To achieve high-entropy design of RE disilicates (RE₂Si₂O₇), we integrate high-throughput experiments with machine learning techniques to identify promising high-temperature wave-transparent materials. The workflow comprises 3 components: high-throughput experiments, feature screening, and machine learning. In the high-throughput experiments (Fig. 1A), 66 compositions of powders were initially prepared for constructing the phase diagram. Subsequently, based on the phase diagram, 24 β- and γ-phase RE₂Si₂O₇ were selected and fabricated into bulk ceramics. Their comprehensive properties were measured, with dielectric constant and thermal conductivity being used in the machine learning model. Finally, 5 compositions comprising 5, 6, 7, 8, and 9 elements, respectively, were designed to validate the model’s generalization capabilities. All the samples were prepared through a self-built high-throughput platform (Fig. S1). During feature screening (Fig. 1B), 28 descriptors spanning atomic, electronic, and structural scales were selected (Table 1). To ensure feature independence and validity, Pearson correlation analysis was performed for preliminary self-correlation screening. It is particularly noteworthy that descriptors with Pearson correlation coefficients greater than 0.8 are strictly considered highly correlated, a criterion that aids in identifying and excluding features that may lead to multicollinearity issues. Furthermore, Pearson analysis was utilized to evaluate the relationships between these descriptors and target properties (phase composition, dielectric constant, and thermal conductivity), thereby eliminating low-correlation descriptors and refining the feature set. In the machine learning framework (Fig. 1C), leave-one-out cross-validation (LOOCV) was applied to guarantee reliability. The final models implemented include decision tree and linear regression models. Through feature importance analysis within these models, the most strongly correlated descriptors were successfully identified, and their underlying mechanisms are thoroughly investigated.

RE₂Si₂O₇ exhibits 7 different crystal polymorphs (β-, α-, γ-, δ-, F-, G-, and A-phases), with β- and γ-phases showing particular promise for high-temperature applications due to their stable structures with no phase transformation. However, the key parameters that govern polymorphic structures remain unclear. A comprehensive sample library consisting of 66 different compositions of (5RE_1/5)₂Si₂O₇ (RE = Sc, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu), primarily focusing on heavy RE elements with a wide range of ionic radii from 0.74 to 0.94 Å, was systematically designed to investigate the polymorphic structures. Multiple sample sets were synthesized using high-throughput experimental methods. The XRD patterns for all prepared (5RE_1/5)₂Si₂O₇ samples are shown in Fig. 2A. The (5RE_1/5)₂Si₂O₇ crystallized into β-, γ-, and δ-phases, with some compositions exhibiting multiphase coexistence.

Pearson correlation analysis was employed to thoroughly examine the self-correlation among these descriptors and the correlation between descriptors and phase formation, ensuring their independence and effectiveness. The analysis results are presented in the form of a correlation matrix in Fig. 2B. The specific results of the correlation analysis between descriptors and phase composition can be found in Fig. 2C. By combining the results of self-correlation analysis with the correlations between descriptors and phase composition, a comprehensive evaluation and comparison were conducted, ultimately selecting and identifying the 6 most critical primary descriptors. These 6 descriptors (IR, EN, δ_EN, IE1, δ_IE1, and IE3) were selected as input features for the subsequent models. Detailed information about these 6 descriptors for the 66 compositions can be found in Table S1. In order to classify the phase compositions of RE₂Si₂O₇, 6 machine learning models were employed, namely, Gradient Boosting Decision Tree (GBDT), Random Forest (RF), Support Vector Machine (SVM) with different kernels (linear, polynomial [poly], and radial basis function [rbf]), and k-Nearest Neighbors (KNN). Figure 2D illustrates the relationship between prediction accuracy and hyperparameter settings for different machine learning models. By comparing the prediction accuracies of these models under various hyperparameter configurations, GBDT was found to perform the best among all models, achieving the highest accuracy of 96.97%. Further analysis of the optimal GBDT model yielded an excellent weighted precision of 98.54%, listed in Table S2. Figure 2E illustrates the relationship between the number of features and validation accuracy in the GBDT model. The results demonstrate that when the number of descriptors is reduced to only 1 or 2, the validation accuracy can reach up to 98.48%. This indicates that a particular descriptor possesses exceptionally high importance. Figure 2F presents the feature importance analysis results of the GBDT model, further indicating that IR has the most dominant impact on phase composition, with an importance of 95.08%. Figure 2G displays the relationship between phase composition and ionic radius for the samples, including those from this work and those from the referenced literature [32–37]. The results show that (5RE_1/5)₂Si₂O₇ tends to form the β-phase when the average RE³⁺ ionic radius is below 0.8864 Å and tends to form a mixture of β-phase and γ-phase when the average RE³⁺ ionic radius ranges from 0.8864 to 0.8916 Å; as the average RE³⁺ ionic radius increases to between 0.8916 and 0.9040 Å, the γ-phase becomes more dominant; and when the average RE³⁺ ionic radius exceeds 0.9040 Å, a mixture of γ-phase and δ-phase is more likely to form. It is worth noting that a narrow gap of 0.0052 Å exists between the β-phase boundary (<0.8864 Å) and the γ-phase boundary (>0.8916 Å). The polymorphic phase transformation in high-entropy ceramics is intricately governed by the interplay between thermodynamic stability and diffusion kinetics. As an intrinsic characteristic of these multicomponent systems, the size variations and chemical mismatches among diverse RE³⁺ cations collectively induce severe lattice distortion, which markedly elevates the migration energy. When the average RE³⁺ ionic radius falls precisely within this critical thermodynamic boundary region, the driving force for phase transformation becomes exceptionally weak. The coupling of this marginal thermodynamic driving force with the inherent sluggish diffusion kinetics prevents the system from completely overcoming the energy barrier to transform into a purely single-phase structure under conventional synthesis conditions, thereby trapping it in a multiphase β+γ coexistence [38]. Based on the above analysis, it is evident that the average RE³⁺ ionic radius is a key factor in determining the phase composition of high-entropy RE₂Si₂O₇, thereby laying a solid foundation for further exploring the relationship between material composition and performance.

Owing to the superior high-temperature stability of β- and γ-phase RE₂Si₂O₇, we selected 24 compositions and fabricated bulk samples based on the above results. The x-ray diffraction (XRD) patterns of these samples are presented in Fig. 3A to D. Rietveld refinements of XRD are provided in Figs. S2 to S4, where the well-refined lattice parameters are compared with density functional theory (DFT) calculation results listed in Table S3, confirming that single-phase β- and γ-RE₂Si₂O₇ have been successfully synthesized and validating the reliability of the subsequent theoretical calculations. Figure S5 presents the fracture morphology of the bulk samples. The micrographs indicate a high densification, evidenced by the presence of few pores. Remarkably, the fracture surfaces display step-like patterns characteristic of cleavage fracture, which can be attributed to the heterogeneous bond energy distribution within the [REO₆] polyhedra of RE₂Si₂O₇. Under applied stress, the weaker RE–O bonds preferentially rupture, leading to transgranular fracture along specific crystallographic planes. This process results in the formation of flat platforms, while the simultaneous occurrence of cleavage on different crystal planes with identical Miller indices manifests as the observed step-like patterns.

To further verify the atomic structures of the as-synthesized bulk (5RE_1/5)₂Si₂O₇, we conducted high-angle annular dark-field (HAADF) imaging using scanning transmission electron microscopy (STEM). Figure 3E and H present atomic-resolution HAADF-STEM images of β-(Ho_1/5Er_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇ viewed along the [1\overline{1}2] zone axis and the γ-(Dy_1/5Ho_1/5Er_1/5Tm_1/5Lu_1/5)₂Si₂O₇ along [001], with fast Fourier transform (FFT) patterns matching the respective β and γ polymorphs. Both β- and γ-phase RE₂Si₂O₇ share a skeleton of corner-sharing double [SiO₄] tetrahedra ([Si₂O₇] units) alternating with layers of 6-coordinated RE cations that form cage-like [REO₆] polyhedra. In both polymorphs, the bridging Si–O–Si angle is linear (180°). Their principal difference is the stacking orientation of the [Si₂O₇] units: β-RE₂Si₂O₇ adopts a sandwich-like arrangement with Si–O–Si chains lying in the (0 0 1) planes, whereas γ-RE₂Si₂O₇ shows a wave-like succession of [REO₆] and [Si₂O₇] slabs with these chains parallel to the (1 0 1) planes. The energy dispersive spectroscopy (EDS) maps in Fig. 3F and I show that the selected elements are uniformly distributed in an equimolar (1:1:1:1:1) ratio, as listed in Table S4. These results confirm the successful production of β- and γ-(5RE_1/5)₂Si₂O₇. Figure 3G and J show high-resolution transmission electron microscopy (HRTEM) images and geometric phase analysis (GPA) maps of β-(Ho_1/5Er_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇ and γ-(Dy_1/5Ho_1/5Er_1/5Tm_1/5Lu_1/5)₂Si₂O₇. Compared with γ-(Dy_1/5Ho_1/5Er_1/5Tm_1/5Lu_1/5)₂Si₂O₇, the elastic strains of β-(Ho_1/5Er_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇ are greater, indicating the occurrence of severe lattice distortion.

The electromagnetic wave transmission characteristics of materials are fundamentally determined by their dielectric properties, primarily the dielectric constant and dielectric loss tangent value. Ideal wave-transparent materials require both low dielectric constant and low dielectric loss tangent value. Figure 4A and B show the dielectric constant and the dielectric loss tangent values of (5RE_1/5)₂Si₂O₇ between 12 and 18 GHz. The dielectric constants of (5RE_1/5)₂Si₂O₇ remain stable and below 9, and the dielectric loss tangent values consistently fall within the 10⁻² order. Detailed dielectric constants and dielectric loss tangent values for the 24 β- and γ-phase samples are presented in Figs. S6 to S8. Among these samples, 1-(Ho_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇, 2-(Ho_1/5Er_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇, and 15-(Tb_1/5Ho_1/5Tm_1/5Yb_1/5Sc_1/5)₂Si₂O₇ exhibit relatively low dielectric constants, while samples 19-(Ho_1/5Er_1/5Tm_1/5Yb_1/5Lu_1/5)₂Si₂O₇, 22-(Tb_1/5Dy_1/5Er_1/5Yb_1/5Lu_1/5)₂Si₂O₇, and 23-(Tb_1/5Ho_1/5Er_1/5Tm_1/5Lu_1/5)₂Si₂O₇ exhibit higher dielectric constants. The dielectric constants of 6 samples are presented in Fig. 4C. To identify the primary factors influencing the dielectric properties of materials, machine learning techniques were employed. Pearson correlation analysis was employed to evaluate the self-correlation among the descriptors as well as the correlation between the descriptors and the dielectric constant. The results of correlation analysis between descriptors, as well as between descriptors and dielectric constants, are presented in Figs. S9 and S10, respectively. Ultimately, 6 descriptors (IR, VEC, EN, δ_EN, IE1, and IE2) were selected as input features for the subsequent models. Detailed information about these 6 descriptors for the 24 compositions can be found in Table S5. Models including Linear Regression (Linear), Ridge Regression (Ridge), Lasso Regression (Lasso), and Elastic Net Regression (Elastic Net) were utilized. The reliability of the models was validated by comparing the root mean squared error (RMSE), as demonstrated in Fig. 4D. The results indicate that the Elastic Net model exhibited the lowest RMSE and was therefore selected as the final model. Feature importance analysis of the Elastic Net model reveals that IR (average RE³⁺ ionic radius) holds a dominant position in dielectric constants, as shown in Fig. 4E. Subsequently, the relationship between the average RE³⁺ ionic radius and dielectric properties is plotted in Fig. 4F, showing that compositions containing Sc with the lowest ionic radius exhibit relatively lower dielectric constants, while those without Sc possess higher dielectric constants. Based on the above findings, a further investigation was conducted into the pivotal role of Sc in the dielectric properties.

Although categorized as a RE element, Sc is distinguished by its lack of 4f electrons and its smallest ionic radius. The incorporation of Sc substantially reduces the dielectric constant because its unique electronic configuration and small size synergistically suppress both ionic and electronic polarization. To elucidate this mechanism, charge density difference (CDD) and electron localization function (ELF) analyses are demonstrated in Fig. 4G and H. The CDD mapping reveals a more pronounced charge transfer within the Sc–O bond compared to the Er–O bond, indicating a stronger covalent character. This is further corroborated by first-principles calculations, which confirm a shorter Sc–O bond length, as detailed in Table S6. This enhanced bond strength inherently restricts the off-center displacement of ions under an external alternating electric field, thereby severely mitigating ionic polarization. Furthermore, the ELF reveals a markedly higher degree of electron localization tightly bound around the Sc atomic cores compared to other RE elements. This robust confinement restrains electron cloud deformation, thereby drastically reducing electronic polarization. The wave-transparent mechanism is schematically depicted in Fig. 4I. To sum up, the synergistic effect of Sc doping effectively minimizes the overall polarizability [39,40]. While the relationship between dielectric loss tangent value and RE species is not obvious, the overall dielectric loss tangent values of (5RE_1/5)₂Si₂O₇ remain highly stable. In general, Sc-doped compositions demonstrate dielectric constants superior to both Al₂O₃ (ε ~ 9.6) and Si₃N₄ (ε ~ 7.9) ceramics, and can approach those of BN (ε ~ 4) and SiO₂ (ε ~ 3.8) ceramics. Among these, (Ho_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇ and (Ho_1/5Er_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇—which have the 2 smallest average RE³⁺ ionic radii—exhibit the 2 lowest dielectric constants, 5.4 and 5.0, respectively. Interestingly, our analysis reveals that both the phase structure and the dielectric constant are intrinsically governed by this average RE³⁺ ionic radius. Fundamentally, a smaller average radius leads to globally shorter RE–O bonds. Structurally, these shorter bonds drive the transition toward the more densely packed β phase, restricting local ionic polarizability. Consequently, coupled with the unique suppression of electronic polarization by Sc discussed above, the Sc-doped β-phase compositions exhibit a more pronounced advantage in dielectric performance. These results indicate that Sc-doped high-entropy ceramics show great promise for wave-transparent applications.

To effectively block external heat flux and protect the antenna from thermal damage while ensuring stable transmission of electromagnetic waves, wave-transparent materials should possess low thermal conductivity. Figure 5A and B illustrate the thermal diffusivities and thermal conductivities of (5RE_1/5)₂Si₂O₇ at various temperatures. At 323 K, the thermal conductivity of (5RE_1/5)₂Si₂O₇ ranges from approximately 2.0 to 5.1 W·m⁻¹·K⁻¹. At 1,273 K, the thermal conductivity of (5RE_1/5)₂Si₂O₇ ranges from approximately 1.3 to 2.5 W·m⁻¹·K⁻¹. Detailed thermal diffusivities and thermal conductivities for the 24 samples are presented in Figs. S11 to S13. Among these samples, 1-(Ho_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇, 4-(Tb_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇, and 12-(Tb_1/5Ho_1/5Tm_1/5Lu_1/5Sc_1/5)₂Si₂O₇ exhibit relatively low thermal conductivities, while samples 19-(Ho_1/5Er_1/5Tm_1/5Yb_1/5Lu_1/5)₂Si₂O₇, 22-(Tb_1/5Dy_1/5Er_1/5Yb_1/5Lu_1/5)₂Si₂O₇, and 23-(Tb_1/5Ho_1/5Er_1/5Tm_1/5Lu_1/5)₂Si₂O₇ exhibit higher thermal conductivities, as shown in Fig. 5C. Notably, the lowest experimental value of 1.3 W·m⁻¹·K⁻¹ closely approaches the theoretical minimum thermal conductivity (κ_min) of 1.25 W·m⁻¹·K⁻¹, which was calculated via the classical Clarke model, with the detailed calculation parameters summarized in Table S7. Machine learning techniques were employed to identify the key factors impacting the thermal properties of materials. Figure S14 presents the Pearson correlation analysis results between descriptors and thermal conductivities. In the end, 5 key descriptors are identified: δ_IR, EN, δ_EN, IE1, and IE3. Detailed information about these 5 descriptors for the 24 compositions can be found in Table S8. The same 4 machine learning models as above were adopted (Linear, Ridge, Lasso, and Elastic Net). Figure 5D demonstrates the validation of model reliability through a comparison of RMSE values. Among the models evaluated, the Lasso model achieves the lowest RMSE and is thus chosen as the final model. As shown in Fig. 5E, feature importance analysis of the Lasso model reveals that δ_IR (the deviation of RE³⁺ ionic radii) played a dominant role in thermal conductivity.

Subsequently, the relationship between the deviation of RE³⁺ ionic radii and thermal conductivities is presented in Fig. 5F, demonstrating that compositions containing Sc exhibit notably lower thermal conductivities. An intrinsic feature of high-entropy structures is the local chemical fluctuation arising from the mutual interaction of constituent elements. Because Sc possesses a drastically smaller atomic mass and ionic radius compared to other RE elements, its incorporation remarkably amplifies such local chemical fluctuations. To quantitatively elucidate this mechanism, the mass fluctuation (Γ_m) and lattice distortion (Δ_d) of the 6 representative samples in Fig. 5C were calculated, as displayed in Fig. 5G and H, confirming that Sc doping substantially elevates both parameters. These theoretical calculations are further experimentally supported by the Raman spectra shown in Fig. S15, where the Sc-doped composition shows a remarkable peak broadening and continuous overlapping compared to the Sc-free composition. Generally, ideal harmonic crystals exhibit extremely sharp Raman peaks. Although the Sc-free composition displays a certain degree of peak broadening, the incorporation of Sc induces a more pronounced broadening. This phenomenon reveals that the Sc-induced extreme mass fluctuations and severe lattice distortion profoundly disrupt the lattice harmonicity, thereby leading to markedly intensified phonon scattering. Based on the phonon transport theory, the phonon lifetime (τ) is inversely proportional to the full width at half maximum (FWHM) of the Raman peak, expressed as [41]:where c is the speed of light and Γ is the FWHM. Consequently, the observed extensive peak broadening and continuous overlapping provide robust spectroscopic evidence for a substantially shortened phonon lifetime, fully confirming that these pronounced local chemical fluctuations disrupt the long-range periodic arrangement of atoms and induce strong scattering of heat-carrying phonons, ultimately leading to a minimized lattice thermal conductivity in the Sc-doped high-entropy ceramics, as shown in Fig. 5I. It should be clarified that the phase structure has no direct correlation with the thermal conductivity. Crystallographically, despite their distinct stacking modes, both the β- and γ-phases feature identical cage-like [REO₆] coordination environments and analogous [SiO₄] structural backbones, which leads to fundamentally equivalent intrinsic phonon scattering mechanisms. As reported in a previous study [42], the thermal conductivities of the β- and γ-phases in RE₂Si₂O₇ are comparable, given identical elemental compositions. Instead, the exceptionally low thermal conductivity achieved in our materials is dominated by the severe lattice distortions and mass fluctuations inherently introduced by the high-entropy compositional design.

To validate the generalization capability of the aforementioned machine learning model, we designed 5 groups of high-entropy RE₂Si₂O₇ compositions comprising 5 [(Er_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇], 6 [(Ho_1/6Er_1/6Tm_1/6Yb_1/6Lu_1/6Sc_1/6)₂Si₂O₇], 7 [(Dy_1/7Ho_1/7Er_1/7Tm_1/7Yb_1/7Lu_1/7Sc_1/7)₂Si₂O₇], 8 [(Tb_1/8Dy_1/8Ho_1/8Er_1/8Tm_1/8Yb_1/8Lu_1/8Sc_1/8)₂Si₂O₇], and 9 [(Gd_1/9Tb_1/9Dy_1/9Ho_1/9Er_1/9Tm_1/9Yb_1/9Lu_1/9Sc_1/9)₂Si₂O₇] components, respectively, based on the previously identified key descriptors. Specifically, the dielectric properties are governed by the average RE³⁺ ionic radius, and the thermal conductivity is primarily influenced by the deviation of RE³⁺ ionic radii. The experimental dielectric constants and thermal conductivities, together with the predicted values, are shown in Fig. S16, thereby confirming the model’s generalization ability. This successful validation demonstrates that the machine learning framework, built upon the key descriptors of average ionic radius and deviation of ionic radii, is not only accurate for the 5-component systems but also effectively extendable to more complex compositions with up to 9 distinct RE cations. The establishment of the model facilitates the development of high-entropy RE₂Si₂O₇ wave-transparent materials with superior wave-transparency and thermal-insulation performance.

For wave-transparent materials, the coefficient of thermal expansion plays a crucial role in maintaining structural stability across varying temperature conditions. Figure S17 displays the coefficients of thermal expansion of (5RE_1/5)₂Si₂O₇, derived from linear fitting of the thermal expansion curves, presented in Figs. S18 to S20. The coefficients of thermal expansion range between 3.3 × 10⁻⁶ K⁻¹ and 4.4 × 10⁻⁶ K⁻¹. The coefficients of thermal expansion of (5RE_1/5)₂Si₂O₇ show insensitivity to RE species. Previous research [43] has reported that [SiO₄] tetrahedra in RE₂Si₂O₇ have near-zero thermal expansion coefficients, while [REO₆] polyhedra dominate thermal expansion behavior. Hazen and Prewitt [44] found that thermal expansion of different crystal structures is primarily determined by the coordination number of RE–O. The RE–O coordination number is identical in both β- and γ-phases, resulting in similar thermal expansion coefficients for these 2 phases and insensitivity to RE species. Becerro et al. used high-temperature x-ray methods to measure the thermal expansion coefficients of several pure-phase RE₂Si₂O₇. The thermal expansion coefficients of β-Lu₂Si₂O₇, β-Yb₂Si₂O₇, β-Er₂Si₂O₇, β-Y₂Si₂O₇, γ-Y₂Si₂O₇, and γ-Ho₂Si₂O₇ were approximately 4 × 10⁻⁶ K⁻¹, while that of β-Sc₂Si₂O₇ was 5.5 × 10⁻⁶ K⁻¹. The (5RE_1/5)₂Si₂O₇ exhibits lower thermal expansion coefficients than β-Sc₂Si₂O₇ due to marked lattice distortion caused by multi-RE-principal-component doping. Such distortion increases low-frequency vibration modes of RE atoms, enhancing the proportion of phonons contributing to negative expansion within the lattice, thus diminishing the thermal expansion coefficient [45].

Hardness is a crucial property for wave-transparent materials, as it quantifies their resistance to external damage. Figure S21 illustrates the Vickers hardness of (5RE_1/5)₂Si₂O₇, which range from approximately 5 to 10 GPa under higher loads. Compared to single-component RE₂Si₂O₇ such as Yb₂Si₂O₇ and Lu₂Si₂O₇, the hardness exhibits an enhancement. The improved hardness in high-entropy RE₂Si₂O₇ may arise from strain field fluctuations induced by lattice distortion, thereby enhancing the resistance to deformation. Notably, the Vickers hardness exhibits an inverse relationship with applied load, with smaller loads yielding higher hardness values due to the indentation size effect. Analysis of the Vickers hardness against the RE species under various loads reveals no clear correlation. The hardness of the 24 samples under different loads is specifically plotted in Figs. S22 to S24.

The radar chart facilitates easy comparison of all of the attributes of interest. Detailed comprehensive properties of the 24 samples are presented in Figs. S25 to S27. Aided by radar chart analysis, the (Ho_1/5Tm_1/5Yb_1/5Lu_1/5Sc_1/5)₂Si₂O₇ high-entropy ceramic was identified as the optimal candidate. As illustrated in Fig. S28, a direct comparison with mainstream wave-transparent materials (e.g., Al₂O₃, Si₃N₄, BN, and SiO₂) demonstrates that our design effectively mitigates the traditional trade-off inherent in structural–functional integration. Specifically, Al₂O₃ and Si₃N₄ are limited by their high dielectric constants, BN suffers from poor mechanical properties, and SiO₂ exhibits an excessively low coefficient of thermal expansion that severely hinders structural matching. In contrast, this Sc-doped ceramic exhibits a low dielectric constant, a low dielectric loss tangent value, low thermal conductivity, good hardness, and moderate thermal expansion coefficients, making it a highly competitive candidate for demanding high-temperature wave-transparent applications. Overall, the introduction of Sc markedly reduces both the dielectric constant and thermal conductivity by decreasing ionic and electronic polarization and enhancing phonon scattering through mass fluctuation and lattice distortion. Therefore, high-throughput experimental methods combined with machine learning approaches facilitate the systematic investigation of the synthesis and properties of RE-based high-entropy wave-transparent ceramics.

A high-throughput method was employed to prepare high-entropy RE₂Si₂O₇. First, RE₂O₃ powders (>99.9%, Guangzhou Jianfeng Rare Earth Co. Ltd., China) and SiO₂ powders (99.7%, Sinopharm Chemical Reagent Co. Ltd., China) were mixed in a molar ratio of nRE₂O₃:SiO₂ = 1:2, with each RE₂O₃ added in equal molar proportions. In order to compensate for SiO₂ volatilization at elevated temperatures, 2% excess SiO₂ was added. The mixed powders were ball-milled at 70 r/min for 2 h in a high-throughput polyethylene jar, using agate balls and anhydrous ethanol as the milling medium. After being dried at 80 °C for 6 h, the powders were sieved through a 60-mesh screen. The fine powders were sintered at 1,700 °C for 3 h (heating rate: 5 °C/min; cooling rate: 2 °C/min) to obtain high-entropy RE₂Si₂O₇ in a muffle furnace. The high-entropy RE₂Si₂O₇ was further ball-milled using the same method as above. The as-prepared powders were uniaxially pressed under a uniform pressure of 5 MPa using a high-throughput mold capable of simultaneously fabricating up to 32 samples, followed by cold isostatic pressing at 200 MPa for 15 min. Finally, the bulk samples were sintered at 1,600 °C for 20 h (heating rate: 5 °C/min) to obtain dense high-entropy RE₂Si₂O₇ ceramics.

The density of the sintered bulk samples was determined by the Archimedes’ method. The phase composition was characterized by XRD (Bruker D8 ADVANCE, Germany). The surface grain morphology was observed using a scanning electron microscope (Hitachi SU5000, Japan). The dielectric constant and the dielectric loss tangent of the samples were measured using a vector network analyzer (Keysight N5227B PNA, America) combined with the waveguide method, with sample dimensions of 16 mm × 8 mm × 2 mm. The Vickers hardness was determined using a Vickers hardness tester (Veiyee HV-30AT, China). The thermal expansion coefficient was measured using a pushrod dilatometer (NETZSCH DIL 402 CLASSIC, Germany). The thermal diffusivity was determined using a laser flash analyzer (NETZSCH LFA 427, Germany) with the sample dimensions of a diameter of 6 mm and a thickness of 1 mm. The thermal conductivity (κ) can be obtained using the following formula [46]:where ρ is the density and C_p represents the heat capacity. C_p was calculated according to the Neumann–Kopp law.

The high-entropy structures were generated using the special quasi-random structure (SQS) method implemented in the Alloy Theoretic Automated Toolkit [47,48]. A 5 × 1 × 1 supercell was constructed for calculations. The electronic structure calculations were performed through first-principles simulations within DFT using the Vienna Ab initio Simulation Package (VASP) [49]. The projector augmented wave (PAW) pseudopotentials method was employed to describe the electron–ion interactions. For the exchange and correlation interactions, the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional was used [50]. The Brillouin zone was sampled using the Monkhorst–Pack scheme with a k-point spacing of 0.3 Å⁻¹ [51]. The cutoff energy for the plane wave basis set was set to 520 eV [52]. To ensure convergence, the energy criterion was set to 10⁻⁶ eV. The charge distribution and transfer in the optimized structures were analyzed using the CDD and ELF. The Young’s modulus (E) required for the Clarke model was derived using the energy-strain method. The theoretical minimum thermal conductivity (κ_min) was then calculated according to the Clarke model [53]:where k_B is the Boltzmann constant, N_A is Avogadro’s constant, n is the number of atoms in the primitive cell, ρ is the density, E is the Young’s modulus, and M is the molecular weight.

A comprehensive machine learning approach was developed using Python and the scikit-learn library to predict phase composition and material properties [54,55]. Before model training, the mean and difference values of the considered fundamental physical parameters from the atomic level and the monolithic level were calculated as follows:where pam_i and c_i are the considered fundamental physical parameters and the ratios of the ith constituents, respectively. Meanwhile, all features were subjected to min–max normalization to scale the values into the range of [0,1], which was calculated as follows:where x_i^{\mathrm{norm}} is the ith normalized value of the input feature x. x_{\max }, x_{\min }, and x_i are the maximum, minimum, and ith value of the input feature x, respectively. In order to reduce potential overfitting introduced by strongly correlated features, the Pearson correlation coefficient (r) between features was calculated as:where x_i and y_i are the ith values of 2 different input features, respectively. \overline{x} and \overline{y} are the mean values of the 2 different input features, respectively. The prediction framework consisted of multiple machine learning algorithms. For phase formation ability, the implemented algorithms included GBDT, RF, SVM with different kernels (linear, poly, and rbf), and KNN. The training dataset for phase prediction was constructed from 66 synthesized high-entropy samples, comprising 48 single-phase and 18 multiphase samples. For the classification task, the phase structures, namely, β, β+γ, γ, and γ+δ were categorically encoded as “0”, “1”, “2”, and “3”, respectively. The classification performance of these models was assessed by calculating the predictive accuracy via LOOCV. For our multiclass phase classification problem, the overall accuracy (Acc) can be mathematically expressed as:where N and {\mathrm{Acc}}_i are the number of folds and the validation accuracy of each fold, respectively. The dataset was split into N folds, and the number of folds was equal to the dataset size. In each fold, only one sample was included in the validation set, while the rest belonged to the training set. Thus, Acc_i strictly equals 1 for a correct prediction and 0 for an incorrect one. Furthermore, to account for the inherent class imbalance in the dataset, the weighted precision was also calculated for the optimal model. For dielectric and thermal properties, Linear, Ridge, Lasso, and Elastic Net were applied, and their predictive performance was evaluated using the RMSE derived from LOOCV. Rather than enforcing a consistent model across both properties, we evaluated multiple algorithms and strictly selected the optimal one for each target based on achieving the lowest RMSE. Ultimately, feature importance analysis was conducted across both the classification and regression models, aiming to elucidate the underlying physical mechanisms.