## 3 Results and Discussion
This differs fundamentally from traditional database queries, which typically rely on preset structural descriptors (such as space group numbers and cell parameter thresholds) or empirical peak position similarity (such as Euclidean distance), and cannot distinguish between symmetry breaking due to differences in Bravais lattice types (such as P\(\rightarrow\)C) and pseudo-symmetry caused by atomic primitive perturbations. Additionally, the deep learning method also compensates for the lack of local-global feature coupling, as traditional methods usually handle lattice parameters and atomic positions independently, while deep models can achieve multi-scale aggregation, establishing dynamic associations between local chemical environment features in cluster expansion and global symmetry features. To further analyze the physical information learned by the model, t-SNE [46] was used to reduce the dimensionality and visualize the hidden layer of the model. From Figure 4, it can be seen that while some local clustering effects of crystal systems are retained during contrastive learning training, the overall clustering is not significantly constrained by crystal systems. In traditional methods, the feature distributions of the seven crystal systems exhibit discrete hard boundaries (such as the sharp division between cubic and hexagonal systems), which stem from the linear dependence of handcrafted features on cell parameters (such as the strong constraint of \(a=b=c\) in cubic systems). Through the soft clustering mechanism of contrastive learning, high-symmetry crystal systems (such as cubic) still  
Figure 3: The structure inferred from the actual XRD diffraction pattern and its corresponding image, from bottom to top, are the actual XRD pattern (which is also the pattern of the model-recommended structure); the Top-2 candidate structures and their corresponding pattern; the Top-3 candidate structures and their corresponding pattern. maintain local clustering, indicating that the features of high-symmetry crystal systems still have strong distinguishability in the model's latent space, while low-symmetry crystal systems (monoclinic, triclinic) form a gradual transition zone. For example, the distribution of potential vectors for monoclinic and orthorhombic systems partially overlaps, as the lower symmetry of monoclinic and triclinic systems results in a larger range of variations in cell parameters and atomic arrangements, making them susceptible to lattice distortions and leading to the blurring of feature distributions. Due to this personalized recognition approach, contrastive learning is not limited by the influence of strict hard-classification categories. We randomly selected some space group samples with small sample sizes and samples of the space group \(Fm\bar{3}m\) with a large number of test samples to test the space group accuracy after Top-1 recognition, as shown in Table 2.  
From Table 2, it can be seen that the individual characteristics of small samples are well preserved. The complementary nature of cluster expansion with E3 invariance and contrastive training in the feature space provides a dual guarantee for model performance improvement: the former ensures feature interpretability through a clear physical parameterization process, while the latter mines implicit cross-scale correlation patterns through a data-driven approach. This hybrid paradigm not only retains prior knowledge in the field of materials science but also fully leverages the nonlinear fitting advantages \begin{table}
\begin{tabular}{c c c} \hline
**Space Group** & **Frequency** & **Accuracy** \\ \hline
225 & 808 & 95.92\% \\
3, 24, 34, 37, 39 & & \\
41, 48, 50, 95, 97 & & \\
112, 116, 120, 132, 138 & 1 (each) & 96.00\% \\
143, 157, 159, 180, 192 & & \\
195, 197, 202, 203, 214 & & \\ \hline \end{tabular}
\end{table} 
Table 2: Space group frequency and accuracy. Figure 4: Visualization of hidden layers in contrastive learning vs. seven crystal systems.