-line, and MgI line, does exist. The reasons for these variations remain unclear, and the complex variations of emission lines and absorption lines in the spectra may be one factor.

Fig. 7 In-domain reconstruction and cycle reconstruction. This figure shows the residuals between the 5000 synthetic spectra in the test set and themselves after in-domain reconstruction and cycle reconstruction. These spectra cover most of the stellar parameter range. The blue part indicates the residuals after reconstruction of each spectrum in the wavelength range 4000–5500 Å. The colour depth is positively correlated with the number of spectra in this error. The red line in the figure indicates the average residuals, and the orange line indicates the average absolute residuals. The upper panel shows the in-domain reconstruction residuals as the relative residuals between 𝒳synth→synth and the original spectra 𝒳synth, and the lower panel shows the cycle reconstruction residuals as the relative residuals between the spectra 𝒳synth→obs→synth and the original spectra 𝒳synth. The residuals for both the in-domain and cycle reconstructions are very small and can be neglected, as can be seen in the figure.

3.1.3 Visualisation of domain distribution using t-SNE

In this section, we use the method introduced in Sect. 2.3 to pair the synthetic and observed spectra in the test set. The matched spectral pairs are fed into the encoder in the SOST for feature extraction. The synthetic spectra provide content features, the observed spectra provide style features, and the two features are combined to achieve domain transfer from the Kurucz synthetic spectra to the LAMOST observed spectra. The t-SNE (Van der Maaten & Hinton 2008) is a non-linear dimensionality reduction and data visualisation algorithm that maps high-dimensional data onto a low-dimensional space in order to be able to better present the similarities and differences between the data. We used t-SNE to visualise the cross-domain translation process.

As shown in Fig. 8, we compressed the synthetic and observed spectra pairs, the content features of the synthetic and observed spectra, the stylistic features, and the synthetic and observed spectra pairs after the cross-domain translation to two dimensions using t-SNE, respectively. In the upper-left panel, one can see that there is a more significant difference between the synthetic spectra and the observed spectra. This is the synthetic gap mentioned earlier, which is the gap we aim to overcome. The upper-right panel shows the common content features extracted by the content encoder. The lower-left panel shows that the style encoder extracts the style features of each of the two domains, which matches our hypothesis. The lower-right panel shows that the data distributions of the two domains are mixed together after the cross-domain translation, which proves that SOST overcomes the differences between the two domains better.

Fig. 8 Distribution of t-SNE visualisation domain. Spectra and features are represented visually using the t-SNE algorithm. The top-left and bottom-right panels show the visualised domain distribution of the synthetic spectra and observed spectra before and after cross-domain translation. The top-right and bottom-left show the corresponding feature distributions of the synthetic spectra and observed spectra extracted by the content encoder and the style encoder. The 𝒳-axis and y-axis coordinates represent the positions of data points in the new coordinate system after dimensionality reduction. The positions after dimensionality reduction can reflect the similarity between data points.

Fig. 9 Comparison of the distribution of parameters of the ELODIE empirical template library with that of the SOST-constructed spectral template library. The horizontal and vertical coordinates are the effective Teff and log g, respectively, and the [Fe/H] is indicated by colour.

3.2 Estimation of stellar spectral parameters

In previous experiments, the successful transfer of the synthetic spectra across domains was demonstrated by t-SNE visualisation. In this section, we explore the effect of the corrected synthetic spectra on parameter estimation. In this experiment, template spectra were generated using SOST and parameter estimation was performed using a method based on 𝒳² minimization for template-matching. The estimated parameters were compared with the stellar atmospheric parameters published by the PASTEL database and LAMOST DR8. Different spectroscopic surveys usually employ different spectral analysis methods to obtain the stellar atmospheric parameters and chemical compositions and to compare them with different spectroscopic surveys in order to assess the reliability of the generated spectra (Kassounian et al. 2019).

3.2.1 Stellar spectral template library construction

We used the low-resolution A, F, G, and K type stellar parameter catalogues published by LAMOST as a reference. Grids were constructed with a range of steps of 150 K, 0.25 dex, and 0.15 dex between the T_eff of 3500–8000 K, log g of 0.0– 5.0 dex, and [Fe/H] of −2.0–0.5 dex, respectively. In order to minimise the effect of LASP measurement errors, we expelled grids with fewer than five stars. A stellar parameter was randomly selected as the parameter of the template spectra within the grid, and a uniform and wide-coverage parameter distribution was ultimately constructed. The metallicity considered in our study is [Fe/H], which is consistent with the official data from LAMOST. We generated synthetic spectra corresponding to this parameter distribution using the PAYNE model. The pairs of synthetic spectra and observed spectra were then matched by the 𝒳² distance. Our test set included a wide range of atmospheric parameters, and to reduce the computational effort, we used the data from the test set directly for matching. The synthetic spectra were then transferred across domains to observed spectra using the SOST, where the spectral atmospheric parameter values after the cross-domain translation were inherited from the synthetic spectra. The corrected synthetic spectra form a stellar spectral template library with a broad and uniform distribution of stellar parameters. The stellar spectral template library contains 2413 spectra, and the spectra in the library have a resolution of R ~ 1800 covering the wavelength range 3900–8700 Å.

Our method has several advantages over previous methods of constructing synthetic template libraries and empirical template libraries. Firstly, the introduction of the PAYNE model made it easy for us to generate synthetic spectra data with any parameter. Second, the SOST better enables correction of the synthetic spectra, overcomes the synthetic gap, and does not require the a priori knowledge of the atmospheric parameters of the observed spectra. Finally, the advantages based on the first two points allowed us to easily construct a stellar spectral template library with a wider parameter coverage and more uniform parameter distributions.

The atmospheric parameter distributions of the ELODIE (Wu et al. 2011) library, which LASP derives stellar atmospheric parameters, and the SOST spectral library are shown in Fig. 9. As can be seen from the figure, our constructed stellar spectral template library has a wider parameter coverage and more uniform parameter distribution. The influence of clusters and vacancies on the parameter distribution of the traditional empirical template libraries of the measurement results has been overcome. Figure 10 shows some samples of SOST spectral templates. All spectra in the SOST spectral templates are normalised by pseudo-continuum normalisation. For spectra with [Fe/H] greater than 0.3, the absorption lines appear deeper. Next, we use the SOST spectral library for parameter estimation in order to further demonstrate the validity of this stellar spectral template library.

Fig 10 Some samples of SOST spectral templates. This figure shows 15 spectra of normal stellar types from the library from top to bottom based on temperature, from highest to lowest.

3.2.2 Accuracy assessment by the PASTEL database

PASTEL is a database of stellar atmospheric parameters brought together from multiple sources. The parameters it provides are obtained from detailed analyses of high-resolution high-S/N spectra, thus providing more accurate measurements, and are often used to assess the accuracy of parameter measurements. In this study, the first step to assessing our obtained parameters was to remove the null values of the parameters in the PASTEL database. Then, a cross-match with the low-resolution spectral data from the eighth LAMOST release was performed with a matching radius of 3 arcsec. Stars with T_eff between 3500 K and 8000 K were selected. Finally, for T_eff, log ɡ, and [Fe/H], 2190, 1872, and 1837 samples that can be compared were obtained.

In this subsection, we use the template spectral before and after the SOST correction to parameterise the observed spectra in the PASTEL database. Typically, one assesses the quality of the predictions using the deviation and scatter between the predictions and the labels of the stars. Good predictions usually have a low bias and scatter. Importantly, the bias quantifies the accuracy, whereas the scatter quantifies the precision of the method.

As can be seen in Fig. 11, after SOST correction, the predictions of the stellar atmospheric parameters T_eff, log ɡ, and [Fe/H] show a significant improvement. The bias decreases from 144 K, −0.08 dex, and −0.33 dex before correction to −20 K, −0.01 dex, and 0 dex, respectively, while the dispersion decreases from 140 K, 0.31 dex, and 0.21 dex to 121 K, 0.26 dex, and 0.13 dex, respectively, before correction. The significant improvement of these metrics suggests that the SOST-corrected spectra are closer to the observed spectra, and therefore more accurate measurements of stellar parameters can be obtained. This further demonstrates the effectiveness of the SOST construction template library.

Fig 11 Comparisons of the results of parameter measurements using the Kurucz and SOST spectra libraries with the PASTEL database. The figure shows the comparison of the parameters Teff, log ɡ, and [Fe/H] from left to right. The samples are colour-coded according to their density. In the top row, the horizontal coordinates are the measurements provided by the PASTEL database, the vertical coordinates are the measurements from the Kurucz spectra library, and the diagonal dashed line is the reference line. The bias and scatter of the samples are labelled in the upper-left corner of the picture in the upper panel. In the bottom row, the vertical coordinates represent the results of the measurements using the SOST spectra library. As is shown, a lower bias and scatter are obtained when using the SOST spectra library compared to PASTEL database.

Fig 12 Comparison of our results with the parameters of LAMOST DR8. A total of 6 542 528 spectral data were compared.

3.2.3 Comparison with the results of the LAMOST DR8 data survey

LAMOST DR8 is the eighth data release of LAMOST, a large sky area multi-object fibre-optic spectroscopic telescope developed by China for large-scale spectroscopic surveys. LAMOST uses LASP to estimate the T_eff, log ɡ, [Fe/H], and other parameters of stars. In our study, we selected a total of 6 542 528 spectral data in the T_eff range of 3500–8000 K for parameter measurements and compared the results with those released by LAMOST DR8. Subsequently, we utilised a template-matching method based on 𝒳² minimisation, using the SOST spectral library, to derive the stellar parameters of the LAMOST DR8 data. This method selects two templates with the minimum 𝒳² for the target spectrum, assigns weights based on their 𝒳² distances, and then calculates the weighted average as the measurement result. From Fig. 12, it can be seen that compared with the parametric results published by LAMOST DR8, our results have a measurement deviation of −40 ± 121 K for T_eff, −0.05 ± 0.22 dex for log ɡ, and −0.01 ± 0.16 dex for [Fe/H]. Overall, our results are in good agreement with the parameter results published by LAMOST DR8, indicating that the template library we constructed achieves satisfactory results in stellar parameter measurements.

The relationship between the parameter difference and S/N of the measurements using the SOST and LAMOST DR8 is shown in Fig. 13. We chose spectra with S/N between 6 and 250. This choice was due to the fact that the small number of spectra with S/N greater than 250 lose statistical significance. From the figure, it can be seen that with the increase in S/N, the deviation and scatter of our predictions from the LASP residuals do not change significantly, and they still agree more closely with the LASP results. When the S/N is low (less than ten), the accuracy of the corresponding three parameters is 130.04 K, 0.24 dex, and 0.17 dex. The accuracy of the three parameters decreases to 118.42 K, 0.21 dex, and 0.15 dex when the S/N is high (greater than 200), and the accuracy of the measurement results is further improved with the increase of the S/N.

To further examine the differences between the SOST measurements and the parameters of LAMOST DR8, in Fig. 14 we mesh the T_eff, log ɡ, and [Fe/H] parameter measurements released by LAMOST DR8. Specifically, we meshed in steps of 400 K and 0.5 dex in the region where T_eff ranges from 3500 to 8000 K and log ɡ ranges from 0 to 5 dex. For [Fe/H], we divided it into the following ranges: [−3.0, −1.5]; [–1.5, −0.6]; [−0.6, −0.2]; [−0.2, 0.2]; and [0.2, 1.0]. In each grid, the difference between the SOST measurements and the LAMOST DR8 parameters was calculated, summed, and divided by the number of spectra to obtain the mean value. The following can be observed from Fig. 14. For stars with higher T_eff (>6700 K), our estimates show differences in temperature compared to those published by LAMOST. The LASP provides higher values of the T_eff for hot stars compared to our results. Ren et al. (2016) pointed out that there is a systematic overestimation of T_eff measurements for stars at higher temperatures on external calibration of LASP atmospheric parameters. This discrepancy is consistent with our results. For stars with log ɡ less than two, our SOST method gives lower values compared to LASP, and the difference is more pronounced. This discrepancy may be due to the small number of stars with log g less than two in ELODIE and the template library used by LASP, which leads to the deviation of the results (Chen et al. 2022). For stars with [Fe/H] in the range [–3.0, –1.5], our SOST method generally gives higher values, and the differences are more pronounced. This may be due to limitations in the parameter coverage of spectral data currently released using LAMOST DR8. Our template library lacks adequate coverage of low-metallicity stars, causing the results to be pulled towards more metal-abundant quantities during template matching.

Taken together, our measurements differ in some cases from the parameters of LAMOST DR8, and these differences may be influenced by the coverage of the template library and stellar properties. It is important to note that despite the differences, our estimates still show some degree of agreement with the atmospheric parameters published by LAMOST.

Fig 13 SOST and LAMOST DR8 parameter differences as a function of the spectral S/N. The black dashed line in the figure is the reference line where the difference is zero. We binned the S/N ([6, 50], [50, 100], [100, 150], [150, 200], and [200, 250]). The mean deviation is shown in orange, and the scatter of the residuals is indicated by the red line in the figure.

4 Conclusions

In traditional stellar spectral template library construction efforts, the construction is usually based on observed spectra or synthetic spectra. We have proposed a template library construction method, called SOST, that is different from previous methods. This method is based on a GAN and uses an auto-encoder to separate and recombine the content and style features of spectra. The content features of synthetic spectra are cross-domain combined with the style features of observed spectra, achieving a cross-domain transfer of synthetic spectra. This process accomplishes the correction of synthetic spectra, and the corrected synthetic spectra can have a good match with the observed spectra that are to be measured. The distinctive feature of this method lies in its elimination of the need for spectral expertise and manual feature extraction.

With this approach, we calibrated the Kurucz synthetic spectra using LAMOST observed spectra, which in turn led to the construction of a widely distributed and uniform template library of stellar spectra. The template library includes a total of 2413 spectra. It covers a parameter space ranging from 3500 to 8000 K for effective temperature, 0.0 to 5.0 dex for surface gravity, and −2.0 to 0.5 dex for metallicity, while T_eff, log ɡ, and [Fe/H] were constructed in grids with step ranges of 150 K, 0.25 dex, and 0.15 dex, respectively. We used the library to perform parameter measurements of the PASTEL database, and after method correction, the accuracy of the predictions of the atmospheric parameters T_eff, log ɡ, and [Fe/H] was significantly improved, decreasing from 140 K, 0.31 dex, and 0.21 dex to 121 K, 0.26 dex, and 0.13 dex. Finally, we also used our stellar spectral template library to estimate the atmospheric parameters of more than 6.5 million spectral data released by LAMOST DR8.

The automatic calibration principle of SOST is versatile and can be applied to reduce discrepancies among various astronomical projects. Moreover, we plan to explore its applicability to various astronomical projects. The work in constructing the template library contributes to stellar parameter measurements, establishing a solid foundation for future studies on the stellar populations, kinematics, and chemistry of the Galactic disc, along with its evolutionary history. Our results further demonstrate the significance of deep learning in advancing astronomical techniques.

Fig 14 Distribution of the mean value of the difference between our results and the parameters of LAMOST DR8 in the Teff−log ɡ plane. The three columns of subplots in the figure are the Teff difference, the log ɡ difference, and the [Fe/H] difference (from left to right), and the four rows are the different [Fe/H] ranges (from top to bottom): [−3.0, −1.5), [−1.5, −0.6), [−0.6, −0.2), [−0.2, 0.2), and [0.2, 1.0]. Each dot in the figure represents the mean value obtained by summing the parameter difference of a grid with a Teff of 400k and a log ɡ of 0.5 dex size range and dividing it by the number of spectra, with the size of the dots representing the size of the difference, where red means that our results are larger than those published in LAMOST DR8 and blue means the opposite.

Acknowledgements

We thank the anonymous referee for valuable suggestions. The work is supported by the National Natural Science Foundation of China (Grant Nos. U1931209, 62272336, 62306204), the Projects of Science and Technology Cooperation and Exchange of Shanxi Province (Grant Nos. 202204041101037, 202204041101033), the Fundamental Research Program of Shanxi Province (Grant Nos. 20210302123223, 202103021224275, 202103021223267), the Taiyuan University of Science and Technology Scientific Research Initial Funding (No. 20212053), the Reward for Outstanding Doctoral Work in Shanxi (No. 20222107). Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.

Appendix A SOST architecture

Here, we summarise the detailed architecture of each sub-module of SOST in Table A.1. Each row represents the operations applied by each network layer, including convolution, normalisation, and activation functions. The CONV layer denotes a standard 1D convolutional layer; the DCONV layer represents a transposed convolutional layer; FC stands for a fully connected layer; IN stands for InstanceNorm1d; ReLU and LeakyReLU are used as activation functions; InstanceNorm1d and AdaIN are used for normalisation; AdaptiveAvgPool1d and AvgPool1d layer denote a pooling layer. For each layer, N, K, S, and P represent the number of filters, the size of the filter kernel, the stride of the convolution operation, and the size of the padding for the convolution operation, respectively. The input spectrum has 4800 pixels. After dimensionality reduction by the encoder, the content code dimension was 256 × 298, while the style code dimension was 8 × 1.

References

All Tables

All Figures

Fig. 1 Architecture of SOST. The style encoder utilizes strided convolutional layers followed by a global average pooling layer and a fully connected layer. In contrast, the content encoder employs strided convolutional layers alongside residual blocks. In the decoder, a multi-layer perceptron is employed to compute a set of AdaIN parameters based on the style code. Subsequently, the content code is processed through residual blocks with AdaIN layers to achieve cross-domain spectral translation.

In the text

	Fig. 2 Auto-encoder model. The SOST has two auto-encoders. The blue and yellow colours represent the synthetic and observed spectra domains, respectively. The auto-encoder extracts the corresponding content features and style features from the input spectra and generates the corresponding spectra through feature combinations.
In the text

Fig. 3 Reconstruction of cross-domain features. In this figure, the dashed arrows point to the target where the reconstruction loss is to be performed. The combination of content features and style features belonging to the two domains in the figure achieves the cross-domain transfer of the spectrum. After the cross-domain translation, the spectrum should be able to reconstruct the corresponding content and style features when the encoder extracts the features again.

In the text

	Fig. 4 Graphical representation of cycle consistency loss. This figure demonstrates the process of spectra being converted from one domain to another and then reversed again to be converted back to the original domain. At this point, the spectra should be consistent with those before the cross-domain translation.
In the text

Fig. 5 Generative adversarial framework. There are respective discriminators, Dsynth and Dobs, in the synthetic spectra domain and the observed spectra domain. They give the corresponding confidence level for the incoming spectral data. The discriminators try to discriminate the original spectra as zero (i.e. true) and the spectra transferred across the domains as one (i.e. false) during the training process.

In the text

	Fig. 6 Cross-domain transfer of synthetic spectra. The cross-domain transfer of the synthetic spectra is achieved by the generation of cross-domain combination of the content features extracted from the synthetic spectra and the style features extracted from the observed spectra.
In the text

Fig. 7 In-domain reconstruction and cycle reconstruction. This figure shows the residuals between the 5000 synthetic spectra in the test set and themselves after in-domain reconstruction and cycle reconstruction. These spectra cover most of the stellar parameter range. The blue part indicates the residuals after reconstruction of each spectrum in the wavelength range 4000–5500 Å. The colour depth is positively correlated with the number of spectra in this error. The red line in the figure indicates the average residuals, and the orange line indicates the average absolute residuals. The upper panel shows the in-domain reconstruction residuals as the relative residuals between 𝒳synth→synth and the original spectra 𝒳synth, and the lower panel shows the cycle reconstruction residuals as the relative residuals between the spectra 𝒳synth→obs→synth and the original spectra 𝒳synth. The residuals for both the in-domain and cycle reconstructions are very small and can be neglected, as can be seen in the figure.

In the text

Fig. 8 Distribution of t-SNE visualisation domain. Spectra and features are represented visually using the t-SNE algorithm. The top-left and bottom-right panels show the visualised domain distribution of the synthetic spectra and observed spectra before and after cross-domain translation. The top-right and bottom-left show the corresponding feature distributions of the synthetic spectra and observed spectra extracted by the content encoder and the style encoder. The 𝒳-axis and y-axis coordinates represent the positions of data points in the new coordinate system after dimensionality reduction. The positions after dimensionality reduction can reflect the similarity between data points.

In the text

	Fig. 9 Comparison of the distribution of parameters of the ELODIE empirical template library with that of the SOST-constructed spectral template library. The horizontal and vertical coordinates are the effective Teff and log g, respectively, and the [Fe/H] is indicated by colour.
In the text

	Fig 10 Some samples of SOST spectral templates. This figure shows 15 spectra of normal stellar types from the library from top to bottom based on temperature, from highest to lowest.
In the text

Fig 11 Comparisons of the results of parameter measurements using the Kurucz and SOST spectra libraries with the PASTEL database. The figure shows the comparison of the parameters Teff, log ɡ, and [Fe/H] from left to right. The samples are colour-coded according to their density. In the top row, the horizontal coordinates are the measurements provided by the PASTEL database, the vertical coordinates are the measurements from the Kurucz spectra library, and the diagonal dashed line is the reference line. The bias and scatter of the samples are labelled in the upper-left corner of the picture in the upper panel. In the bottom row, the vertical coordinates represent the results of the measurements using the SOST spectra library. As is shown, a lower bias and scatter are obtained when using the SOST spectra library compared to PASTEL database.

In the text

	Fig 12 Comparison of our results with the parameters of LAMOST DR8. A total of 6 542 528 spectral data were compared.
In the text

	Fig 13 SOST and LAMOST DR8 parameter differences as a function of the spectral S/N. The black dashed line in the figure is the reference line where the difference is zero. We binned the S/N ([6, 50], [50, 100], [100, 150], [150, 200], and [200, 250]). The mean deviation is shown in orange, and the scatter of the residuals is indicated by the red line in the figure.
In the text

Fig 14 Distribution of the mean value of the difference between our results and the parameters of LAMOST DR8 in the Teff−log ɡ plane. The three columns of subplots in the figure are the Teff difference, the log ɡ difference, and the [Fe/H] difference (from left to right), and the four rows are the different [Fe/H] ranges (from top to bottom): [−3.0, −1.5), [−1.5, −0.6), [−0.6, −0.2), [−0.2, 0.2), and [0.2, 1.0]. Each dot in the figure represents the mean value obtained by summing the parameter difference of a grid with a Teff of 400k and a log ɡ of 0.5 dex size range and dividing it by the number of spectra, with the size of the dots representing the size of the difference, where red means that our results are larger than those published in LAMOST DR8 and blue means the opposite.

In the text