## 2 Methodology  
### Dataset Obtaining and Parameter Calculation  
The alloy phase data were derived from the alloy phase data disclosed in references [29; 30; 31; 32; 33]. Since the as-cast samples were stable, all alloy samples in the dataset were as-cast. The alloy phase dataset was collated, and the obtained dataset contained a total of 1009 samples of solid solution phase binary alloys and medium and HEAs, of which the dataset contains 391 BCC, 392 FCC, 74 BCC+FCC, and 152 HCP data. Feature engineering is one of the most important steps, which determines whether the model can accurately map the formation pattern of alloy phases. Due to the complex formation mechanism of alloy phases, it is challenging to select an appropriate feature set for model construction. It was believed that a large feature set may help to find out some undiscovered empirical parameters that can be used to help predict alloy phases. The materials-agnostic platform for informatics and exploration (Magpie) [34] was therefore used to generate 143 features directly according to alloy composition and molar ratio. Among these features, the features describing atomic radius, electronegativity, and electron concentration in Magpie have been reported to have a close relationship with the formation of alloy phases [20; 35]. The features and their calculations included in Magpie are as follows:  
1. The L\({}_{\text{P}}\) norm features of HEA The L\({}_{\text{P}}\) norm features of the HEA are calculated as shown in Equation (1): \[\text{L}_{\text{P}}=\left(\sum_{i=1}^{\text{n}}\lvert\text{c}_{i}\rvert^{\text {P}}\right)^{1/\text{p}}\] (1) where c\({}_{i}\) represents the molar ratio of metal elements of the HEA; p is taken as 3, 5, 7, and 10 to construct four L\({}_{\text{P}}\) parametric features, namely L\({}_{3}\), L\({}_{5}\), L\({}_{7}\), and L\({}_{10}\), respectively;
2. Elemental molar-ratio weighted features of HEAs For each element in the HEA, the atomic number (Num), Mendeleev Number (MN), Atomic Weight (AW), Melting Temperature (Tm), Column (Col), Row, Covalent Radius (R), Electronegativity (\(\chi\)), the number of filled s valence electrons, the number of filled p valence electrons, the number of filled d valence electrons, the number of filled f valence electrons, the number of filled s valence orbitals, the number of unfilled p valence orbitals, the number of unfilled d valence orbitals, the number of unfilled f valence orbitals, the number of unfilled d valence orbitals, Specific Volume of 0 K Ground State, Band Gap Energy of 0 K Ground State, Magnetic Moment (per atom) of 0 K Ground State, and Space Group Number of 0 K Ground State for the following six calculation operations: \[\text{F}=\sum_{i=1}^{\text{n}}\text{c}_{i}\text{F}_{i},\] (2) \[\text{max\_F}=\max_{i=1,\ldots,n}(\text{c}_{i}\text{F}_{i}),\] (3) \[\text{min\_F}=\min_{i=1,\ldots,n}(\text{c}_{i}\text{F}_{i}),\] (4)\[\Delta\text{F}=\text{max\_F}-\text{min\_F}, \tag{5}\]  
\[\delta\text{F}=\sqrt{\sum\
olimits_{i=1}^{n}\text{c}_{i}\times\left(\text{F}- \text{F}_{i}\right)^{2}}, \tag{6}\]  
\[\text{most\_F}=\frac{1}{n}\text{\sum\
olimits_{i=1}^{n}\text{F}_{i}}, \tag{7}\]  
where F represents the above features and c\({}_{i}\) represents the molar ratio of each element of the HEA. 3. Valance orbital occupation features  
Fractions of filled s valence electrons, filled p valence electrons, filled d valence electrons, and filled f valence electrons are calculated as follows:  
\[\text{frac\_OV}}=\frac{\sum\
olimits_{i=1}^{n}\text{c}_{i}\text{N}_{i}^{ \text{O}}}{\sum\
olimits_{i=1}^{n}\text{c}_{i}\text{N}_{i}}, \tag{8}\]  
where O denotes the s, p, d, and f orbits, N\({}_{i}^{\text{O}}\) is the number of filled valence orbitals for a specific orbital, and N\({}_{i}\) is the number of filled valence orbitals for all orbitals. 4. Ionic compound features  
Three features are included. The first feature (CanFormIonic) is used to describe whether it is possible to form a neutral ionic compound, assuming each element takes exactly one of its common charge states (see Equation (9)).