Identifiability assessment and physics-informed residual machine learning for calculated Ti–Zr–Nb–Ta–X thermodynamics (X = Mo, Hf)

Calculated alloy properties can produce misleading measures of machine-learning accuracy when the prediction target is generated by the same equation used as the model prior. This study examines that dependence in 140 Ti–Zr–Nb–Ta–X records (X = Mo, Hf), reduced to 129 unique compositions. The tabulated mixing enthalpies agree with a fixed regular-solution prior to within rounding precision. With complete elemental-enrichment trajectories withheld in turn, the prior and a regularized residual learner both give an RMSE of 0.00285 kJ mol \(^{-1}\) ; the maximum learned correction on the candidate grid is \(1.18\times 10^{-6}\) kJ mol \(^{-1}\) . This agreement will be inherited from the analytical prior and will not be an independent physical validation or measurable improvement in accuracy of machine learning. In the other hand, the composition-only support vector regression changes from the RMSE of 0.0202 (shuffled validation) to 1.5687 kJ mol \(^{-1}\) (omitting a complete enrichment trajectory). The features of each pair are ranked with 12 for 15 in post hoc singular-value analysis. There are two null directions, one being the absence of the Mo–Hf interaction, and two others being sampled interaction. There are 3051 possible compositions of the grid, of which 3005 satisfy the empirical thermodynamic criteria, 126 satisfy both proposed geometric restrictions, and 36 satisfy both restrictions and the empirical thermodynamic criteria and form a resulting Pareto set. The global equal-weight compromise is Ti \(_{15}\) Zr \(_{15}\) Nb \(_{25}\) Ta \(_{15}\) Mo \(_{30}\) . The fact that the calculations were performed with fixed priors is reflected in the results of the sensitivity tests and in the 300 interior compositions separately produced, which indicate that the geometric restrictions do not guarantee the reliability of the calculations. Controlled synthetic tests distinguish a learnable smooth deviation from unpredictable point offsets. The contribution is a reproducible assessment of analytical dependence, validation design and coefficient identifiability in a deterministic dataset. Biomedical suitability is not evaluated.