2.1. LAMOST

2.1.2. Data selection

In order to compare the ability of the model to estimate stellar parameters for VMP stellar spectra and non-VMP stellar spectra, and the accuracy of identifying VMP stars, we construct a dataset including VMP stars and non-VMP stars ([Fe/H]>-2.0) as positive and negative samples. To ensure the reliability of the stellar parameters, we combine two stellar parameter sources. For the VMP stellar sample, we choose the catalog containing 10 008 VMP stars obtained from LAMOST DR3 by Li et al. (Reference Li, Tan and Zhao2018) since it still provides researchers with the largest pool of bright and accurate VMP candidates to date. The stellar atmospheric parameters of these VMP stars are determined by line indices and by comparison with a grid of synthetic spectra, with metallicity ranging from −4.5 dex to −2.0 dex. For the negative sample dataset, we randomly select data with [Fe/H]>-2.0 from the LAMOST DR8 dataset that is recently made public to the world, where the stellar atmospheric parameters are determined by LASP (Wu et al. Reference Wu, Du, Luo, Zhao, Yuan, Heavens, Starck and Krone-Martins2014). To ensure that the accuracy of the stellar parameters of this dataset is comparable to that of the VMP stars dataset, we obtain 16 638 non-VMP stars with minor uncertainties and a signal-to-noise ratio larger than 10 at the g-band. In total, we construct a dataset including 26 646 stars. The resolution of all the spectral data is $R\sim 1\,800$ . The parameters of these stars range from 3 824.88 K< $T_{\textrm{eff}}$ <8 866.15 K, 0.213 dex< $\log$ g<4.897 dex, and −4.55 dex<[Fe/H]<0.699 dex. The errors of the parameters range from 0 K< $\sigma(T_{\textrm{eff}})$ <399 K, 0 dex< $\sigma(\log g)$ <0.94 dex, and 0 dex< $\sigma ([\textrm{Fe/H}])$ <0.4 dex. For subsequent model training and testing, we divide this dataset into a training set and a test set in the ratio of 7:3, including 18 652 and 7 994 stars, respectively. Fig. 1 depicts more clearly the distribution of the parameters in the training and test sets.

Figure 1. The distribution of $T_{\textrm{eff}}$ (top panel), $\log$ g (middle panel), and [Fe/H] (bottom panel) in the LAMOST training set and test set. The division of the training and test sets ensures the consistency of the data distribution.

To estimate the carbon abundance and identify CEMP stars in VMP stars, we need a dataset of VMP stars containing [C/Fe] labels. Since the VMP stars catalog mentioned above does not contain [C/Fe] values, Yuan et al. (Reference Yuan2020) cross-matched this catalog with Gaia DR2 and obtained a modified catalog including 9 690 stars with parameters derived by SEGUE Stellar Parameter Pipeline (SSPP, Lee et al. Reference Lee2008). We cross-match the VMP stars catalog (Li et al. Reference Li, Tan and Zhao2018) with the modified catalog (Yuan et al. Reference Yuan2020) and find 8 117 of 10 008 VMP stars with [C/Fe] labels. The range of the [C/Fe] labels is −2.018 dex<[C/Fe]<4.803 dex. For the aim of model training and testing, we divide the dataset into a training set and a test set in the ratio of 7:3, containing 5 681 and 2 436 stars, respectively.

Figure 2. The plots of spectra with resolution 1 800 (left) and 200 (right). Spectra with lower resolution have fewer characteristic points.

2.2. MARCS dataset

To further test the validity of the CNN model for estimating stellar parameters, we use the MARCS synthetic spectra (Gustafsson et al. Reference Gustafsson, Edvardsson, Eriksson, Jørgensen, Nordlund and Plez2008). It is a grid of about $10^{4}$ model atmospheres with nearly 52 000 stellar spectra containing F, G, and K types of stars. This grid of one-dimensional LTE model atmospheres can be combined with atomic and molecular spectral line data and software to generate stellar spectra, which has been widely used in a variety of studies (Roederer et al. Reference Roederer, Preston, Thompson, Shectman, Sneden, Burley and Kelson2014; Lu et al. Reference Lu, Li, Lin and Qiu2018; Reggiani et al. Reference Reggiani2019; VandenBerg et al. Reference VandenBerg, Edvardsson, Casagrande and Ferguson2021; Salsi et al. Reference Salsi, Nardetto, Plez and Mourard2022).

We select 9 644 data from MARCS for the experiment. The range of the stellar parameters is 2 500 K< $T_{\textrm{eff}}$ <8 000 K, −0.5 dex< $\log$ g<5.5 dex, −5 dex<[Fe/H]<−1 dex, and the step size of the parameters is 2 500 K for $T_{\textrm{eff}}$ , 0.5 dex for $\log$ g, and 0.25 dex for [Fe/H]. We perform the same interpolation and normalization operations on the data and obtain spectral data with 746 feature points. The dataset is divided into a training set and a test in the ratio of 7:3, including 6 750 and 2 894 stars, respectively. Section 4.3 shows the results of estimating stellar parameters of the MARCS synthetic spectra using the CNN model.

3.1. Introduction to the convolutional neural network (CNN)

In this paper, we construct a CNN model and test its performance in estimating stellar labels for low-resolution ( $R\sim200$ ) spectra and identifying VMP stars. This can enable us to better handle low-resolution spectra of CSST in the future and search for VMP stars.

The concept of deep learning originated from the study of artificial neural networks and was proposed by Hinton (Reference Hinton2008). CNN (Lecun et al. Reference Lecun, Bottou, Bengio and Haffner1998) is a typical supervised model of deep learning that has been widely applied to various fields in recent years. It is the first actual multilayer structure learning algorithm that uses spatial relative relationships to reduce the number of parameters to improve training performance. It adds a feature part to the original multilayer neural network; i.e. a convolutional layer and a pooling layer (dimensionality reduction layer) are added before the fully connected layer, and the network selects the features itself. CNN is a deep feedforward neural network that is widely used for supervised learning problems in image processing and natural language processing, such as computer vision (Krizhevsky et al. Reference Krizhevsky, Sutskever and Hinton2012), semantic segmentation (Ronneberger, Fischer, & Brox Reference Ronneberger, Fischer and Brox2015), object recognition (Redmon et al. Reference Redmon, Divvala, Girshick and Farhadi2016), etc.

While the CNN model can directly process one-dimensional sequences such as spectral data, most current CNN models demonstrate their power in processing two-dimensional image data. In the LAMOST dataset, the length of the 1D spectral data ( $1\times410$ ) we obtain is long, and collapsing it into 2D data ( $21\times21$ ) to simulate image data can perform cross-correlation operations and extract spectral features better and faster, thus speed up the network learning speed and parameter optimization efficiency. For MARCS spectral data ( $1\times746$ ), we also collapse them into a two-dimensional matrix ( $28\times28$ ) as the input to the CNN model. The following is a brief description of the structure and principles of the 2D CNN.

A complete CNN model must include convolutional layers, non-linear activation functions, pooling layers, and fully connected layers.

1. (1) Convolutional layer:

   The convolutional layer is the core layer for establishing the CNN model, which can act as a filter and reduce the number of parameters. The discrete form of two-dimensional convolution is used.

   Given a figure $\textbf{X}\in \textbf{R}^{M\times N}$ and a convolutional kernel $\textbf{W}\in\textbf{R}^{U\times V}$ . In general, U<M, V<N. The convolution between them can be denoted as

   (2) \begin{equation} \textbf{Y}=\textbf{W}*\textbf{X},\end{equation}
   (3) \begin{equation} y_{ij}=\sum_{u=1}^{U}\sum_{v=1}^{V}w_{uv}x_{i-u+1,j-v+1}.\end{equation}

   An example of the convolution process is shown in Fig. 3. A two-dimensional input array ( $3\times3$ ) performs a mutual correlation operation with a two-dimensional convolutional kernel array ( $2\times2$ ), resulting in a two-dimensional array ( $2\times2$ ). The convolution kernel slides over the input array from left to right and top to bottom.

   The convolutional layer extracts features from local regions, and different convolutional kernels are equivalent to various feature extractors. Based on the standard definition of convolution, strides and zero padding of the convolutional kernel can be introduced to increase the diversity of convolution. Strides refer to the number of steps each convolutional kernel moves when performing a convolutional operation. Set stride=k, which means convolving k rows and k columns from left to right and from top to bottom. Zero padding represents adding zeros to the outer side of the image. Setting padding=d means supplementing d layers of zeros around the input vector. Zero padding allows us to obtain more detailed feature information and control the network structure.

   

   Figure 3. Convolution process.

   After the convolutional layer, an activation function is usually added as a non-linear factor, which can deal with problems that cannot be solved by linear models and enhance the ability of the network to interpret the model. A commonly used activation function is ReLU (Equation (4)). It has a small computational effort, and can effectively alleviate the gradient disappearance and gradient explosion problem because its derivative is maintained at 1.

   (4) \begin{equation} f(x)=\max\!(0,x). \end{equation}

2. (2) Pooling layer:

   The pooling layer (subsampling layer) is designed to reduce the number of features in the network, thereby reducing the number of parameters and avoiding overfitting. Local translation invariance is an essential property of pooling layers, which indicates that pooling is approximately invariant in its representation of the input when a small number of translations are performed on the input. Max-pooling is the most commonly used which means extracting the maximum value within the neighbourhood (see Fig. 4). It can reduce the offset of the estimated mean value caused by the parameter error of the convolutional layer, and retain more texture information.

3. (3) Fully connected layer:

   The purpose of the fully connected layer is to connect the results of the last pooling layer to the output nodes and map the feature representation learned by the network to the label space of the samples. It acts as the ‘classifier’ of the network. It is important to note that when encoding the model, the last pooling layer needs to be flattened to a one-dimensional vector before connecting to the fully connected layer.

CNN extracts high-level semantic information from the input data gradually through a series of operations in these layers, formalizes the target task as an objective function, and generates the predicted values. This process is called Forward Propagation. The parameter optimization in the model is performed by Back Propagation. By calculating the loss between the predicted values and the true values, the CNN feeds the loss from the previous layer to the next layer, calculates the gradient of the loss on the parameters of each layer, and updates the corresponding parameters. Once all the parameters have completed one round of updates, the feedforward operation is performed again. This process is repeated iteratively until the model converges.

Figure 4. Max-pooling process.

3.2. The method of preventing overfitting

When the number of parameters in a model is large compared to the available training samples, overfitting can occur. This is manifested by high prediction accuracy on the training set and a significant decrease in accuracy on the test set.

The advent of dropout (Hinton et al. Reference Hinton, Srivastava, Krizhevsky, Sutskever and Salakhutdinov2012) has greatly alleviated this problem. During training, each neuron is retained with probability p and stops working with probability $1-p$ , and a different set of neurons is retained for each forward propagation. This approach reduces the reliance of the model on certain local features and has better generalization performance.

Figure 5. The structure of the proposed CNN model. The 1D spectral data is transformed into a 2D image-like matrix and is then input into the neural network consisting of three convolutional layers, three max-pooling layers, and two fully connected layers to extract the stellar parameters.

Another way is to add Batch Normalization (BN, Ioffe and Szegedy Reference Ioffe and Szegedy2015) layers after the convolutional and fully connected layers. During the training process of neural networks, parameter changes can lead to unstable distribution of activation values, which hinders the ability of subsequent layers to learn useful features. In addition to normalizing the input data of each layer and promoting a stable distribution of activation values, the BN layer can accelerate the convergence thus preventing model overfitting.

3.3. The structure of the proposed CNN model

After parameter tuning of the model, the final CNN model we construct is shown in Fig. 5. The convolutional layers filter the processed 2D spectra using a filter of size $9\times9$ , and the input is filled with zero space on the boundary so that the size of the output layer of the convolution is equal to the size of the input layer. The $9\times9$ convolution kernel can acquire a larger field of perception and therefore can capture more characteristics. The kernels of each convolutional layer are 64, 128, and 256, respectively. Each convolutional layer is followed by a max-pooling layer of size $2\times2$ with a step size set to two. Afterwards, two fully connected layers with 128 and 64 channels are added to combine the features previously extracted by the model. To prevent overfitting, a dropout layer can be set after each fully connected layer with a value of 0.2 to avoid the over-regularization of the model. BN and ReLU activation function layers are added between each layer to reduce overfitting and enhance the expressiveness of the model. The final output layer is the predicted values derived from the model.

3.4. Experimental procedure

To test whether our model can better estimate stellar parameters and identify VMP stars, we conduct experiments using Python 3.8 on NVIDIA GeForce RTX GPU. As mentioned in Section 2, both the LAMOST dataset and the MARCS dataset are divided into training and testing sets in the ratio of 7:3 to train the model and test its effectiveness. The training and testing process is performed on $T_{\textrm{eff}}$ , $\log$ g, [Fe/H], and [C/Fe] respectively.

We set a total of 1 000 epochs for model training, and L1loss (MAE, Equation (5)) is used as the loss function. The training process is performed in batches, with the size of each batch set to 128. The Adam algorithm (Kingma and Ba Reference Kingma and Ba2014), with an initial learning rate set to 0.001, is chosen to be the optimizer, which is an extension of the stochastic gradient descent method. Adam optimizer can speed up convergence by adapting the learning rate, making it well-suited for deep learning problems. An early stopping mechanism is set when the loss function no longer decreases beyond 250 epochs, which can effectively prevent the model from overfitting.

3.4.1. Evaluation metrics

We use three main evaluation metrics to test the effectiveness of the model in estimating stellar parameters, Mean Absolute Error (MAE, Equation (5)), Standard Deviation (STD, Equation (6)), and R squared ( $R^2$ , Equation (7)).

Suppose that N is the number of samples contained in the test set, y denotes the true values, and $\hat y$ denotes the predicted values derived by the proposed model. Let $e_{i}$ be $y_{i}-\hat y_{i}$ , and $\bar e_{i}$ be the average value of $e_{i}$ .

1. (1) Mean absolute error (MAE): MAE is a loss function used in regression models, which can express the fitting ability of the model more intuitively.

   (5) \begin{equation} MAE(y,\hat y)=\frac{1}{N}\sum_{i=0}^{N-1}|y_{i}-\hat y_{i}|. \end{equation}

2. (2) Standard deviation (STD): Standard deviation reflects the degree of dispersion of a dataset.

   (6) \begin{equation} STD(y,\hat y)=\sqrt {\frac{1}{N}\sum_{i=1}^{N}(e_{i}-\bar e_{i})^2}. \end{equation}

3. (3) R squared ( $R^2$ ): The numerator is the error between the predicted value and the true value, and the denominator is understood as the dispersion of the true value. The division of the two can eliminate the effect of the dispersion of the true value. The closer $R^2$ is to 1, the better the model fits the data.

   (7) \begin{equation} R^2(y,\hat y)=1-\frac{\sum\!(y-\hat y)^2}{\sum\!(y-\bar y)^2}. \end{equation}

Three other metrics can be used to test the effectiveness of the model in classifying stars, precision (Equation (8)), recall (Equation (9)), and accuracy rate (Equation (10)).

1. (1) True Positive (TP): VMP stars predicted as VMP stars.

2. (2) True Negative (TN): Common stars predicted as common stars.

3. (3) False Positive (FP): Common stars predicted as VMP stars.

4. (4) False Negative (FN): VMP stars predicted as common stars.

(8) \begin{equation}Precision = \frac{TP}{TP+FP}.\end{equation}

(9) \begin{equation}Recall = \frac{TP}{TP+FN}.\end{equation}

(10) \begin{equation}Accuracy = \frac{TP+TN}{TP+FP+TN+FN}.\end{equation}