## 2 Methodology and Experimental Database  
The advancement of modern techniques, especially in computer science, has enabled the processing of vast datasets that were once unmanageable. ANN modeling, a machine learning method, is utilized across various research fields, including materials science [32], mechanical engineering [33], architecture [34], medicine [35], and business [36], to estimate predicted values or results, which are difficult to obtain through conventional methods.  
In comparison to our prior work on the Ms temperature of high-carbon steels [37], the construction of the ANN model in this study is similar but not identical. A typical neural network, containing several nodes, consists of three layers: input, hidden, and output. Each node in the input layer represents an independent variable value. The hidden layer nodes perform computations, and the output layer nodes calculate the final output as the dependent variable. The parameters for the input and output layers are set according to the project requirements, while others of the model, such as the number of hidden layers, nodes per layer, learning rate, momentum term, and iterations, are determined by the researchers' expertise and through trial and error across different scenarios.  
The ANN modeling process consists of the following steps:  
1. Collection of data.  
The ANN model requires a detailed description of the chemical composition of steel sets (including ten alloying elements--carbon (C), manganese (Mn), silicon (Si), chromium (Cr), nickel (Ni), molybdenum (Mo), vanadium (V), aluminum (Al), tungsten (W) and copper (Cu)--which may vary depending on different cases) and their corresponding experimental Ms temperature values. A reference search yielded a total of 1533 samples that meet the criteria, out of which 313 sets were classified as medium-carbon steel in accordance with the definition, as shown in Table S2 in Supplementary Materials.  
2. Grouping of data.  
The dataset, comprising 313 entries, was divided into two groups: training and testing, with the ratio typically ranging from 6:4 to 8:2. Training with a higher portion can increase the model accuracy but the selection of too large a training set may result in overfitting, which limits the model generalizability. Therefore, a ratio of 6:4 was adopted to ensure the model's broad applicability, assigning sets #1 to #188 for training and the remaining sets #189 to #313 for testing.  
3. Training of ANN.  
The training dataset was used to optimize the ANN configuration, minimizing deviations in the untrained test data through a trial-and-error process. The process involved selecting various hidden layers with different numbers of neurons and optimizing the learning rate, momentum term, and number of iterations to achieve the best prediction for unseen data. Figure 1 illustrates the entire process of trial and error for the modeling in different parameters.  
The ideal ANN model was achieved under the following conditions: two hidden layers within 4-4 neurons, a learning rate of 0.05, a momentum term of 0.05, and 8000 iterations. With this configuration, the model's performance on the testing dataset resulted in a mean absolute error (MAE) of 10.74 K, an average absolute error (AAE) of 0.76 K, a mean squared error (MSE) of 0.00189 K\({}^{2}\), and a mean percentage error (MPE) of 0.17%. The calculation equations could be easily found in a statistics textbook and are listed below.  
\[\text{MAE}=\frac{1}{n}\text{$\sum_{i=1}^{n}$}|(y_{i}-\hat{y}_{i})| \tag{2}\]  
\[\text{AAE}=\frac{1}{n}\text{$\sum_{i=1}^{n}$}(y_{i}-\hat{y}_{i}) \tag{3}\]\[\text{MSE} =\frac{1}{n}{\sum_{i=1}^{n}}(y_{i}-\hat{y}_{i})^{2} \tag{4}\] \[\text{MPE} =\frac{100\%}{n}{\sum_{i=1}^{n}}\frac{y_{i}-\hat{y}_{i}}{\hat{y}_ {i}} \tag{5}\]  
in which \(n\) is the number of different steels for which the Ms temperature is predicted, \(y_{i}\) is the predictive value of the Ms temperature, and \(\hat{y}_{i}\) is the actual value of the Ms temperature being predicted.