A hybrid cluster-expansion–informed machine learning framework for predicting enthalpy of mixing in BCC refractory binary alloys

Abstract The enthalpy of mixing is a key thermodynamic quantity governing alloy stability and serves as a foundational input for high-entropy alloy (HEA) design. In this work, we focus on developing and validating an efficient predictive framework at the binary subsystem level, which constitutes the fundamental building blocks of multicomponent HEAs. Effective Hamiltonians based on cluster expansion (CE) can reproduce configurational energies from density functional theory, but they often involve a large number of interaction parameters and exhibit system specific constraints across chemical systems. Conversely, purely empirical machine-learning (ML) models are computationally efficient but may lack explicit configurational physics. To bridge these approaches, we construct a hybrid framework that combines a minimal set of physically grounded CE-derived configurational descriptors, primarily first nearest-neighbor interactions and selected short-range multi-site clusters, with compact chemical descriptors including valence electron concentration, electronegativity difference (Δχ), atomic size mismatch ( δ ), and molar volume ( V ). We systematically compare five modeling strategies (full CE, restricted CE, configurational-only ML, empirical-only ML, and hybrid ML) across six body-centered cubic refractory binary systems. Ten ML algorithms and exhaustive feature combinations were evaluated using cross-validation. Within the binary systems considered, the hybrid ML approach consistently matches or exceeds full-CE accuracy ( R 2 > 0.93, frequently > 0.98) while employing substantially fewer fitted parameters. It also demonstrates improved stability and cross-system consistency relative to purely configurational or purely empirical models. Although the present validation is limited to binary subsystems, the physically grounded descriptor construction provides a scalable framework that can be systematically extended to ternary and higher-order systems. The results, therefore, establish a robust and computationally efficient foundation for future multicomponent HEA modeling, rather than constituting a direct validation on quaternary random solid solutions.