Distance and stellar parameter estimations of solar-like stars from the LAMOST spectroscopic survey

High-quality low-resolution spectra of solar-like stars are selected from the LAMOST LRS Stellar Parameter Catalog of A, F, G and K Stars of LAMOST DR9 v1.1 (hereafter LAMOST LRS AFGK Catalog),¹¹1https://www.lamost.org/dr9/ which contains 7,060,679 spectra with determined stellar parameters ($T_{\rm eff}$, log $\it g$, $\rm[Fe/H]$) by the official LAMOST Stellar Parameter Pipeline (LASP). A total of 4,067,525 solar-like stellar spectra were selected according to the criteria in Table 1, which is similar to the criteria used by Zhang et al. (2022).

Solar-like stars of the late F-type, G-type, and early K-type are included in this study. The effective temperature range for these stars ranges from $4800$ K to $6800$ K, which encompasses the effective temperature of the Sun ($\sim$ 5800K) with a fluctuation of $\pm$ 1000K. The surface gravity log $\it g$ is constrained by the empirical formula proposed by Zhang et al. (2022) in the $T_{\rm eff}$-log $\it g$ Kiel diagram, as shown in Fig. 1 and Table 1. This empirical formula distinguishes between main-sequence stars and giants. The metallicity range is set to $-1.0<\hbox{$\rm[Fe/H]$}<1.0$ dex, which is around the metallicity of the Sun ($\hbox{$\rm[Fe/H]$}=0.0$).

In the LAMOST LRS AFGK catalog, there are 4,022,857 solar-like stars with Gaia data. To find the most reliable distance for training, we considered three primary sources of distance data: inverse parallax (including raw parallax data and parallax after zero-point correction), distance from the General Stellar Parametrizer from Photometry (GSP-Phot; Andrae et al., 2023), and distance obtained by Bailer-Jones et al. (2021) using the geometric methods. Before calculating the inverse parallax, the zero-point correction for parallax is determined separately for five-parameter and six-parameter astrometric solutions, as explained in Lindegren et al. (2021). The method for calculating zero points follows the Python code.²²2https://gitlab.com/icc-ub/public/gaiadr3_zeropoint

According to the distribution of distances shown in Fig. 2, as $\varpi/\sigma_{\varpi}$ increases, the amount of distance data decreases, especially for distances greater than 1 kpc. Compared to the corrected inverse parallax, the raw inverse parallax has a larger dispersion and relatively overestimates distances in panel (a). In panel (b), the geometric distance has a similar distribution to the corrected inverse parallax. In panel (c), the GSP-Phot distance has a smaller dispersion at close distances and tends to underestimate the distance.

In Fig. 3 GSP-Phot distances and geometric distances are shown for $\varpi/\sigma_{\varpi}>5$. GSP-Phot distances are significantly underestimated when compared to geometric distances, and this underestimation worsens at larger distances. However, for larger $\varpi/\sigma_{\varpi}$, indicating higher data quality, the underestimation is mitigated with increasing distance. In this study, geometric distances, which are derived purely through a Bayesian approach that excludes the influence of photometric data, are chosen to determine absolute magnitudes.

We calculated the absolute magnitude from the known apparent magnitude g_mean_mag in the Gaia DR3 catalog and the geometric distance $d_{\rm{geo}}$ (r_med_geo) from Bailer-Jones et al. (2021):

Here $M_{\rm{G}}$ is the calculated absolute magnitude, which is used as the label for the training process and $A_{\rm{G}}$ (ag_gspphot) is the extinction in G band from GSP-Phot Aeneas best library using BP/RP spectra. Additionally, we use the extinction information from GSP-Phot (ebpminrp_gspphot) to determine $(\rm{{BP}-{RP}})_{0}$ of the color index $(\rm{{BP}-{RP}})$ (bp_rp).

Finally, a dataset containing 18,573 stars with stellar atmosphere parameters provided by APOGEE is used for training in this study. Table 2 details the step-by-step filtering process of the training data, including the filtering criteria and the remaining number of spectra at each stage. The color-magnitude diagram of the training data, color-coded by metallicity from APOGEE, is shown in the inset of Fig. 1. Additionally, the number density distribution of each parameter is presented in Fig. 17; further details on how the reference and test sets are distinguished can be found in Sect. 3.2.

The radial velocity of the object causes the spectral lines to undergo the Doppler effect. Hence, for the input spectrum, we first needed to convert the wavelengths to the rest frame.

Second, to ensure alignment, we interpolated the raw spectra to a uniform wavelength range. Typically, both ends of the spectrum are noisy and contain less information about the spectral features. Therefore, we interpolated the entire sample set to a wavelength range of 3925–8800Å, using 1Å as the sampling point, resulting in a total of 4875 data points for spectral flux data.

Finally, each stellar spectrum in the training set was normalized by dividing it with its pseudocontinuum. The pseudocontinuum was obtained by fitting a polynomial to the spectrum. For this work, we found that despite FIBERMAS=0, there are still bad pixels that were not completely removed, leading to incorrect spectral lines and extremely high pixel values that affected the model prediction. To address this, we implemented two additional steps. First, the original spectrum was smoothed using the Savitzky-Golay filter, and then the smoothed spectrum was normalized. Using this approach, the normalized spectra outperformed those provided in the official LAMOST pipelines. Second, the pixels were thresholded in the normalized spectrum by simply setting to zero those above 1.2 and below 0.1. This straightforward threshold processing is effective because the anomalous pixels appear at random wavelengths. It is believed that as long as their values are not too far from the normalized flux range ($0-1$), individual pixels do not significantly affect the prediction ability of the neural network. Two examples illustrating these preprocessing steps are shown in Fig. 18.

A neural network has three components: an input layer, some hidden layers, and an output layer. Given a training set of input spectra with known labels ($T_{\rm eff}$, log $\it g$, $\rm[Fe/H]$, $M_{\rm G}$, $(\rm{{BP}-{RP}})_{0}$), a neural network model can generate a function that maps the input spectra to the output parameters. This nonlinear capability is attributed to the activation function, which determines the significance of each neuron in the network. The trained model can then be used to predict the physical properties of spectra with unknown stellar parameters in another dataset.

In this work, we propose a one-dimensional convolutional neural network (CNN) regression model. The architecture of the CNN model is shown in Fig. 4. It consists of three convolutional blocks for feature extraction and two fully connected layers for regression. The activation function for both the convolutional and fully connected layers is Leaky ReLU. The output layer uses a linear activation function and has five neurons, each corresponding to a stellar parameter.

We tested different network architectures specifically for our dataset, including various combinations of convolutional blocks and fully connected layers, similar to Fig. 3 in Zhang et al. (2019). We found that the architecture with three convolutional blocks and two fully connected layers is well-suited for our dataset. It is important to note that for different tasks and datasets, the model must be modified or fine-tuned.

The training process was carried out using the dataset described in Sect. 2.1, which was divided into reference and test sets in an 8:2 ratio. A total of 14,858 stars were used for training and cross-validation in the reference set, while 3,715 stars were used in the test set to evaluate the performance of the trained models. Before training, the input pseudo-continuous spectra were already normalized, while the output labels were standardized using Z-score standardization.

To ensure robust model performance, we employed five-fold cross-validation, dividing the reference set into five non-overlapping subsets. One subset served as the validation set, and the remaining four served as the training set. This strategy ensured the efficient use of data and the proper examination of the model’s generalization ability by utilizing the entire dataset for both training and validation.

For the model optimization we used the mean squared error (MSE) loss function, defined in Eq. 2, to measure the difference between the target and predicted parameters:

Here $n$ is the number of samples, $y_{i}$ is the true value, and $\hat{y_{i}}$ is the model’s predicted value. The Adam optimizer (Kingma & Ba, 2014) was employed for gradient descent, with an initial learning rate of 0.0001. This rate automatically decreases using the adaptive learning rate tuning method, ReduceOnPlateau (Ioffe & Szegedy, 2015), which halves the learning rate if the MSE value does not improve after five epochs. This approach allows the model to find a more precise solution and avoids premature convergence to a local optimum.

The CNN model was trained using the NN library Keras (Chollet et al., 2015), which provides a high-level API for the TensorFlow (Abadi et al., 2015) machine intelligence package. The train and test split, standardization, and $k$-fold cross-validation were performed using the scikit-learn library (Pedregosa et al., 2011).

The performance of the CNN model was evaluated on the test set, as shown in Fig. 5. The results indicate that the CNN model’s predictions for $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$ closely align with the APOGEE parameters, while $M_{\rm G}$ and $(\rm{{BP}-{RP}})_{0}$ closely match the Gaia parameters. There is no significant bias for these parameters. The scatter is 86 K for $T_{\rm eff}$, 0.07 dex for log $\it g$, 0.06 dex for $\rm[Fe/H]$, 0.25 mag for $M_{\rm G}$, and 0.03 mag for $(\rm{{BP}-{RP}})_{0}$. To calculate the distance, we assumed $M_{\rm{true}}\sim\mathcal{N}(\mu_{M},\sigma_{M})$, where $\mu_{M}$ is the $M_{\rm G}$ predicted by CNN, and $\sigma_{M}$ is the scatter between the labeled values and the predicted values, as defined in Eq. 3. The predicted absolute magnitudes $M_{\rm G}$ were then used to estimate the distance distribution, as shown in Eq. 5:

These estimated distances are illustrated in the lower right panel of Fig. 5, where the predicted distances are compared to the geometric distances, with a scatter of 65 pc. We also calculated the fractional error, defined as the difference between the predicted and geometric distances divided by the geometric distances. The median fractional distance error is 3.7% with a standard deviation of 8.2%, as shown in the fourth panel in Fig. 7. It should be noted that our dataset contains some contamination from unresolved binaries. These stars tend to have higher residuals because the model predicts a dimmer luminosity for them, leading to a slight bimodal distribution in the fractional distance error. However, since our focus is on studying solar-like stars, and these potential unresolved binaries are fairly uniformly distributed across the parameter space, their influence on the model is naturally averaged out across each type of spectrum during the training process. Moreover, as shown in Fig. 2 of El-Badry et al. (2018), for LAMOST-like spectra with $\rm{S/N=30}$, the typical systematic error caused by unresolved binaries is approximately 100 K in $T_{\rm eff}$, while the errors in log $\it g$ and $\rm[Fe/H]$ are smaller than 0.1 dex and 0.05 dex, respectively. This degree of systematic error is acceptable for most scientific objectives.

To further evaluate, we compared the fractional distance error and residual of the CNN predictions to the labeled values with $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$ separately, as shown in Fig. 6. In the left three panels, the mean fractional distance errors (gray dots) do not exhibit significant deviations from $T_{\rm eff}$. However, the standard deviation of fractional error (represented by the error bars on the gray dots) shows a slight increase at higher $T_{\rm eff}$, lower log $\it g$, and lower $\rm[Fe/H]$. At lower log $\it g$ ($\hbox{log\,$\it g$}<4.0$) and $\rm[Fe/H]$ ($\hbox{$\rm[Fe/H]$}<-0.5$), our model shows larger deviations from the Gaia distances. These deviations likely arise because the quality of the model predictions depends on the number of stars in the training set that span the parameter space of the test set. Consequently, we suggest that the deviations are due to an insufficient number of stars in our reference set, which is a common issue in data-driven models. Additionally, the precision of predicted distances improves with smaller distances due to Gaia’s higher accuracy in these ranges.

The right three panels of Fig. 6 show the residuals of $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$, respectively. The model predictions show excellent agreement with the ASPCAP results for the high $\rm{S/N_{g}}$ spectra. However, larger scatters are observed between CNN and ASPCAP for the lower $\rm{S/N_{g}}$ spectra. From the projected residual distributions, $\rm[Fe/H]$ is the most sensitive parameter to the signal-to-noise ratio, while $T_{\rm eff}$ and log $\it g$ are less sensitive. The metallicity depends on the strength of the spectral lines, which are more affected by the signal-to-noise ratio, whereas $T_{\rm eff}$ and log $\it g$ rely on the continuum shape and are less affected. Meanwhile, for lower effective temperatures ($\hbox{$T_{\rm eff}$}<5000$ K), lower surface gravity ($\hbox{log\,$\it g$}<4.1$), and lower metallicity ($\hbox{$\rm[Fe/H]$}<-0.5$), the CNN model tends to predict higher values compared to ASPCAP. This discrepancy is likely due to the nature of the spectra themselves, where weaker spectral features would make it challenging for the CNN model to identify the most significant features during training. Similarly, the parameters determined by ASPCAP have larger intrinsic uncertainties, which in turn affect the model performance.

In this work we find no strong evidence that the derived distance errors are affected by surface gravity, as reported in previous studies (Carlin et al., 2015; Leung & Bovy, 2019b; Stone-Martinez et al., 2024); this is due to the limited surface gravity range of our dataset, which is mostly concentrated around $3.5-4.5$ dex for dwarf stars. In the left three panels of Fig. 7, the fractional distance error is compared to the errors in $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$. There is no significant dependence of distance error on the uncertainties in these three stellar parameters. The second panel shows a slight correlation between $\Delta\hbox{log\,$\it g$}$ and log $\it g$: the model tends to overestimate log $\it g$ at lower values and underestimate it at higher values, which is also reflected in the second panel on the right in Fig. 6. The last panel shows the distribution of fractional distance error, with a median of 3.7% and a standard deviation of 8.2%, which demonstrate that this CNN model has a profound ability for distance estimations.

We also compared our CNN model with other ML algorithms, such as LightGBM boosted trees (Ke et al., 2017), Random Forests, and $k$-Nearest Neighbors. We find that our CNN model is a better choice in both precision and efficiency; the details are described in Appendix C.

Based on the solar-like stars obtained using the second screening condition in Table 2, we selected spectra with ${\rm{S/N_{g}}}\geq 10$. Among these, we identified spectra with problematic distance measurements: those where Gaia did not provide parallaxes, those where Gaia GSP-Phot did not provide distances, and those where the parallaxes had large errors ($\texttt{parallax\_over\_error}\leq 5$, indicating fractional error larger than 20%). This selection process resulted in a total of 521,455 spectra, representing 13% of the LAMOST solar-like sample. More details are shown in the Venn diagram in Fig. 8. Even with data lacking parallax values, it can be seen that GSP-Phot can provide distances in some circumstances; however, only 9,449 spectra meet this condition. Additionally, larger parallax errors lead to less precise distance estimates (Ulla et al., 2022). Fortunately, our model offers a solution for these problematic cases.

Using the same preprocessing described in Sect. 2.2, we obtained 521,424 normalized spectra spanning wavelengths from 3925Å to 8800Å. Based on the trained CNN model, we predicted the stellar parameters $T_{\rm eff}$, log $\it g$, $\rm[Fe/H]$, $M_{\rm G}$, and $(\rm{{BP}-{RP}})_{0}$ for 521,424 solar-like stars. The results are shown in Fig. 9, which displays a color–magnitude diagram of these stars using the predicted $M_{\rm G}$ and $(\rm{{BP}-{RP}})_{0}$. The CNN predictions are compared to stellar isochrones to show the expected trend in the parameter relationships. The four lines are the theoretical isochrones from the PAdova and TRieste Stellar Evolution Code (PARSEC) model (Bressan et al., 2012) with $\hbox{$\rm[Fe/H]$}=-0.8,-0.5,0.0,$ and $0.3$ dex at an age of 1.5 Gyr, as marked by the corresponding colors. The trend aligns well with the isochrones, though the large scatter is due to the multiple populations in the sample. The results including stellar parameters and distance are saved in a CSV format table and can be found online.³³3https://nadc.china-vo.org/res/r101400/ The column descriptions of the final catalog are given in Table 6.

To ensure the reliability of the distances obtained by our CNN model, we compared them with those from other studies, including Queiroz et al. (2018) (hereafter StarHorse), Stone-Martinez et al. (2024) (hereafter DistMass), and Leung & Bovy (2019b) (hereafter astroNN).

StarHorse (Santiago et al., 2016; Queiroz et al., 2018, 2020) is a Bayesian isochrone-fitting code that derives distances, extinctions, and astrophysical parameters for APOGEE stars, with a typical distance uncertainty of $\sim 5\%$. DistMass (Stone-Martinez et al., 2024) is a simple neural network that predicts distances and masses of stars in APOGEE DR17. The distances are trained from stellar parameters using Gaia DR3 distances and literature distances for star clusters, achieving a median fractional distance error of $\sim 10\%$ at higher log $\it g$ and a standard deviation of $\sim 11\%$. The deep learning model astroNN (Leung & Bovy, 2019a, b) uses APOGEE DR17 spectra and Gaia eDR3 data to determine stellar parameters, distances, and other stellar properties. These three catalogs are provided as the VACs in APOGEE DR17.⁴⁴4https://www.sdss4.org/dr17/data_access/value-added-catalogs/

We cross-matched the 521,424 stars in our results with three catalogs, resulting in 5443, 5240, and 5648 common stars for StarHorse, DistMass, and astroNN, respectively. The top three panels in Fig. 10 show the comparisons, demonstrating the consistency of the distances obtained by our CNN model with those from other studies. The scatter of the difference between this work and astroNN is 117 pc with an 18.74% standard deviation of fractional error, which is less than the 123 pc for StarHorse and 165 pc for DistMass. The mean fractional error between this work and both DistMass and astroNN is less than 5%, while for StarHorse it is 7%. Except for StarHorse, our results appear to be slightly underestimated compared to DistMass and astroNN. This discrepancy may be due to the different training sets and methodologies used in the models.

To provide a comprehensive comparison, we include pairwise comparisons between these studies in the bottom three panels of Fig. 10, involving 4401, 4788, and 5242 common stars, respectively. DistMass and astroNN appear to overestimate compared to StarHorse, while DistMass shows higher estimates compared to astroNN. Among these three comparisons, astroNN and DistMass exhibit the best agreement with a 3.8% mean fractional error, compared to 9.4% between astroNN and StarHorse, and 15.2% between DistMass and StarHorse. However, astroNN and StarHorse show the smallest scatter of 90 pc, while it is 118 pc between astroNN and DistMass, and 157 pc between DistMass and StarHorse. The pairwise comparison shows that astroNN and DistMass, both neural network models, exhibit greater consistency with each other than with StarHorse, a Bayesian isochrone-fitting model, reflecting the methodological differences. In summary, our distance estimation agrees well with those of other studies.

To ensure the accuracy of the stellar parameters obtained with our CNN model, we compared our results with those from high-resolution observations, specifically the APOGEE DR17 (Abdurro’uf et al., 2022) and the third data release of the GALactic Archaeology with HERMES surveys (GALAH DR3; Buder et al., 2021), two of the most comprehensive spectroscopic surveys.

APOGEE DR17 (Abdurro’uf et al., 2022) published medium-high-resolution (R $\sim 22,500$) near-infrared spectra of over 650,000 stars from the APOGEE-North and APOGEE-South surveys. The stellar parameters were derived by ASPCAP using MARCS model atmospheres with new spectral grids accounting for non-LTE level populations. GALAH DR3 (Buder et al., 2021) includes 768,423 high-resolution (R $\sim 28,000$) optical spectra from 342,682 stars. The parameters were estimated using the Spectroscopy Made Easy (SME) model-driven approach and one-dimensional MARCS model atmospheres. Additionally, GALAH DR3 mitigated spectroscopic degeneracies using astrometry from Gaia DR2 and photometry from 2MASS.

To perform the comparison, we cross-matched our results with APOGEE DR17 and GALAH DR3, identifying 6,040 and 6,329 stars with corresponding stellar parameters. To ensure the reliability of the comparison, we applied STARFLAG $=0$ for the APOGEE spectra; flag_sp $=0$, flag_fe_h $=0$, and red_flag $=0$ for the GALAH spectra; and snrg $>50$ for the LAMOST spectra. Moreover, we impose constraints on the errors of $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$ given by ASPCAP and GALAH, setting them to less than 200 K, 0.2 dex, and 0.2 dex, respectively. Finally, we obtain 2,677 common stars with APOGEE and 2,218 with GALAH.

Fig. 11 shows the differences of $T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$ between this work and the two surveys. In terms of effective temperature, our results are more consistent with those of GALAH, with a scatter of 97 K, while APOGEE shows a scatter of 121 K. This difference may be attributed to the wavelength ranges observed by the surveys, as LAMOST and GALAH both observe in the optical range, while APOGEE in the near-infrared. For surface gravity, the scatter for GALAH is 0.14 dex, which is slightly larger than the 0.12 dex for APOGEE; however, the bias for APOGEE is smaller than that for GALAH. Our results slightly overestimate log $\it g$ compared to both surveys. Regarding metallicity, this work exhibits more consistency with APOGEE than with GALAH, with APOGEE having a scatter of 0.06 dex compared to 0.13 dex for GALAH. Nonetheless, there is no significant bias for either survey. In summary, our results are in good agreement with both APOGEE and GALAH, demonstrating the reliability of our CNN model.

Using the 521,424 solar-like stars, we investigated the radial and vertical metallicity gradients to explore the utility of our distance and metallicity measurements for Galactic structure studies. Previous studies indicate that the Milky Way’s disk shows a negative radial metallicity gradient, with the inner Galaxy (where the galactocentric radius $R$ is less than the solar value) is typically more metal-rich than the outer disk. This gradient ranges from $-0.1<\Delta\hbox{$\rm[Fe/H]$}/\Delta R<0.0\ {\rm{dex\ kpc^{-1}}}$ in the Galactic plane. Additionally, the Galactic disk exhibits a negative absolute vertical metallicity gradient, varying in the range $-0.25<\Delta\hbox{$\rm[Fe/H]$}/\Delta Z<-0.10\ {\rm{dex\ kpc^{-1}}}$, depending on the tracer population and survey volume (Önal Taş et al., 2016; Yan et al., 2019; Gaia Collaboration et al., 2023a; Wang et al., 2023a; Hawkins, 2023; Imig et al., 2023, and references therein). These negative metallicity gradients strongly suggest that the Galactic disk formed in an inside-out manner, with the inner Galaxy forming early and rapidly, followed by the outer Galaxy later on (Frankel et al., 2019).

Based on our distances, we calculated the galactocentric Cartesian coordinates using astropy.⁵⁵5https://docs.astropy.org/en/stable/coordinates/index.html In Fig. 12 we show the X-Z (top panel) and R-Z (bottom panel) spatial distribution of the solar-like stars in our results. From these we selected a subsample of 458,061 stars within $7\leq R\leq 14$ kpc and $|Z|\leq 2$ kpc to investigate the metallicity gradients. In Fig. 13 we display the metallicity distribution in the same planes as Fig. 12. The top panel shows that the inner Galaxy has a higher metallicity than the outer regions (indicating a negative radial metallicity gradient), while the bottom panel reveals that metallicity decreases as $|Z|$ increases (indicating a negative vertical metallicity gradient).

For this work we chose a single linear model for both radial and vertical metallicity gradients. The radial metallicity gradient is defined as

where $\hbox{$\rm[Fe/H]$}_{R}$ is the metallicity at a particular galactocentric radius $R$, $\Delta\hbox{$\rm[Fe/H]$}/\Delta R$ is the radial metallicity gradient, and $b_{R}$ is the intercept. The vertical metallicity gradient is defined as

where $\hbox{$\rm[Fe/H]$}_{Z}$ is the metallicity at a particular absolute vertical height $|Z|$, $\Delta\hbox{$\rm[Fe/H]$}/\Delta Z$ is the vertical metallicity gradient, and $b_{Z}$ is the intercept. However, since the metallicity gradients are not the main focus of this work, we applied a simple linear least-squares regression to estimate the gradients as well as their uncertainties ($\sigma_{\Delta\hbox{$\rm[Fe/H]$}/\Delta R}$, $\sigma_{b_{R}}$; $\sigma_{\Delta\hbox{$\rm[Fe/H]$}/\Delta Z}$, $\sigma_{b_{Z}}$) using the scipy library.⁶⁶6https://docs.scipy.org/doc/scipy-1.14.0/reference/generated/scipy.stats.linregress.html To calculate the radial metallicity gradient, we divided our sample into equally spaced vertical bins with a width of 0.2 kpc, covering the range $0<|Z|<2$ kpc. Similarly, for the vertical metallicity gradient, the stars were divided into equally spaced radial bins from $7<R<14$ kpc, each with a bin width of 1 kpc.

The results are summarized in Tables 3 and 4. The radial metallicity gradient ranges from $-0.05$ to $0.0\ {\rm{dex\ kpc^{-1}}}$, while the vertical metallicity gradient ranges from $-0.26$ to $-0.07\ {\rm{dex\ kpc^{-1}}}$. These results are consistent with previous studies (Gaia Collaboration et al., 2023a; Hawkins, 2023; Imig et al., 2023), as shown in Fig. 14. In the top panel our result show the best agreement with those of Gaia Collaboration et al. (2023a), which is based on FGK dwarfs and giants from the Gaia DR3 survey. Imig et al. (2023) and Hawkins (2023) used red giants from APOGEE and OBAF-type stars from LAMOST as tracers, respectively, and found steeper gradients compared to ours. These differences likely arise from the distinct tracer populations and observational surveys used. In the bottom panel, our results are also generally consistent with those of Hawkins (2023) and Imig et al. (2023). The discrepancies between these studies can be attributed to variations in binning methods and sample sizes. Additionally, it can be seen that the radial and vertical gradients generally flatten at larger distances from the Galactic center and greater vertical heights, respectively, which aligns well with previous findings. Our results for the metallicity gradients serve as a simple demonstration, illustrating that our distance and stellar parameter estimations have properties similar to what is expected based on our knowledge of metallicity distributions of Milky Way components.

To examine which parts of the spectra the CNN is weighting when predicting stellar parameters, we analyzed the feature importance through Shapley values. Using the Python package SHAP (Shapley Additive Explanations) (Lundberg & Lee, 2017), we calculated the Shapley values of each feature for our CNN model, where each feature corresponds to a 1Å wavelength.

We calculated the Shapley values for 2,000 stars. To quantify the impact, we computed the normalized average absolute Shapley values for each parameter, where larger values at a given wavelength indicate a greater influence on the parameter predictions. Comparisons are then made between hot ($\hbox{$T_{\rm eff}$}>5000$ K) and cool stars ($\hbox{$T_{\rm eff}$}\leq 5000$ K), and between metal-poor stars ($\hbox{$\rm[Fe/H]$}<0.0$ dex) and metal-rich stars ($\hbox{$\rm[Fe/H]$}\geq 0.0$ dex), as shown in Figs. 15 and 16. Only the results for the wavelength range of $3925-5500$Å are demonstrated, as the $5500-8800$Å range does not show significant importance. A few notable features include the following:

1. (i)

   The helium line plays a crucial role in the determination of surface gravity and color, as well as the metallicity estimation for hot stars.

2. (ii)

   Atomic metal lines, such as FeI, MgI, CrI, Ca, and Mg, significantly contribute to the determination of temperature and metallicity throughout our stellar parameter range.

3. (iii)

   Spectral lines that substantially impact a particular parameter also influence other parameters, although their significance may vary. For instance, the FeI line at 4272Å has a pronounced effect on absolute magnitude, but a weaker influence on surface gravity.

4. (iv)

   Unidentified or weaker lines, including those in the 5500–8800Å range that may not be prominent individually, collectively contribute significantly to the measurement of stellar parameters. Therefore, employing full spectrum analysis might be preferable to using spectral indices.

5. (v)

   In Fig. 15 for hot stars, the CH line plays an important role in determining temperature and absolute magnitude, while the helium line contributes substantially to the estimation of metallicity.

In this work we proposed a convolutional neural network model to derive distances and stellar parameters ($T_{\rm eff}$, log $\it g$, and $\rm[Fe/H]$) directly from LAMOST low-resolution spectra, combined with photometric data from Gaia. The model is trained on the LAMOST DR9 spectra, using APOGEE DR17 stellar parameters as well as Gaia DR3 for solar-like stars. For spectra with $\rm{S/N_{g}}>10$, the test shows a scatter of 86 K for $T_{\rm eff}$, 0.07 dex for log $\it g$, 0.06 dex for $\rm[Fe/H]$, 0.25 mag for $M_{\rm G}$, and 0.03 mag for $(\rm{{BP}-{RP}})_{0}$. The distance predictions have a scatter of 65 pc, indicating a median fractional error of less than 4% and a standard deviation of 8%.

Using the CNN model, we determined the distance and stellar parameters of 521,424 solar-like stars from LAMOST DR9, thus compensating for those that lack precise distance measurements. The reliability of the distance estimation was evaluated by comparing it with three other studies: StarHorse, DistMass, and astroNN. The comparisons yield mean fractional errors of $7\%$, $-4\%$, and $-4\%$, respectively, indicating good agreement with these studies. Additionally, the accuracy of the stellar parameters is validated by comparison with APOGEE DR17 and GALAH DR3, showing minor biases and different degrees of scatter but overall consistency with these datasets.

Moreover, using a subsample of solar-like stars from our dataset, we calculated the radial and vertical metallicity gradients of the Milky Way. The radial gradient ranges from $-0.05<\Delta\hbox{$\rm[Fe/H]$}/\Delta R<0.0\ {\rm{dex\ kpc^{-1}}}$ for stars located at $0<|Z|<2$ kpc, while the vertical gradient ranges from $-0.26<\Delta\hbox{$\rm[Fe/H]$}/\Delta Z<-0.07\ \rm{dex\ kpc^{-1}}$ for stars situated at $7<R<14$ kpc. These results align well with previous studies, indicating that our distance and metallicity measurements are reliable for Galactic structure studies.

In the future, we plan to extend this model to include a broader range of stellar types, beyond solar-like stars. This includes improving the determination of distances and stellar parameters for a larger number of stars in spectroscopic surveys.