Feature dimensionality reduction techniques are commonly employed to map high-dimensional data into a lower-dimensional space for more intuitive visualization of sample distributions across classes in classification tasks, such as PCA. These methods primarily retain directions of maximum variance to highlight inter-class differences. However, high variance does not necessarily correspond to discriminative or physically meaningful information. For instance, noise may exhibit greater variance than critical target features. In such cases, reducing the input dimensionality can lead to the loss of important information, thereby damaging the classification performance of the model^[49]. The proposed 2D-SFG method not only reduces the input dimensionality by constructing new features but also incorporates nonlinear combinations among features, thereby effectively preserving critical information from the original data while enhancing model visualization and improving predictive accuracy. To evaluate its effectiveness, we compared its performance with existing results in perovskite structural categorization and HEA phase classification, as reported in high-impact journals including Nature Communications and Acta Materialia.

Balachandran et al. developed a support vector machine-based binary classification model using a dataset of 167 perovskite structures and five input features, achieving an average accuracy of 85.6%^[30]. Based on HEA data gathered from references^[50-52], a phase classification model was built using ten influential features selected via a feature screening method, resulting in a prediction accuracy of 84.5%. The material features employed in the two classification tasks are presented in Table 2. Comprehensive details can be found in the Supplementary Materials.

Figure 2A and B illustrate the iterative feature generation process for the two classification cases. As the iterations proceed, the model accuracy gradually increases until it converges to a stable value. This trend indicates that the two new features generated by the 2D-SFG method incorporate more meaningful information to improve model performance. The explicit mathematical relationships of the evolved features after 100 iterations are shown in Equations (1) to (4).

Perovskites:

(1) $$ \boldsymbol{X_{c1}=f_3-f_3^2-f_2\times\frac{f_3}{f_1}} $$

(2) $$ \boldsymbol{X_{c2}=f_4-f_2} $$

HEAs:

(3) $$ \boldsymbol{X_{c3}=\delta MR\times \delta AV\times \left ( 2\times \overline{MAC}\times\frac{\overline{EA}}{\overline{LC}}+\overline{RM}+\delta AV \right ) } $$

(4) $$ \boldsymbol{X_{c4}=\overline{MN}-Xc+\frac{\delta AV}{\delta CE}+\frac{\overline{MN}}{\overline{EA}}} $$

where X_C1, X_C2, X_C3, and X_C4 denote the constructed symbolic features. Definitions of all other parameters are provided in Table 2.

A support vector classifier was established using the constructed features, and the corresponding classification probability maps are shown in Figure 2C and D. On the one hand, these maps provide an intuitive representation of the ML model, in which the red arrows indicate the direction of increasing probability for the target property, forming a “navigation map” for alloy design. Based on the value range associated with the positive class, the compositional design window for ferroelectric perovskites without secondary phases and single solid-solution HEAs can be determined through either forward screening or inverse calculation. On the other hand, underlying patterns in the data can be revealed by analyzing the influence of the constructed features on the target property, particularly the transition boundaries. This approach offers possibilities for extracting physical interpretability and discovering new knowledge.

In regression tasks, the “curse of dimensionality” also poses a significant challenge. Although reducing the dimensionality of input variables can visualize the relationship between inputs and outputs, it is impossible to obtain the relationship between features before and after dimensionality reduction. The proposed 2D-SFG method can not only preserve meaningful information and eliminate redundancies during dimensionality reduction, but also provide explicit mathematical expressions that clarify the connections between the original and transformed features, as well as their influence on material properties. This approach significantly enhances model interpretability and offers valuable support for in-depth investigations into how key features affect performance mechanisms in materials science. To further assess the effectiveness of our approach, we selected studies on ML-assisted alloy design published in high-impact journals, including Nature Communications and Acta Materialia, for comparative validation.

Xue et al. established a support vector regression prediction model using six input features to design NiTi-based shape memory alloys with low thermal hysteresis, achieving a prediction error of 2.9 K^[53]. The authors^[9] aimed to optimize both hardness and electrical conductivity in precipitation-strengthened copper alloys. Five key features influencing alloy hardness and six features affecting electrical conductivity were identified through feature selection. By constructing corresponding support vector regression prediction models, prediction errors of 5.9% and 9.7% were achieved, respectively. The material features used in these two regression tasks are summarized in Table 3, with additional details provided in the Supplementary Materials.

Figure 3A-C illustrates the iterative feature generation process for the two regression cases. As the number of genetic iterations increases, the 10-fold cross-validation error progressively decreases, indicating that the two constructed features incorporate increasingly meaningful information and thereby contribute to model improvement. Notably, the error for the hardness regression task continued to decrease beyond 100 iterations. However, further increasing generations resulted in overly complex expressions. This finding reflects the need for more sophisticated mathematical representations to accurately model the underlying strengthening mechanisms and integrate the substantial information embedded within the original features. To balance model accuracy and expression complexity, the evolutionary process was terminated after 100 generations. The final mathematical expressions of the evolved features after 100 generations are provided in Equations (5) to (10).

ΔT:

(5) $$ \boldsymbol{X_{R1}=(arc+dor)\times cs-dor-ven} $$

(6) $$ \boldsymbol{X_{R2}=\frac{en}{mr\times (2cs+arc+mr+en+dor)}} $$

HV:

(7) $$ \boldsymbol{X_{R3}=\frac{M.S3+O12}{V.A8}+V.S7-O12-\frac{V.A8}{M.S12}} $$

(8) $$ \boldsymbol{X_{R4}=M.S3\times V.S7+M.S3-M.S12-\frac{V.S7-1}{V.A8\times O12-M.S3}} $$

EC:

(9) $$ \boldsymbol{X_{R5}=M.A10\times M.S6-M.A10} $$

(10) $$ \boldsymbol{X_{R6}=\frac{M.A10}{M.S6\times HV}+3\times M.E4} $$

where X_R1, X_R2, X_R3, X_R4, X_R5, and X_R6 denote the constructed symbolic features. Definitions of all other parameters can be found in Table 3.

Support vector regression models were established using the constructed features as input to predict alloy properties across different values, as visualized in Figure 3D-F. The red arrows in each contour plot indicate the gradient variation of the target property, serving as guidance for efficiently approaching the desired property objectives. For instance, thermal hysteresis is minimized when features X_R1 and X_R2 approach around 145 and 4.2 × 10^-5, respectively. Accordingly, NiTi-based shape memory alloys with low thermal hysteresis can be designed by forward screening or inverse calculation of compositions that satisfy this specific feature range. Similarly, higher alloy hardness is achieved near X_R3 ≈ -4 and X_R4 ≈ -246, while superior electrical conductivity is attained around X_R5 ≈ 12.5 and X_R6 ≈ 14.2. These identified value ranges offer practical and interpretable design guidelines for developing copper alloys with an improved balance between hardness and electrical conductivity.

As illustrated in Figures 2 and 3, the 2D-SFG method effectively visualizes phase boundaries and property contours, enabling intuitive identification of target regions and variation trends. A direct correspondence between composition design windows and target properties was established by mapping candidate alloy compositions into 2D symbolic features. In this work, we employed the 2D-SFG method to construct visualization models that can compute symbolic features for unknown alloy compositions. These models were subsequently used to evaluate and predict target properties, successfully screening refractory HEAs with single solid-solution phases, as well as advanced copper alloy compositions with superior performance. In the V-Nb-Ti-W-Zr-Mo-Hf-Ta HEA system, a genetic algorithm was used to define a composition space with 1 at.% increments. The ten-dimensional features listed in Table 2 were computed for each composition and projected into a 2D visualization space using Equations (3) and (4), thereby identifying candidate compositions with a high probability of forming solid-solution phases, as summarized in Table 4.

Figure 4A displays the classification probability and evolutionary path of alloys across two generations, showing a clear design pathway toward higher solid-solution probability. The selected alloys were synthesized using vacuum arc melting and characterized by X-ray diffraction (XRD). As shown in Figure 4B, all as-cast alloys exhibit a single body-centered cubic (BCC) solid-solution phase, consistent with the predictions of the 2D-SFG method. Figure 4C compares the predictions of different visualization modeling approaches. While most methods struggle to accurately identify the phase structures of alloys with low solid-solution probabilities, as shown in Figure 4A, only the 2D-SFG method correctly predicts the phase structures of both alloys. These results underscore the strong generalization capability and classification accuracy of the proposed approach for phase identification in HEAs.

Figure 5 illustrates the application of the constructed hardness and conductivity visualization models to the unexplored composition space of the Cu-Ni-Si system. A promising copper alloy composition, Cu-1.87Ni-0.55Si-0.47Co-0.11Mg-0.07Zr-0.21Zn, was identified on the performance Pareto front as exhibiting a favorable combination of high hardness and medium conductivity. The designed alloy was synthesized via vacuum induction melting, achieving a peak hardness of 275 HV and a corresponding electrical conductivity of 47.2 %IACS (International Annealed Copper Standard). As summarized in Table 5, the prediction performance of different design methods was compared. The 2D-SFG model demonstrated superior generalization capability, with all errors between experimental measurements and model predictions remaining below 7%, confirming its effectiveness in supporting the rational design of high-performance alloys.

To evaluate the performance of the 2D-SFG method in classification tasks, a comprehensive assessment was conducted using multiple metrics, including accuracy, precision, recall, F1 score, and area under the curve (AUC), as shown in Figure 6A and B. The model constructed using 2D-SFG features achieved accuracy and precision rates above 90% on both the training and test sets, with only minor differences between them, indicating no significant overfitting. The F1 score remained consistently above 0.9, reflecting a well-balanced trade-off between precision and recall. Furthermore, the AUC values for both the training and test sets were significantly higher than the 0.9 threshold, demonstrating the model’s strong discriminative ability for classification boundaries and reliability for practical applications.

A comparative analysis was conducted between direct modeling and the 2D-SFG approach, as illustrated in Figure 6C. The direct modeling method utilized the initial feature set from Table 2 as input, while the 2D-SFG approach employed the new features defined in Equations (1) to (4). It should be noted that both methods were evaluated using the same training and test datasets and underwent hyperparameter optimization. The 2D-SFG model achieved prediction accuracies of 94.2% for perovskite structure classification and 88.4% for HEA phase classification, outperforming the initial feature-based model, which achieved 85.6% and 84.5%, respectively. These results demonstrate that the 2D-SFG method effectively extracts meaningful information from the original features while reducing interference from redundant or irrelevant data.

From the perspective of feature reduction, the proposed 2D-SFG method can also be regarded as a dimensionality reduction technique. To evaluate its effectiveness, this work compared it against several common dimensionality reduction methods, including PCA, kernel principal component analysis (KPCA)^[54], locally linear embedding (LLE)^[55], Isomap, and multidimensional scaling (MDS)^[56]. All methods were configured to reduce the original features to two dimensions before model construction. As shown in Figure 6D, models using features from conventional reduction methods exhibited varying degrees of accuracy degradation, indicating the loss of critical information during the reduction process. In contrast, the 2D-SFG approach not only preserved essential information but also enhanced predictive accuracy, demonstrating its superior applicability for classification tasks.

For the regression tasks, a similar comparative analysis was performed among direct modeling, 2D-SFG modeling, and models built using common dimensionality reduction methods, evaluated through 10-fold cross-validation. The evaluation was based on RMSE and percentage error metrics. As shown in Figure 7, models using conventional dimensionality reduction methods showed either no significant improvement or a slight increase in error, indicating the substantial limitations of these methods for regression tasks. In contrast, the 2D-SFG approach achieved prediction errors of 1.1 K, 3.7%, and 6.2% for the thermal hysteresis of shape memory alloys, and the hardness and electrical conductivity of copper alloys, respectively. These results are lower than the corresponding errors of 2.9 K, 5.9%, and 9.7% obtained using the original feature set, demonstrating the strong applicability of the 2D-SFG method to regression tasks.

Although our primary analysis and discussion focused on the support vector machine algorithm due to its robustness in small-sample materials science problems, the core advantage of 2D-SFG, as a feature generation method for physics-informed knowledge discovery, lies in its ability to generate mathematical descriptors that explicitly encode physical mechanisms. This capability alleviates the burden on black-box models to implicitly extract features, thereby enhancing both predictive accuracy and generalization capability. To verify whether these performance gains are algorithm-agnostic, we conducted additional comparative studies using random forest (RF) and gradient boosting (GB) algorithms. The results, presented in Supplementary Table 3, demonstrate that models trained on symbolic features generated by 2D-SFG consistently outperform those based on original feature inputs across all tested algorithms. These findings indicate that 2D-SFG serves as a versatile and seamlessly compatible module that can be integrated into diverse ML workflows. Extending the 2D-SFG framework to guide more complex architectures, such as Deep Neural Networks, will be a primary focus of our future work.

The occurrence frequency of each original feature in the explicit expressions generated over 100 iterations was counted to assess feature importance, as illustrated in Figure 8. A higher frequency indicates greater importance of the corresponding feature. For instance, feature f₀ does not appear in Equation (2), suggesting that its information is either redundant with that of the other four features or has limited influence on the target property. Consequently, it was discarded in later iterations, resulting in an overall frequency of less than 20. In the case of copper alloy hardness, all features exhibited occurrence frequencies above 100, reflecting their substantial individual importance and joint contribution to the target property. This also explains why the final explicit expression for hardness is mathematically more complex. In contrast, the electrical conductivity of precipitation-strengthened copper alloys is governed by a relatively simpler mechanism compared to hardness. As a result, meaningful information could be extracted from the original features using simpler mathematical operations, leading to a less complex symbolic expression for conductivity.

This work successfully constructed new features for various material classification and regression problems through the 2D-SFG method. These features not only significantly enhance the predictive performance of ML models but also maintain explicit quantitative relationships with the original features, thereby improving physical interpretability in specific materials research problems. The 2D-SFG method is capable of constructing meaningful nonlinear combinations that capture complex physical interactions. Analysis of the specific mathematical forms of the generated symbolic features, such as products, ratios, and linear corrections, can further elucidate how feature co-occurrence influences target properties.

In the classification tasks, the new feature X_C1 couples the bonding strength driven by electronegativity (f₂) with the valence electron number (f₁). This formulation reflects the electronic shielding effect, whereby a lower valence electron count amplifies the impact of charge rearrangement induced by electronegativity differences on lattice distortion. The formula demonstrates that perovskite formation requires not only geometric matching but also weighted correction of chemical bond polarity strength by electron concentration^[57]. The new feature X_C3 employs a multiplicative form to quantify the nonlinear geometric distortion driving force arising from dual deviations in metal radius (δMR) and atomic volume (δAV). The formation of HEA solid solutions requires not only minimal geometric distortion but also sufficient intrinsic electronic properties and mechanical stiffness to effectively buffer and accommodate inevitable lattice strains^[58,59].

In the regression tasks, the original features atomic radius (arc) and pseudopotential radius (dor), both associated with atomic size mismatch, frequently co-occur. Increases in these values can raise the distortion energy barrier, thereby increasing thermal hysteresis. The new feature X_R1 multiplies atomic size-related features by a chemical scaling parameter to capture the synergistic effect between atomic size mismatch and the chemical bonding environment, while subtracting the valence electron number (ven) to correct for the shielding effect on lattice distortion in systems with high electron concentrations. This formulation comprehensively reflects their combined influence on thermal hysteresis^[60]. Furthermore, the new feature X_R2 is expressed as a ratio of electronegativity to multi-scale structural parameters. Physically, a larger denominator indicates stronger lattice distortion and greater resistance to the phase transformation driving force, while a larger numerator reflects more intense electron transfer and a higher phase transformation energy barrier. Consequently, a smaller X_R2 value facilitates easier phase transformation and corresponds to lower thermal hysteresis^[61,62]. The new feature X_R5 is formulated as the product of the mass attenuation coefficient and the extranuclear electron distance. This construction captures the coupling between electron scattering efficiency and atomic size effects, where the extranuclear electron distance governs electron cloud distribution. Enhanced electron cloud overlap driven by this interaction leads to significantly improved conductivity. Additionally, the new feature X_R6 incorporates measured Vickers hardness and absolute electronegativity. This combination accounts for the effects of lattice distortion and dislocation density. The formulation aligns well with the established understanding that electrical conductivity in metallic systems is collectively determined by the interplay among lattice defects, solute atoms, and polarization effects^[63-65].

The 2D-SFG method demonstrates a unique capability to automatically discover nonlinear descriptors that are difficult to predefine empirically. Although the resulting feature expressions may appear mathematically complex, each term corresponds to established physical models or empirical relationships in materials science. Furthermore, these explicit mathematical expressions effectively overcome the lack of physical insights inherent in black-box models. However, it is important to acknowledge certain limitations of the generated features: the physical universality of these expressions requires validation across broader material systems, and certain high-order interaction terms might introduce singularities. Future work should focus on enhancing the physical consistency and stability of the feature engineering process.

To further evaluate whether the symbolic feature generation method successfully integrates and learns universal knowledge, the authors collected a dataset of aluminum alloys subjected to solution and aging treatments. This dataset includes alloy composition, processing parameters, hardness, and electrical conductivity. The initial features listed in Table 3 were calculated, with the solid solubility of elements in aluminum used as a substitute for their solubility in copper. As shown in Table 6, calculating features X_R3 to X_R6 using the aluminum alloy dataset and retraining the model (Method A) yielded better predictive performance than the model trained with the original aluminum alloy features (Method B). For hardness prediction, Method A achieved cross-validation, training, and test errors of 9.9%, 7.5%, and 8.4%, respectively, representing an average reduction of 2.0% compared to Method B (10.9%, 9.6%, and 11.3%). Similarly, for electrical conductivity prediction, the errors decreased from 3.8%, 3.6%, and 5.3% to 2.9%, 1.1%, and 2.4%, corresponding to an average improvement of 2.1%. When a RF classifier was used instead, the 2D symbolic features still reduced the average hardness prediction error by 1.8% and the conductivity prediction error by 2.2%, confirming that the advantage stems from the features themselves rather than from algorithmic coincidence.

The distributional differences in feature values between copper and aluminum alloys pose a challenge for out-of-distribution generalization when copper alloy models are directly applied to aluminum alloy data. Although Method B was retrained using aluminum alloy data, the original features may still fail to capture the complex nonlinear relationships governing material properties. In contrast, the symbolically generated features derived from copper alloy data preserve the underlying mappings associated with alloy performance, as evidenced by the superior predictive performance of the aluminum alloy regression model (Method A) built with these features. The strong performance of Method A confirms both the applicability and effectiveness of the symbolically generated features, which not only retain key physical information learned through the 2D-SFG method but also adapt effectively to aluminum alloy data through refitting. The demonstrated predictive capability across both copper and aluminum alloy systems validates the transferability of the 2D-SFG method, offering a novel approach for cross-system materials modeling.