A dataset comprising 205 experimentally verified HEA samples was compiled for this study. The data were primarily curated from peer-reviewed literature and public materials databases^[2,36-38]. This dataset encompasses diverse alloy systems, predominantly including 3d Transition Metal Systems (e.g., Al-Co-Cr-Fe-Ni and its derivatives, spanning FCC/BCC/Multi-phase structures) and Refractory HEA Systems (e.g., W-Mo-Ta-Nb-V-Ti systems, typically displaying a BCC structure)^[39,40]. Each data entry records the complete chemical composition (in atomic percentage) alongside the corresponding experimental Vickers hardness (HV). To facilitate the training of the physics-informed model, the feature space was not limited to raw elemental ratios. Instead, 12 key physical and thermodynamic descriptors were calculated for each alloy utilizing CALPHAD methods and empirical formulas^[8,23]. Core features include: Atomic Size Difference (δ_r), which quantifies the degree of lattice distortion and serves as a critical parameter for SSS; Mixing Enthalpy (ΔH_mix) and Mixing Entropy (ΔS_mix), which characterize thermodynamic phase stability; valence electron concentration (VEC), a key electronic parameter for predicting FCC/BCC phase stability; and Shear Modulus (G), reflecting the material’s resistance to elastic deformation.

To eliminate noise and ensure training stability, a strict cleaning process was executed: Missing and Outlier Treatment: Samples with missing physical parameters or hardness records were removed. Outliers significantly deviating from physical common sense were excluded based on the 3σ principle^[12,41]. Physical Consistency Check: All physical descriptors were verified to ensure inputs complied with the basic assumptions of the SSS theory.

Given the immense dimensional differences among physical descriptors (e.g., ΔH is roughly 10¹ kJ/mol, while δ_r is 10^-2), direct input into a neural network would cause convergence difficulties or weight bias. Therefore, Z-Score Standardization was applied^[3,42]:

(1) $$ x'=\frac{x-\mu }{\sigma } $$

where μ is the mean and σ is the standard deviation. This step is crucial for the artificial neural network (ANN) model and distance-based SHapley Additive exPlanations (SHAP) analysis, ensuring that the weight contribution of each physical feature in the loss function is of the same order of magnitude.

To objectively evaluate generalization, the cleaned dataset was randomly split into a Training Set (80%) and a Test Set (20%)^[5,42]. To ensure reproducibility, the random seed was fixed (Random_State = 42). The training set was used for fitting the physical baseline and updating residual network parameters, while the test set was reserved strictly for final evaluation and was never involved in gradient updates.

To ensure a rigorous mapping relationship within the framework, the inputs and output are explicitly defined as follows: Model Inputs (X_aug): A total of 17 physics-augmented descriptors are utilized. These include 14 fundamental chemical and thermodynamic features (e.g., VEC, ΔH_mix, ΔS_mix, δ_r, and shear modulus G) calculated via CALPHAD and empirical formulas, alongside 3 critical physics-based terms, with the LS_term (G·δ_r^2/3) being the primary prior. Model Output (Y): The primary and sole direct output of the Hybrid PIML model is the HV. Derived Property: The yield strength (σ_y) is not a direct output of the ML model but is subsequently derived from the predicted HV based on the phase-dependent physical correlation analysis established in this study.

As SSS originates from the lattice elastic stress field induced by solute atoms, the atomic size mismatch (δ_r) is identified as the primary geometric parameter. In parallel, Mixing Enthalpy (ΔH_mix), Mixing Entropy (ΔS_mix), and the Ω parameter were selected to assess the thermodynamic stability of solid solution phases against IMCs. The corresponding calculation formulas are detailed below^[23,43]:

Atomic size mismatch (δ_r):

(2) $$ \delta _r=\sqrt{\sum_{i=1}^nc_i(1-\frac{r_i}{\bar{r}})^2} $$

where c_i denotes the atomic fraction of element i, r_i is the atomic radius, and r = Σc_ir_i represents the average atomic radius. The parameter δ_r directly quantifies the severity of lattice distortion and serves as a core input for the subsequent physics-based strengthening model.

Mixing Enthalpy (ΔH_mix) and Mixing Entropy (ΔS_mix):

(3) $$ \mathrm{\Delta H_{mix}=\sum_{i < j}4\Delta H_{ij}^{mix}c_{i}c_{j}} $$

(4) $$ \mathrm{\Delta S_{mix}=-R\sum_{i=1}^{n}c_{i}lnc_{i}} $$

where ΔH_ij^mix represents the binary mixing enthalpy between elements i and j, and R is the gas constant.

Thermodynamic Parameter (Ω):

(5) $$ \Omega =\frac{T_m\Delta S_{mix}}{|\Delta H_{mix}|} $$

This parameter integrates the competition between entropic and enthalpic effects and is typically used to determine the stability of solid solutions (where Ω > 1.1 favors solid solution formation)^[44,45]. To visualize the coverage and phase stability trends of the dataset, the distribution of the 205 compiled HEA samples in the δ_r - ΔH_mix parameter space is illustrated in Figure 1.

Experimental studies indicate that the crystal structure of HEAs exhibits a strong dependence on the VEC. Since different lattice structures possess varying numbers of slip systems, their hardness baselines differ vastly (BCC is generally harder than FCC)^[3]. Therefore, introducing VEC assists the model in implicitly learning the “Phase Structure-Hardness” correlation^[46]:

(6) $$ \mathrm{VEC=\sum_{i=1}^{n}c_i(VEC)_i} $$

Where (VEC)_i is the number of valence electrons for the i-th element.

This section highlights the key distinction between the proposed Hybrid PIML framework and traditional “composition-property” models. To allow the neural network to explicitly learn SSS laws, this study calculated mechanical modulus parameters directly related to dislocation motion resistance based on the SSS theory:

Average Shear Modulus (G):

(7) $$ \mathrm{G} =\sum_{i=1}^{n}c_iG_i $$

The shear modulus determines the magnitude of dislocation line tension and serves as the “substrate” for measuring material stiffness^[31].

Physics-Based Strengthening Factor (LS_term): To provide direct physical guidance to the Hybrid PIML model, a theoretical strengthening term was pre-constructed. Based on the classical Labusch model and its subsequent extensions for multi-principal element alloys (e.g., by Senkov and Toda-Caraballo), the SSS increment is highly dependent on the coupled effect of lattice distortion and modulus mismatch. Accordingly, we explicitly embedded this physical prior by formulating the composite feature:

(8) $$ \mathrm{LS_{term}} =\propto \mathrm{G\cdot \delta _r^{2/3}} $$

This composite feature physically couples lattice distortion (δ_r) with matrix stiffness (G)^[11,38], capturing the theoretical baseline of resistance to dislocation motion. In the subsequent model training, this feature is input into the neural network as the most critical prior knowledge, enabling the model to perform residual correction directly on the basis of physical laws, thereby substantially enhancing the physical consistency of predictions^[9].

First, a Pearson correlation coefficient matrix (r) was employed to evaluate the pairwise correlations among all candidate descriptors, as illustrated in Figure 2. In HEAs, certain physical parameters (e.g., atomic radius difference δ_r and lattice mismatch parameters) may exhibit high mathematical collinearity (|r| > 0.9)^[9,22]. The screening principle adopted was as follows: for highly correlated feature pairs, priority was given to retaining descriptors with clearer physical significance and tighter linkage to the SSS theory^[23]. For instance, when the simple atomic size difference (δ_r) was found to be highly correlated with the theoretical strengthening factor (LS_term), the latter was retained because LS_term incorporates shear modulus information, thereby better characterizing the comprehensive resistance to dislocation slip.

Following the removal of redundant features, a recursive feature elimination (RFE) algorithm based on RF was applied^[19,43]. Utilizing the “Gini Impurity” metric inherent to tree models, the remaining features were preliminarily ranked by importance. Noise features making negligible contributions to hardness prediction (Importance < 0.01) - such as the atomic number features of certain non-critical elements - were eliminated.

Representing the crux of the feature selection strategy, this step introduces domain knowledge intervention, distinguishing the approach from purely data-driven methods^[36]. Irrespective of RFE rankings, the retention of three core physical descriptors was enforced for the final model input: LS_term(G·δ_r^2/3) as the theoretical baseline for SSS^[38]; VEC as the indicator of phase structural stability; and ΔH_mix as the thermodynamic measure of interatomic binding forces. This three-step process reduced the high-dimensional feature space to 17 key descriptors. Consequently, the computational load for ANN training was minimized while ensuring the input information retained clear metallurgical meaning, a prerequisite for the subsequent SHAP mechanistic analysis.

As seen in Figures 5A and 6A, while these two pure data-driven models can identify some key features, their decision logic relies primarily on statistical correlation rather than physical causality. Spurious Statistical Correlation: In the XGBoost model, atomic number or the content of specific elements (e.g., Co or Ni) are often assigned excessively high weights. Although these elements are associated with phase structure to some extent, they are not direct determinants of hardness in SSS theory. Ambiguity of Physical Mechanisms: Although the standard ANN model captured the importance of VEC, correctly reflecting VEC’s control over BCC/FCC phase stability, it lacked sensitivity to the interaction between lattice distortion (δ_r) and shear modulus (G). Consequently, when predicting high-distortion alloys (such as refractory HEAs), the model often infers results by “memorizing” composition data rather than “understanding” the lattice stress field, limiting its extrapolation capability in unknown compositional spaces.

Conversely, the Hybrid PIML model, augmented with the physics-based feature (LS_term = G·δ_r^2/3), demonstrates superior physical consistency. SHAP analysis highlights the following mechanisms: (1) Dominance of Physical Descriptors: The engineered LS_term dominates the feature importance ranking [Figure 7B], surpassing individual elemental features. This confirms a shift in the model’s attention from statistical composition to physical fundamentals, identifying the coupling of shear modulus (G) and atomic size mismatch (δ_r) as the core determinant of HEA hardness; (2) Theoretically Aligned Strengthening Direction: Figure 7A shows that high LS_term values correlate positively with SHAP values, indicating that increased lattice distortion and modulus stiffness lead to higher predicted hardness. This behavior is consistent with classical SSS theory (H∝G·δ_r^2/3), validating that the model has internalized physical causality rather than relying solely on data fitting; (3) Non-linear Threshold Effect: Figure 7C illustrates a non-linear contribution of the physical term to hardness. While the relationship is linear in low-distortion regions, a slope change is observed in high-distortion regions (e.g., refractory HEAs). This deviation implies complex coupling between SSS and phenomena such as SRO or nano-cluster strengthening under high lattice stress - non-linear details effectively captured by the PIML framework.

In summary, the comparative analysis shows that while pure data-driven models (XGBoost/ANN) perform acceptably in numerical precision, they suffer from the defect of “knowing the what but not the why” regarding physical interpretability. The Hybrid PIML framework, however, not only achieved the highest prediction accuracy (R² = 0.976) but, more critically, established a decision-making process consistent with physical metallurgy principles. This “physics-aware” capability makes the PIML model more reliable in new material discovery, effectively avoiding erroneous predictions that violate thermodynamic common sense.

In the virtual screening funnel, the ‘optimization performance’ is defined by two hierarchical criteria: Maximization of HV (Target Output): Identifying compositions with predicted HV > 900 HV; Phase Stability Constraint: Filtering for BCC-dominant solid solution structures while maintaining a balance between high hardness and the avoidance of brittle intermetallic phases. Traditional experimental trial-and-error methods are often cost-prohibitive and limited to the central regions of phase diagrams. Leveraging the Hybrid PIML model, a Screening Funnel was established to navigate the expanded design space, as illustrated in Figure 9. First, the target system (e.g., the refractory W-Mo-Ta-Nb-V-Ti system) is defined with a step size of 5 at.%, generating approximately 53,000 potential non-equiatomic combinations. Next, physics-based physical descriptors (e.g., δ_r, ΔH_mix, VEC) are calculated for each virtual composition. The trained Hybrid PIML model is then deployed to predict the hardness of all candidates, ranking them by predicted value. Finally, phase formation laws implicitly learned by the model (as shown in the SHAP analysis) are applied to filter out compositions with extremely low predicted hardness - implying the formation of soft FCC phases - or abnormal physical parameters that suggest a risk of amorphous formation.

To validate the screening strategy, the Co-Cr-Ti-Mo-W system was subjected to retrospective analysis. While the traditional equiatomic design (Co₂₀Cr₂₀Ti₂₀Mo₂₀W₂₀) exhibits limited performance, the Hybrid PIML model successfully identified optimized non-equiatomic regions [Figure 10]. The analysis suggests that reducing Ti/W content and increasing the Co/Cr matrix fraction optimizes the trade-off between SSS and brittle phase suppression, specifically within the range of (Co + Cr + Mo) ≈ 0.9 and (W + Ti) ≈ 0.1. These predictions are strongly supported by recent experimental literature^[11,41,49]. Evidence shows that alloys in this optimized interval surpass the equiatomic benchmark, maintaining ultra-high hardness (> 900 HV) while mitigating brittle intermetallic phases (e.g., σ or Laves phase) precipitation induced by severe lattice distortion^[50]. This finding accords with Wen et al. on the synergistic enhancement of strength and ductility via non-equiatomic design^[51]. Ultimately, this case study demonstrates that the Hybrid PIML model offers both robust interpolation accuracy and the extrapolation capability required to navigate complex compositional spaces and locate local optima.

To conclusively address the critical necessity for experimental validation regarding phase constitutions (e.g., confirming the solid-solution-only condition) and macroscopic strengthening mechanisms, we cross-verified our PIML-optimized compositions against extensive experimental efforts documented in recently published, independent literature.

To address the critical necessity for experimental validation regarding both macroscopic hardness and microscopic phase constitutions, we cross-verified our PIML-optimized compositions against recent, independent high-throughput experimental efforts on the exact Co_xCr_yTi_zMo_uW_v system^[52]. While direct in-house synthesis and microstructural characterizations [scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS)] are beyond the scope of this computational methodology study, the rigorous experimental data provided by Liu et al. offers robust out-of-sample validation^[52]. Crucially, their experimental microstructural analysis [X-ray diffraction (XRD)/SEM] of equimolar CoCrTiMoW revealed the precipitation of Co-Cr-Ti IMCs alongside the BCC solid-solution dendrites due to excessive Ti content. This experimental finding perfectly justifies the optimization logic internalized by our Hybrid PIML framework. As revealed by our SHAP analysis, the model heavily relies on the solid-solution strengthening term (LS_term). To maximize this solid-solution contribution and suppress brittle second-phase precipitations, our model correctly predicted that the atomic ratios of severe lattice distorters must be significantly constrained [i.e., (W + Ti) ≈ 0.1 with an elevated Co/Cr matrix]. Furthermore, Liu et al. experimentally confirmed that non-equiatomic alloys designed following similar compositional constraints successfully maintain structural integrity and exhibit ultra-high HV exceeding 900 HV (up to 971 HV)^[52]. This exceptional agreement between independent experimental observations and our mathematical outputs not only validates the predicted hardness values but also explicitly proves that the Hybrid PIML framework successfully captured the true physical metallurgy principles governing solid-solution optimization in complex HEA design spaces. Future work will entail targeted in-house synthesis to further map the phase boundaries of these optimized candidates.